%PDF-1.2 % 113 0 obj << /Linearized 1 /O 115 /H [ 688 436 ] /L 242333 /E 6571 /N 30 /T 239954 >> endobj xref 113 14 0000000016 00000 n 0000000631 00000 n 0000001124 00000 n 0000001282 00000 n 0000001412 00000 n 0000001617 00000 n 0000002303 00000 n 0000002512 00000 n 0000003572 00000 n 0000005446 00000 n 0000006132 00000 n 0000006340 00000 n 0000000688 00000 n 0000001102 00000 n trailer << /Size 127 /Info 112 0 R /Root 114 0 R /Prev 239943 /ID[<1b327eb78b6e637b36e6fbae0e1f68b2><1b327eb78b6e637b36e6fbae0e1f68b2>] >> startxref 0 %%EOF 114 0 obj << /Type /Catalog /Pages 109 0 R >> endobj 125 0 obj << /S 387 /Filter /FlateDecode /Length 126 0 R >> stream Hb```f``d`d@ A+s 8@8uì!\yBZ600|)P2i-(0%BYDXvwY%p)]6^iqsh9XfKB3,ZjHJhHJ@ҦsYBuƱ9넕 +7GQ%-& 1V*I͞.K\=& 0X)ґyaIUZVhXѰ .h@qj@aX@`p*( gi. 97CoBʅdUxa`2R` TZ endstream endobj 126 0 obj 327 endobj 115 0 obj << /Type /Page /Parent 108 0 R /Resources 116 0 R /Contents 121 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 116 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R /F4 122 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 117 0 obj << /Type /FontDescriptor /Ascent 733 /CapHeight 692 /Descent -281 /Flags 34 /FontBBox [ -166 -283 1021 927 ] /FontName /Palatino-Roman /ItalicAngle 0 /StemV 84 /XHeight 469 >> endobj 118 0 obj << /Type /Font /Subtype /Type1 /FirstChar 32 /LastChar 181 /Widths [ 250 278 402 500 500 889 833 227 333 333 444 606 250 333 250 296 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 667 722 833 611 556 833 833 389 389 778 611 1000 833 833 611 833 722 611 667 778 778 1000 667 667 667 333 606 333 606 500 333 500 611 444 611 500 389 556 611 333 333 611 333 889 611 556 611 611 389 444 333 611 556 833 500 556 500 310 606 310 606 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 500 0 0 0 0 0 747 0 0 0 0 0 0 0 606 0 0 0 611 ] /Encoding /WinAnsiEncoding /BaseFont /Palatino-Bold /FontDescriptor 119 0 R >> endobj 119 0 obj << /Type /FontDescriptor /Ascent 726 /CapHeight 681 /Descent -271 /Flags 262178 /FontBBox [ -152 -266 1000 924 ] /FontName /Palatino-Bold /ItalicAngle 0 /StemV 122 /XHeight 471 >> endobj 120 0 obj << /Type /Font /Subtype /Type1 /FirstChar 32 /LastChar 255 /Widths [ 250 278 371 500 500 840 778 208 333 333 389 606 250 333 250 606 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 611 709 774 611 556 763 832 337 333 726 611 946 831 786 604 786 668 525 613 778 722 1000 667 667 667 333 606 333 606 500 333 500 553 444 611 479 333 556 582 291 234 556 291 883 582 546 601 560 395 424 326 603 565 834 516 556 500 333 606 333 606 0 778 778 709 611 831 786 778 500 500 500 500 500 500 444 479 479 479 479 287 287 287 287 582 546 546 546 546 546 603 603 603 603 500 400 500 500 500 606 628 556 747 747 979 333 333 0 944 833 0 606 0 0 500 603 0 0 0 0 0 333 333 0 758 556 444 278 606 0 500 0 0 500 500 1000 250 778 778 786 998 827 500 1000 500 500 278 278 606 0 556 667 167 500 331 331 605 608 500 250 278 500 1144 778 611 778 611 611 337 337 337 337 786 786 0 786 778 778 778 287 333 333 333 333 250 333 333 380 313 333 ] /Encoding /MacRomanEncoding /BaseFont /Palatino-Roman /FontDescriptor 117 0 R >> endobj 121 0 obj << /Length 1819 >> stream BT /F2 1 Tf 14 0 0 14 135 636 Tm 0 g /GS1 gs -0.0114 Tc 0 Tw (Color Constancy for Scenes with Varying Illumination)Tj 2.1429 -3.4286 TD -0.0123 Tc (\(Color Constancy with Varying Illumination\))Tj /F3 1 Tf -1.5 -5.1429 TD 0.0051 Tc 0.051 Tw (Kobus Barnard, Graham Finlayson, and Brian Funt)Tj -0.7857 -1.7143 TD 0.005 Tc 0.0499 Tw (School of Computing Science, Simon Fraser University)Tj 6.3571 -1.7143 TD 0.0019 Tc 0.0195 Tw (Burnaby, British Columbia)Tj 2.2143 -1.7143 TD 0.0022 Tc 0.0224 Tw (Canada V5A 1S6)Tj 12 0 0 12 180 402 Tm 0.0168 Tc 0 Tw (kobus@cs.berkeley.edu)Tj 0 -1.5 TD 0.0138 Tc (graham@sys.uea.ac.uk)Tj T* 0.0248 Tc (funt@cs.sfu.ca)Tj 12 0 TD -0.0004 Tc 3.0844 Tw [(604-291-3126 604-291-3045 )3084.7(\(fax\))]TJ -15 -5.5833 TD 0.0078 Tc 0.078 Tw (The content of this paper very similar to that which appeared in:)Tj ET 1 i 141 314 1 -1 re f 141 314 1 -1 re f 142 314 368 -1 re f 510 314 1 -1 re f 510 314 1 -1 re f 141 313 1 -18 re f 510 313 1 -18 re f BT 12 0 0 12 144 273 Tm 0.0103 Tc 0.1038 Tw (Kobus Barnard, Graham Finlayson, and Brian Funt, Colour)Tj ET 141 295 1 -26 re f 510 295 1 -26 re f BT 12 0 0 12 144 259 Tm 0.0141 Tc 0.1406 Tw (constancy for scenes with varying illumination, )Tj /F4 1 Tf 23.2491 0 TD 0.0409 Tc 0 Tw (Computer)Tj ET 141 269 1 -14 re f 510 269 1 -14 re f BT 12 0 0 12 144 245 Tm 0.0177 Tc 0.1769 Tw (Vision and Image Understanding)Tj /F3 1 Tf 14.5898 0 TD 0.0033 Tc 0 Tw (, )Tj /F2 1 Tf 0.5392 0 TD 0.0392 Tc (65)Tj /F3 1 Tf 1.0785 0 TD 0.0121 Tc 0.1208 Tw (, 2, pp. 311-321 \(1997\).)Tj -16.2074 -2.3333 TD (Copyright 1997 by Academic Press)Tj ET 141 201 m 142 201 l 142 200 l 142 240 l 141 240 l 141 241 l f 141 201 m 142 201 l 142 200 l 142 240 l 141 240 l 141 241 l f 142 201 368 -1 re f 510 201 1 -1 re f 510 201 1 -1 re f 141 255 1 -54 re f 510 255 1 -54 re f endstream endobj 122 0 obj << /Type /Font /Subtype /Type1 /FirstChar 32 /LastChar 181 /Widths [ 250 333 500 500 500 889 778 333 333 333 389 606 250 333 250 296 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 500 747 722 611 667 778 611 556 722 778 333 333 667 556 944 778 778 611 778 667 556 611 778 722 944 722 667 667 333 606 333 606 500 333 444 463 407 500 389 278 500 500 278 278 444 278 778 556 444 500 463 389 389 333 556 500 722 500 500 444 333 606 333 606 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 500 0 0 0 0 0 747 0 0 0 0 0 0 0 606 0 0 0 556 ] /Encoding /WinAnsiEncoding /BaseFont /Palatino-Italic /FontDescriptor 123 0 R >> endobj 123 0 obj << /Type /FontDescriptor /Ascent 733 /CapHeight 692 /Descent -276 /Flags 98 /FontBBox [ -170 -276 1010 918 ] /FontName /Palatino-Italic /ItalicAngle -10 /StemV 84 /XHeight 482 >> endobj 124 0 obj << /Type /ExtGState /SA false /SM 0.02 /TR /Identity >> endobj 1 0 obj << /Type /Page /Parent 108 0 R /Resources 2 0 R /Contents 3 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 2 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 3 0 obj << /Length 1210 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (2)Tj /F2 1 Tf 14 0 0 14 280 660 Tm -0.0166 Tc (Abstract)Tj /F3 1 Tf 12 0 0 12 72 612 Tm 0.0105 Tc 0.1051 Tw (We present an algorithm which uses information from both surface reflectance and)Tj 0 -2 TD 0.0112 Tc 0.112 Tw (illumination variation to solve for color constancy. Most color constancy algorithms)Tj T* 0.0085 Tc 0.0853 Tw (assume that the illumination across a scene is constant, but this is very often not)Tj T* 0.0096 Tc 0.0954 Tw (valid for real images. The method presented in this work identifies and removes)Tj T* 0.0124 Tc 0.124 Tw (the illumination variation, and in addition uses the variation to constrain the)Tj T* 0.0108 Tc 0.1085 Tw (solution. The constraint is applied conjunctively to constraints found from surface)Tj T* 0.0084 Tc 0.0844 Tw (reflectances. Thus the algorithm can provide good color constancy when there is)Tj T* 0.0119 Tc 0.1189 Tw (sufficient variation in surface reflectances, or sufficient illumination variation, or a)Tj T* 0.0109 Tc 0.1093 Tw (combination of both. We present the results of running the algorithm on several)Tj T* 0.007 Tc 0.0702 Tw (real scenes, and the results are very encouraging.)Tj ET endstream endobj 4 0 obj << /Type /Page /Parent 108 0 R /Resources 5 0 R /Contents 6 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 5 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R /F4 122 0 R /F6 96 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 6 0 obj << /Length 3180 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (3)Tj /F2 1 Tf 14 0 0 14 72 708 Tm -0.012 Tc [(I)-13408.7(Introduction)]TJ /F3 1 Tf 12 0 0 12 108 684 Tm 0.0068 Tc 0.0684 Tw [(Many color constancy algorithms have been developed, but all are subject to)]TJ -3 -2 TD 0.0691 Tw [(quite restrictive assumptions and few have been tested on real images. Of the)]TJ T* 0.0688 Tw (existing algorithms we believe that the one by Finlayson [1] is currently the most)Tj T* 0.0081 Tc 0.0808 Tw (general and most thoroughly tested. Nonetheless, it is restricted to scenes in which)Tj T* 0.0103 Tc 0.1032 Tw (the illumination is constant or at least locally constant. This assumption is more)Tj T* 0.0111 Tc 0.1111 Tw (often violated than one might at first suspect given that the incident illumination)Tj T* 0.0071 Tc 0.0711 Tw (from any fixed direction does generally vary slowly as function of position. The)Tj T* 0.0086 Tc 0.0858 Tw (problem is that the surface orientation even of smooth surfaces can vary quite)Tj T* 0.0062 Tc 0.0619 Tw (rapidly with position so that light at nearby surface locations may be received from)Tj T* 0.0093 Tc 0.0937 Tw [(very different regions of the illumination field. Since these different regions of the)]TJ T* 0.01 Tc 0.1002 Tw (illumination field often posses substantially different spectral power distributions)Tj T* 0.0068 Tc 0.0685 Tw (such as is the case in a room in which where there is light from a light bulb mixed)Tj T* 0.0075 Tc 0.075 Tw (with daylight from a window)Tj /F6 1 Tf 13.6035 0 TD 0 Tc 0 Tw ()Tj /F3 1 Tf 1.0383 0 TD 0.0062 Tc 0.0618 Tw (nearby points on the surface in fact can receive very)Tj -14.6418 -2 TD 0.0198 Tc 0.1981 Tw (different incident illumination.)Tj 3 -2 TD 0.0065 Tc 0.0653 Tw (This paper addresses the problem of color constancy in scenes where the)Tj -3 -2 TD 0.0093 Tc 0.0927 Tw [(spectral power distribution of the incident illumination is allowed to vary with)]TJ T* 0.0033 Tc 0.0333 Tw [(scene location. Finlayson et. al. [2], DZmura et. al. [3], and Tsukada et al [4] have)]TJ T* 0.017 Tc 0.1697 Tw (shown that a difference in illumination, )Tj /F4 1 Tf 19.443 0 TD 0.0149 Tc 0.1488 Tw (once it has been identified)Tj /F3 1 Tf 11.3141 0 TD 0.0224 Tc 0.2244 Tw (, provides)Tj -30.7571 -2 TD 0.0059 Tc 0.0589 Tw (additional constraints that can be exploited to obtain color constancy, but they do not)Tj T* 0.0094 Tc 0.0942 Tw [(provide an automatic method of determining when such a difference exists. We)]TJ T* 0.0119 Tc 0.1192 Tw (present a new algorithm that first uncovers the illumination variation in an image)Tj T* 0.0074 Tc 0.0743 Tw (and then uses the additional constraint it provides to obtain better color constancy.)Tj 3 -2 TD 0.0122 Tc 0.1217 Tw (The color constancy problem is the retrieval of an illumination-independent)Tj -3 -2 TD 0.0077 Tc 0.0777 Tw [(description of a scenes surface colors. This is essentially equivalent to modeling the)]TJ T* 0.0124 Tc 0.1245 Tw (illumination incident on the scene, since if the illumination is known the surface)Tj T* 0.0047 Tc 0.0476 Tw [(colors can be calculated. Following Forsyth [5] we interpret color constancy as taking)]TJ ET endstream endobj 7 0 obj << /Type /Page /Parent 108 0 R /Resources 8 0 R /Contents 9 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 8 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R /F6 96 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 9 0 obj << /Length 3152 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (4)Tj 12 0 0 12 72 710 Tm 0.013 Tc 0.1297 Tw (images of scenes under unknown illumination and determining the camera)Tj 0 -2 TD 0.0078 Tc 0.0788 Tw (response to the same scene under a known, canonical light. In a general context this)Tj T* 0.0089 Tc 0.0889 Tw (problem has proven difficult to solve, so to make progress, restrictive assumptions)Tj T* 0.005 Tc 0.0502 Tw [(are made. In particular, it is common to assume that the scene is flat [6, 7, 8, 9], that)]TJ T* 0.0073 Tc 0.0734 Tw (the illumination is constant throughout [10, 5, 11, 12], and that all reflectances are)Tj T* 0.0067 Tc 0.0668 Tw (matte. Finlayson [1] has shown that if we focus on solving only for surface)Tj T* 0.0088 Tc 0.0876 Tw [(chromaticity and forego estimating surface lightness then the restriction to flat)]TJ T* 0.0069 Tc 0.0691 Tw [(matte surfaces can be relaxed. However, the assumption that the chromaticity of)]TJ T* 0.0125 Tc 0.1252 Tw (the illumination does not change is more troublesome.)Tj 3 -2 TD 0.0048 Tc 0.0484 Tw (The Retinex algorithm [13, 14, 6, 7, 9] partially addresses the issue of varying)Tj -3 -2 TD 0.0091 Tc 0.0908 Tw [(illumination. At least in principle)]TJ /F6 1 Tf 15.7302 0 TD 0 Tc 0 Tw ()Tj /F3 1 Tf 1.0522 0 TD 0.0074 Tc 0.0742 Tw (it does not in fact work in practice)Tj /F6 1 Tf 15.8272 0 TD 0 Tc 0 Tw ()Tj /F3 1 Tf 1.0522 0 TD 0.0291 Tc (Retinex)Tj -33.6618 -2 TD 0.0124 Tc 0.1244 Tw (eliminates the variation in illumination and computes surface lightnesses for each)Tj T* [(of the three color channels independently. Since eliminating the illumination and)]TJ T* 0.0097 Tc 0.0965 Tw (recovering the illumination are equivalent problems [15], if Retinex worked it could)Tj T* 0.0966 Tw [(be used to recover the incident illumination. Retinex operates on the principle that)]TJ T* 0.0958 Tw (within a single color channel, small changes in image intensity arise from changes)Tj T* 0.0104 Tc 0.1042 Tw (in illumination while large changes indicate changes in surface color. The small)Tj T* 0.0051 Tc 0.0506 Tw (changes are thresholded away and the big changes are preserved so that the surface)Tj T* 0.0079 Tc 0.0793 Tw (lightness can be reconstructed, essentially by integration. Unfortunately any error in)Tj T* 0.007 Tc 0.0699 Tw [(classifying the intensity changes can lead to serious errors in the recovered result. In)]TJ T* 0.0094 Tc 0.0938 Tw (essence the Retinex algorithm uses a primitive, gradient-based-edge-detection)Tj T* 0.0052 Tc 0.0522 Tw [(strategy to identify the reflectance edges, so given the long history of edge-detection)]TJ T* 0.0072 Tc 0.0724 Tw [(research, it should not be surprising that it does not perform well.)]TJ 3 -2 TD 0.0086 Tc 0.086 Tw (To overcome the weaknesses of Retinexs edge detection method, we)Tj -3 -2 TD 0.0088 Tc 0.0883 Tw (incorporate knowledge about the set of plausible illuminants and from this set)Tj T* 0.0092 Tc 0.0917 Tw (derive information about the kinds of chromaticity change that a change in)Tj T* 0.0119 Tc 0.119 Tw (illumination can induce within a region of uniform reflectance. This constraint is)Tj ET endstream endobj 10 0 obj << /Type /Page /Parent 108 0 R /Resources 11 0 R /Contents 12 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 11 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R /F6 96 0 R /F9 97 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 12 0 obj << /Length 3556 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (5)Tj 12 0 0 12 72 710 Tm 0.0062 Tc 0.0629 Tw [(more global than local edge detection and using both types of constraint together)]TJ 0 -2 TD 0.0099 Tc 0.0988 Tw (yields good results. Once the illumination variation is uncovered it is combined)Tj T* 0.0082 Tc 0.0813 Tw (with the other constraints arising from the set of colors found in the image as will)Tj T* 0.0011 Tc 0.0105 Tw (be discussed below.)Tj /F2 1 Tf 14 0 0 14 72 590 Tm -0.0109 Tc 0 Tw [(I)52.7(I)-9582.2(The Color Constancy Algorithm)]TJ /F3 1 Tf 12 0 0 12 108 566 Tm 0.0082 Tc 0.0818 Tw [(Our color constancy algorithm has two main components: one to extract the)]TJ -3 -2 TD 0.0097 Tc 0.0962 Tw (illumination field and another to combine the constraints provided by the a priori)Tj T* 0.0099 Tc 0.0993 Tw (knowledge of the surface and illumination gamuts with those obtained from the)Tj T* 0.0081 Tc 0.0804 Tw [(observed surfaces and the extracted illumination field. The constraint part of the)]TJ T* 0.0058 Tc 0.058 Tw (algorithm will be described first.)Tj /F2 1 Tf 0 -4 TD -0.0371 Tc 0 Tw (II.1)Tj 12 0 TD 0.0048 Tc 0.0483 Tw (Surface and Illumination Constraints)Tj /F3 1 Tf -9 -2 TD 0.0089 Tc 0.0889 Tw (In order to represent the constraints efficiently, we make the approximation)Tj -3 -2 TD 0.0056 Tc 0.0559 Tw (that the effect of the illumination can be modeled by a diagonal matrix [16, 17].)Tj T* 0.0046 Tc 0.0461 Tw (Specifically, if )Tj /F9 1 Tf 6.6771 0 TD 0.0193 Tc -0.0193 Tw [([r,)-110.9( g,)-110.9( b)18(])]TJ /F3 1 Tf 3.5729 0 TD 0.0093 Tc 0.092 Tw [( is the camera response of a surface under one illumination,)]TJ -10.25 -1.9167 TD 0.0142 Tc 0 Tw (then )Tj /F9 1 Tf 2.5104 0 TD 0.0193 Tc -0.0193 Tw [([r,)-110.9( g,)-110.9( b)18(])29.4(D)]TJ /F6 1 Tf 4.4375 0 TD 0 Tc 0 Tw ()Tj /F9 1 Tf 0.7552 0 TD 0.016 Tc [([r)16(D)]TJ 7 0 0 7 182.7187 323.9375 Tm 0 Tc (11)Tj 12 0 0 12 190.9063 327 Tm 0.1302 Tc -0.1302 Tw [(, g)130.2(D)]TJ 7 0 0 7 213.5313 324 Tm 0 Tc 0 Tw (22)Tj 12 0 0 12 221.875 327 Tm 0.1302 Tc -0.1302 Tw [(, b)130.2(D)]TJ 7 0 0 7 245.1875 324 Tm 0 Tc 0 Tw (33)Tj 12 0 0 12 253.4688 327 Tm (])Tj /F3 1 Tf 0.3776 0 TD 0.0043 Tc 0.0435 Tw (, where )Tj /F2 1 Tf 3.6407 0 TD 0 Tc 0 Tw (D)Tj /F3 1 Tf 0.8618 0 TD 0.0051 Tc 0.0504 Tw [( is a diagonal matrix, is the camera)]TJ -20.0025 -2.0833 TD 0.0087 Tc 0.0878 Tw [(response to the same surface under a second illumination. In other words, each)]TJ 0 -2 TD 0.0072 Tc 0.0718 Tw (camera channel is scaled independently. The accuracy of the diagonal)Tj T* 0.0086 Tc 0.0859 Tw (approximation depends on the camera sensors, which for the camera used in the)Tj T* 0.0098 Tc 0.0973 Tw [(experiments is within 10% \(magnitude of )]TJ /F9 1 Tf 19.5104 0 TD 0.0193 Tc -0.0193 Tw [([r,)-110.9( g,)-110.9( b)18(])]TJ /F3 1 Tf 3.5729 0 TD 0.008 Tc 0.0796 Tw [( difference\) of the general linear)]TJ -23.0833 -2 TD 0.0078 Tc 0.0783 Tw (model. For sensors for which the diagonal model is too inaccurate, it is usually)Tj T* 0.0048 Tc 0.0487 Tw (possible to improve it by spectrally sharpening the sensors [16].)Tj 3 -2 TD 0.0029 Tc 0.0294 Tw [(Following Finlayson [1], we work in the chromaticity space [r/b, g/b]. This)]TJ -3 -2 TD 0.0078 Tc 0.0784 Tw [(space preserves the diagonal model in the sense that if illumination was exactly)]TJ T* 0.0031 Tc 0.0306 Tw (modeled by a diagonal transform applied to [r, g, b], then it will also be exactly)Tj T* 0.0022 Tc 0.0213 Tw [(modeled by a diagonal transform \(now 2D\) applied to [r/b, g/b]. If either the)]TJ ET endstream endobj 13 0 obj << /Type /Page /Parent 108 0 R /Resources 14 0 R /Contents 15 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 14 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 15 0 obj << /Length 3075 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (6)Tj 12 0 0 12 72 710 Tm 0.0125 Tc 0.1252 Tw (illumination is spatially constant or pre-processing has removed all illumination)Tj 0 -2 TD 0.0107 Tc 0.1063 Tw (variation, then transforming the input image to what it would have looked like)Tj T* 0.0096 Tc 0.0959 Tw (under the canonical illuminant requires simply transforming it by a single diagonal)Tj T* 0.0077 Tc 0.0772 Tw [(matrix. The goal of the color constancy algorithm is to calculate this matrix.)]TJ 3 -2 TD 0.0062 Tc 0.062 Tw (The algorithms basic approach is to constrain the set of possible diagonal)Tj -3 -2 TD 0.0068 Tc 0.0686 Tw (maps by adding more and more information so that only a small set of possible)Tj T* 0.006 Tc 0.0604 Tw [(maps remains. The first constraints are the those due to Forsyth [5]. He observed)]TJ T* 0.0068 Tc 0.0674 Tw [(that the set of camera responses that could be obtained from all combinations of a)]TJ T* 0.007 Tc 0.0697 Tw (large set of surfaces viewed under a fixed illuminant is a convex set which does not)Tj T* 0.0052 Tc 0.0521 Tw [(fill all of the [r, g, b] colour space. This set is referred to as that illuminants gamut,)]TJ T* 0.0077 Tc 0.0773 Tw [(and in the case of the canonical illuminant is called the canonical gamut. For a)]TJ T* 0.01 Tc 0.1001 Tw (typical scene under unknown illumination, the camera responses will lie in a subset)Tj T* 0.0092 Tc 0.0919 Tw [(of the unknown illuminants full gamut. Since all possible surfaces are assumed to)]TJ T* 0.0118 Tc 0.1185 Tw (be represented within the canonical gamut, whatever the unknown illuminant is, it)Tj T* 0.0052 Tc 0.0525 Tw [(is constrained by the fact that it is a diagonal map projecting the scenes observed)]TJ T* 0.007 Tc 0.0702 Tw [(response set into the canonical gamut. There will be many possible diagonal maps)]TJ T* 0.0051 Tc 0.0512 Tw (satisfying this constraint because the scenes set is a subset of the full gamut and so it)Tj T* 0.0057 Tc 0.058 Tw [(can fit inside the larger gamut many different ways. Forsyth shows that the)]TJ T* 0.0067 Tc 0.0666 Tw (resulting constraint set of diagonal maps is convex. As shown in [1], all the required)Tj T* 0.0037 Tc 0.0379 Tw [(relationships hold in the [r/b, )37.9(g/b] chromaticity space.)]TJ 3 -2 TD 0.0091 Tc 0.0905 Tw (The second source of constraint arises from considering the set of common)Tj -3 -2 TD 0.0063 Tc 0.0627 Tw [(illuminants as has been formulated by Finlayson [1]. After applying Forsyths surface)]TJ T* 0.0072 Tc 0.0728 Tw [(constraints, the resulting set of diagonal maps typically includes many that)]TJ T* 0.0114 Tc 0.1143 Tw (correspond to quite atypical illuminants. The illumination constraint excludes all)Tj T* 0.0109 Tc 0.109 Tw (the illuminants that are not contained in the set of typical illuminants. Finlayson)Tj T* 0.0098 Tc 0.0978 Tw [(restricted the illumination to the convex hull of the chromaticities of the 6 daylight)]TJ T* 0.0023 Tc 0.0234 Tw (phases provided by Judd et al [18], the CIE standard illuminants A, B, C [19], a 2000K)Tj ET endstream endobj 16 0 obj << /Type /Page /Parent 108 0 R /Resources 17 0 R /Contents 18 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 17 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 18 0 obj << /Length 2867 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (7)Tj 12 0 0 12 72 710 Tm 0.0116 Tc 0.1157 Tw (Planckian radiator, and uniform white. We have improved upon this sampling of)Tj 0 -2 TD 0.0138 Tc 0.1382 Tw (illuminants by using 100 measurements of illumination around the university)Tj T* 0.0123 Tc 0.1236 Tw (campus, including both indoor and outdoor illumination. Some inter-reflected light)Tj T* 0.0075 Tc 0.0747 Tw (was included such as that from concrete buildings and light filtering through trees,)Tj T* 0.009 Tc 0.09 Tw (but illumination that was obviously unusual was excluded. The set of)Tj T* 0.0101 Tc 0.1006 Tw (chromaticities of the measured illuminants is larger in area than the original set,)Tj T* 0.0068 Tc 0.0676 Tw [(but it does not contain it entirely, as the 2000K Planckian radiator is more red than)]TJ T* 0.0145 Tc 0.1452 Tw (what is common.)Tj 3 -2 TD 0.0078 Tc 0.078 Tw (It should be noted that the set of typical illuminants provides a constraint on)Tj -3 -2 TD 0.0095 Tc 0.0951 Tw (mappings from the canonical to the unknown, which is the reverse of that for)Tj T* 0.0078 Tc 0.078 Tw (surfaces discussed above in which the restriction was on mappings from the)Tj T* 0.0125 Tc 0.125 Tw [(unknown illuminant to the canonical illuminant. To make use of the constraint it)]TJ T* 0.0108 Tc 0.1086 Tw (must be inverted which means that the restriction on the set of illuminants)Tj T* 0.0072 Tc 0.0718 Tw [(becomes a non-convex set in the mapping space used for surface constraints. This)]TJ T* 0.0717 Tw (potentially presents a problem since the sets must be intersected in order to combine)Tj T* 0.0103 Tc 0.1031 Tw (constraints and in three-dimensions it is much faster to compute intersections of)Tj T* 0.0117 Tc 0.1165 Tw (convex sets than non-convex ones. While in the two-dimensional case the set)Tj T* 0.0078 Tc 0.078 Tw (intersections can be directly computed, in practice the inverse of the measured)Tj T* 0.0109 Tc 0.1093 Tw (illumination non-convex gamut was found to be close enough to its convex hull)Tj T* 0.0079 Tc 0.0795 Tw (that for convenience the hull could be used anyway.)Tj 3 -2 TD 0.0098 Tc 0.0972 Tw [(Varying illumination provides the third source of constraint. Our use of it)]TJ -3 -2 TD 0.0073 Tc 0.0738 Tw (here generalizes the algorithm presented in [2]. In that work the map taking the)Tj T* 0.0105 Tc 0.1046 Tw (chromaticity of a single surface colour under an unknown illuminant to its)Tj T* 0.011 Tc 0.1099 Tw (chromaticity under the canonical illuminant is constrained to lie on a line.)Tj T* 0.0106 Tc 0.1056 Tw (Effectively this amounts to assuming all the candidate illuminants are)Tj T* 0.0096 Tc 0.0961 Tw (approximately Planckian radiators and their chromaticities lie roughly on the)Tj T* 0.0077 Tc 0.0778 Tw [(Planckian locus. The chromaticity of the same surface viewed under a second)]TJ ET endstream endobj 19 0 obj << /Type /Page /Parent 108 0 R /Resources 20 0 R /Contents 21 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 20 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R /F6 96 0 R /F8 98 0 R /F9 97 0 R /F10 99 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 21 0 obj << /Length 3043 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (8)Tj 12 0 0 12 72 710 Tm 0.0117 Tc 0.1176 Tw [(illuminant defines a second line. If the difference in the illuminations)]TJ 0 -2 TD 0.0107 Tc 0.1079 Tw [(chromaticities is non-trivial, the two lines will intersect, thereby constraining the)]TJ T* 0.0087 Tc 0.0866 Tw (surfaces chromaticity to a unique value.)Tj 3 -2 TD 0.0097 Tc 0.0967 Tw (We extend the idea of using the variation in illumination in two ways. First)Tj -3 -2 TD 0.01 Tc 0.0999 Tw (we use the entire illumination gamut instead of simply the Planckian radiators.)Tj T* 0.0093 Tc 0.0923 Tw [(Second we exploit the illumination variation across the entire image, as opposed to)]TJ T* 0.0103 Tc 0.1027 Tw [(just that between two points on one surface patch. Thus the illumination over the)]TJ T* 0.0072 Tc 0.0728 Tw [(entire image is both used, and solved for. The details follow.)]TJ 3 -2 TD 0.0118 Tc 0.1183 Tw (For the moment assume that we already have the relative illumination field)Tj -3 -2 TD 0.008 Tc 0.0796 Tw (for the image. The relative illumination field for each pixel P is defined by the)Tj T* 0.0089 Tc 0.0893 Tw (diagonal transform required to map the illumination at some chosen base pixel B to)Tj T* 0.0102 Tc 0.1015 Tw [(the illumination at P. The relative illumination field describes all the pixels only)]TJ T* 0.0085 Tc 0.0852 Tw (with respect to one another, so given it the remaining problem is to solve for the)Tj T* 0.0128 Tc 0.1288 Tw (illumination at B and hence establish the illumination everywhere in absolute)Tj T* 0.0321 Tc 0 Tw (terms.)Tj 3 -2 TD 0.0083 Tc 0.0822 Tw [(The approach is motivated by the following argument. Suppose that the left)]TJ -3 -2 TD 0.008 Tc 0.0797 Tw (side of the image is illuminated by a blue light. This means that the relative)Tj T* 0.0095 Tc 0.0948 Tw (illumination field at a pixel on the left side transforms illuminants so that they are)Tj T* 0.009 Tc 0.0903 Tw (more blue. However, the illumination at the center of the image cannot be so blue)Tj T* 0.0095 Tc 0.0948 Tw (that making it even more blue produces an illumination that falls outside the set of)Tj T* 0.0112 Tc 0.1128 Tw [(possible illuminants. Thus the illumination at the center is constrained by the jump)]TJ T* 0.0083 Tc 0.0821 Tw [(towards blue. All entries in the field contribute this sort of constraint. This will now)]TJ T* 0.0102 Tc 0.1013 Tw (be made more formal.)Tj 3 -2 TD 0.0076 Tc 0.0758 Tw (First we verify the intuitive claim that the constraint provided by a value )Tj /F2 1 Tf 33.9692 0 TD 0 Tc 0 Tw (D)Tj /F3 1 Tf -36.9692 -2 TD 0.0089 Tc 0.0894 Tw (in the relative illumination field is the set of possible illuminants scaled by )Tj /F9 1 Tf 35.0625 0 TD 0 Tc 0 Tw (D)Tj /F6 1 Tf 7 0 0 7 501.875 139.3438 Tm ()Tj /F10 1 Tf 0.5446 0 TD (1)Tj /F3 1 Tf 12 0 0 12 510 134 Tm (.)Tj -36.5 -2 TD 0.0123 Tc 0.1236 Tw (Consider the illumination gamut, )Tj /F8 1 Tf 15.9203 0 TD 0.0228 Tc -0.0096 Tw (I )Tj /F3 1 Tf 0.8692 0 TD 0.0094 Tc 0.0942 Tw (which is a convex set:)Tj ET endstream endobj 22 0 obj << /Type /Page /Parent 108 0 R /Resources 23 0 R /Contents 24 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 23 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R /F6 96 0 R /F8 98 0 R /F9 97 0 R /F10 99 0 R /F11 100 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 24 0 obj << /Length 10243 >> stream BT /F3 1 Tf 10 0 0 10 571 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (9)Tj ET q 1 i 108 679 298 40 re W n BT /F8 1 Tf 12 0 0 12 108 679 Tm 0.0314 Tc ( )Tj ET Q BT /F8 1 Tf 12 0 0 12 109.0312 696 Tm (I)Tj /F6 1 Tf 0.7604 0 TD ()Tj /F9 1 Tf 1.3151 0 TD (X )Tj /F10 1 Tf 1.1797 0 TD ( )Tj /F9 1 Tf 0.3958 0 TD (X)Tj /F6 1 Tf 0.974 0 TD ()Tj 12 0 2.64 12 188.4375 696 Tm ()Tj /F9 1 Tf 9 0 0 9 196 692.9062 Tm (i)Tj 12 0 0 12 198.9687 696 Tm (X)Tj 9 0 0 9 208.0625 692.9062 Tm (i)Tj /F10 1 Tf 12 0 0 12 215.5312 696 Tm (where )Tj /F6 1 Tf 12 0 2.64 12 265.75 696 Tm ()Tj /F9 1 Tf 9 0 0 9 273.3125 692.9062 Tm (i)Tj /F6 1 Tf 12 0 0 12 279.9062 696 Tm ()Tj /F10 1 Tf 0.7187 0 TD (1)Tj 9 0 0 9 256.6562 682.3437 Tm (i)Tj /F6 1 Tf 18 0 0 18 251.5312 693.2812 Tm ()Tj /F9 1 Tf 9 0 0 9 179.375 682.3437 Tm (i)Tj /F6 1 Tf 18 0 0 18 174.2187 693.2812 Tm ()Tj ET q 1 i 108 679 298 40 re W n 0 G 0 J 0 j 0.5 w 10 M []0 d 146.844 716.844 m 146.844 681.344 l S Q BT 12 0 0 12 127.9375 706.75 Tm ()Tj 0 -1.0651 TD ()Tj 0 0.7604 TD ()Tj 0 -1.8281 TD ( )Tj 0 0.2969 TD ()Tj 13.8438 1.8359 TD (\012)Tj 0 -1.0651 TD ()Tj 0 0.7604 TD ()Tj 0 -1.8281 TD ()Tj 0 0.2969 TD ()Tj /F10 1 Tf 0.5234 0.9401 TD ( for hull points )Tj /F9 1 Tf 6.888 0 TD (X)Tj 9 0 0 9 392.0937 692.9062 Tm (i)Tj /F6 1 Tf 12 0 0 21.336 377.4999 693.875 Tm 1.3846 Tc (\015)Tj /F3 1 Tf 12 0 0 12 526 696 Tm 0 Tc (\(1\))Tj -37.8333 -4.25 TD 0.0086 Tc 0.0861 Tw (We have the constraint that we can map the illumination by the diagonal map )Tj /F2 1 Tf 36.5447 0 TD 0 Tc 0 Tw (D)Tj /F3 1 Tf -36.5447 -2 TD 0.0058 Tc 0.0581 Tw (and still be in this set:)Tj ET q 1 i 108 596 37 11 re W n BT /F8 1 Tf 12 0 0 12 108 596 Tm 0.0314 Tc 0 Tw ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 109.75 597 Tm 0 Tc 0 Tw (XD)Tj /F6 1 Tf 1.651 0 TD ()Tj /F8 1 Tf 0.7135 0 TD (I)Tj /F3 1 Tf 32.3229 0 TD (\(2\))Tj -37.8333 -2 TD 0.014 Tc 0.14 Tw (This means that:)Tj ET q 1 i 108 531 255 25 re W n BT 12 0 0 12 108 531 Tm 0 Tc 0 Tw ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 109.75 544 Tm 0 Tc 0 Tw (XD)Tj /F6 1 Tf 1.6849 0 TD ()Tj 12 0 2.64 12 153.875 544 Tm ()Tj /F11 1 Tf 7 0 0 7 161.375 541 Tm (i)Tj /F9 1 Tf 12 0 0 12 163.9375 544 Tm (X)Tj /F11 1 Tf 7 0 0 7 172.9688 541 Tm (i)Tj -3.9955 -1.2366 TD (i)Tj /F6 1 Tf 18 0 0 18 139.6562 541.2812 Tm ()Tj /F3 1 Tf 12 0 0 12 180.2813 544 Tm ( for some )Tj /F6 1 Tf 12 0 2.64 12 235.4063 544 Tm ()Tj /F11 1 Tf 7 0 0 7 242.9063 541 Tm (i)Tj /F3 1 Tf 12 0 0 12 245.7187 544 Tm ( with )Tj /F6 1 Tf 12 0 2.64 12 297.0312 544 Tm ()Tj /F11 1 Tf 7 0 0 7 304.5313 541 Tm (i)Tj /F6 1 Tf 12 0 0 12 310.7188 544 Tm ()Tj /F10 1 Tf 0.7187 0 TD (1)Tj /F3 1 Tf 7 0 0 7 288.1562 532.3437 Tm (i)Tj /F6 1 Tf 18 0 0 18 282.8125 541.2812 Tm ()Tj /F3 1 Tf 12 0 0 12 324.875 544 Tm 0.1328 Tc (, )Tj /F6 1 Tf 12 0 2.64 12 332.4688 544 Tm 0 Tc ()Tj /F11 1 Tf 7 0 0 7 339.9688 541 Tm (i)Tj /F6 1 Tf 12 0 0 12 345.9375 544 Tm ()Tj /F10 1 Tf 0.7917 0 TD (0)Tj /F3 1 Tf 14.2135 0 TD (\(3\))Tj -37.8333 -2.75 TD 0.0564 Tc (And)Tj ET q 1 i 108 468 272 26 re W n BT 12 0 0 12 108 468 Tm 0 Tc ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 109.75 481 Tm 0 Tc (X)Tj /F6 1 Tf 0.974 0 TD ()Tj 12 0 2.64 12 145.3437 481 Tm ()Tj /F11 1 Tf 7 0 0 7 152.8437 478 Tm (i)Tj /F9 1 Tf 12 0 0 12 160.375 481 Tm (X)Tj /F11 1 Tf 7 0 0 7 169.4062 478 Tm (i)Tj /F9 1 Tf 12 0 0 12 176.4687 481 Tm (D)Tj /F6 1 Tf 7 0 0 7 185.5937 486.3437 Tm ()Tj /F10 1 Tf 0.5446 0 TD (1)Tj /F6 1 Tf 12 0 0 19.932 156.0312 479.2812 Tm 2.8154 Tc ()Tj /F11 1 Tf 7 0 0 7 136.4687 469.3437 Tm 0 Tc (i)Tj /F6 1 Tf 18 0 0 18 131.125 478.2812 Tm ()Tj /F3 1 Tf 12 0 0 12 197.125 481 Tm ( for some )Tj /F6 1 Tf 12 0 2.64 12 252.25 481 Tm ()Tj /F11 1 Tf 7 0 0 7 259.75 478 Tm (i)Tj /F3 1 Tf 12 0 0 12 262.5625 481 Tm ( with )Tj /F6 1 Tf 12 0 2.64 12 313.875 481 Tm ()Tj /F11 1 Tf 7 0 0 7 321.375 478 Tm (i)Tj /F6 1 Tf 12 0 0 12 327.5625 481 Tm ()Tj /F10 1 Tf 0.7187 0 TD (1)Tj /F3 1 Tf 7 0 0 7 304.9999 469.3437 Tm (i)Tj /F6 1 Tf 18 0 0 18 299.6562 478.2812 Tm ()Tj /F10 1 Tf 12 0 0 12 341.1562 481 Tm 0.1172 Tc (, )Tj /F6 1 Tf 12 0 2.64 12 350.2812 481 Tm 0 Tc ()Tj /F11 1 Tf 7 0 0 7 357.7812 478 Tm (i)Tj /F6 1 Tf 12 0 0 12 363.75 481 Tm ()Tj /F10 1 Tf 0.7917 0 TD (0)Tj /F3 1 Tf 12.7292 0 TD (\(4\))Tj -37.8333 -2.75 TD 0.0056 Tc 0.0556 Tw (So we define a new constraint set )Tj /F8 1 Tf 15.5279 0 TD 0 Tc 0 Tw (V)Tj /F3 1 Tf 0.6647 0 TD 0.0043 Tc 0.0424 Tw [( as:)]TJ ET q 1 i 108 391 271 40 re W n BT /F8 1 Tf 12 0 0 12 108 391 Tm 0.0314 Tc 0 Tw ( )Tj ET Q BT /F8 1 Tf 12 0 0 12 108.7812 408 Tm 0 Tc 0 Tw (V)Tj /F6 1 Tf 1.0885 0 TD ()Tj /F9 1 Tf 1.3151 0 TD (X)Tj /F10 1 Tf 0.8594 0 TD 0.2083 Tc ( )Tj /F9 1 Tf 0.8542 0 TD 0 Tc (X)Tj /F6 1 Tf 0.974 0 TD ()Tj 12 0 2.64 12 193.7812 408 Tm ()Tj /F9 1 Tf 9 0 0 9 201.3437 404.9062 Tm (i)Tj 12 0 0 12 209.2812 408 Tm (X)Tj 9 0 0 9 218.375 404.9062 Tm (i)Tj 12 0 0 12 225.8437 408 Tm (D)Tj /F6 1 Tf 9 0 0 9 234.9375 413.25 Tm ()Tj /F10 1 Tf 0.5486 0 TD (1)Tj /F6 1 Tf 12 0 0 21.432 204.9375 405.9375 Tm 3.0107 Tc ()Tj /F10 1 Tf 12 0 0 12 249.375 408 Tm 0 Tc ( where )Tj /F6 1 Tf 12 0 2.64 12 305.5937 408 Tm ()Tj /F9 1 Tf 9 0 0 9 313.1562 404.9062 Tm (i)Tj /F6 1 Tf 12 0 0 12 319.75 408 Tm ()Tj /F10 1 Tf 0.7187 0 TD -0.0859 Tc [(1,)-203.1( )]TJ /F6 1 Tf 12 0 2.64 12 342.4687 408 Tm 0 Tc ()Tj /F9 1 Tf 9 0 0 9 350.0312 404.9062 Tm (i)Tj /F6 1 Tf 12 0 0 12 356.4062 408 Tm ()Tj /F10 1 Tf 0.7917 0 TD (0)Tj 9 0 0 9 296.5 394.3438 Tm (i)Tj /F6 1 Tf 18 0 0 18 291.375 405.2812 Tm ()Tj /F9 1 Tf 9 0 0 9 184.7187 394.3438 Tm (i)Tj /F6 1 Tf 18 0 0 18 179.5625 405.2812 Tm ()Tj ET q 1 i 108 391 271 40 re W n 0 G 0 J 0 j 0.5 w 10 M []0 d 152.187 428.844 m 152.187 393.344 l S Q BT 12 0 0 12 131.625 418.75 Tm ()Tj 0 -1.0651 TD ()Tj 0 0.7604 TD ()Tj 0 -1.8281 TD ( )Tj 0 0.2969 TD ()Tj 20.0651 1.8359 TD (\012)Tj 0 -1.0651 TD ()Tj 0 0.7604 TD ()Tj 0 -1.8281 TD ()Tj 0 0.2969 TD ()Tj /F3 1 Tf 12.7995 0.9401 TD (\(5\))Tj -37.8333 -3.0833 TD 0.0041 Tc 0.0407 Tw (It is clear that for all )Tj ET q 1 i 185 369 33 12 re W n BT /F8 1 Tf 12 0 0 12 185 369 Tm 0.0314 Tc 0 Tw ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 186.75 371 Tm 0 Tc 0 Tw (X)Tj /F6 1 Tf 0.9401 0 TD ()Tj /F8 1 Tf 0.6927 0 TD (V)Tj /F3 1 Tf 0.9714 0 TD (, )Tj ET q 1 i 224 370 37 11 re W n BT /F8 1 Tf 12 0 0 12 224 370 Tm 0.0314 Tc ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 225.75 371 Tm (XD)Tj /F6 1 Tf 1.651 0 TD ()Tj /F8 1 Tf 0.7135 0 TD (I)Tj /F3 1 Tf 0.5729 0 TD 0.0106 Tc 0.106 Tw (. Furthermore, the argument is reversible. That is,)Tj -15.75 -2 TD 0.0036 Tc 0 Tw (if )Tj ET q 1 i 83 346 58 11 re W n BT /F8 1 Tf 12 0 0 12 83 346 Tm 0.0314 Tc ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 84.7812 347.0001 Tm 0.2173 Tc [(Y=X)217.3(D)]TJ /F6 1 Tf 3.3776 0 TD 0 Tc ()Tj /F8 1 Tf 0.7135 0 TD (I)Tj /F3 1 Tf 0.5938 0 TD (, )Tj ET q 1 i 147 345 33 12 re W n BT /F8 1 Tf 12 0 0 12 147 345 Tm 0.0314 Tc ( )Tj ET Q BT /F9 1 Tf 12 0 0 12 148.75 347 Tm (X)Tj /F6 1 Tf 0.9401 0 TD ()Tj /F8 1 Tf 0.6927 0 TD (V)Tj /F3 1 Tf 0.9714 0 TD 0.0052 Tc 0.0515 Tw [( . It should be noted that the above also shows that we can)]TJ -9 -1.8333 TD 0.0087 Tc 0.087 Tw (identify the convex constraint set with the mapped hull points )Tj /F9 1 Tf 29.3125 0 TD 0 Tc 0 Tw (X)Tj /F11 1 Tf 7 0 0 7 432.7812 322 Tm (i)Tj /F9 1 Tf 12 0 0 12 439.8437 325 Tm (D)Tj /F6 1 Tf 7 0 0 7 448.9687 330.3437 Tm ()Tj /F10 1 Tf 0.5446 0 TD (1)Tj /F3 1 Tf 12 0 0 12 457 325 Tm (.)Tj -29.0833 -2.1667 TD 0.009 Tc 0.0902 Tw (Next, we consider the set of constraints determined from the relative)Tj -3 -2 TD 0.0101 Tc 0.1009 Tw (illumination field and verify that the convex hull of these constraints is just as)Tj T* 0.0086 Tc 0.0857 Tw (powerful as the entire set. The motivation for using the hull is that it saves a)Tj T* 0.0077 Tc 0.0764 Tw (significant amount of processing time. We are free to use the hull regardless, but it)Tj T* 0.0086 Tc 0.0857 Tw (is comforting to know that doing so does not weaken the algorithm. To demonstrate)Tj T* 0.0074 Tc 0.074 Tw (this we need to show that given two diagonal transforms )Tj /F9 1 Tf 26.7292 0 TD 0 Tc 0 Tw (D)Tj /F10 1 Tf 7 0 0 7 401.1875 176 Tm (1)Tj /F3 1 Tf 12 0 0 12 405 179 Tm 0.0024 Tc 0.0239 Tw [( and )]TJ /F9 1 Tf 2.3958 0 TD 0 Tc 0 Tw (D)Tj /F10 1 Tf 7 0 0 7 442.75 175.9999 Tm (2)Tj /F3 1 Tf 12 0 0 12 448 179 Tm 0.0081 Tc 0.0811 Tw (, the)Tj -31.3333 -2 TD 0.0819 Tw (corresponding constraint sets )Tj ET q 1 i 237 151 15 14 re W n BT /F8 1 Tf 12 0 0 12 237 151 Tm 0.0314 Tc 0 Tw ( )Tj ET Q BT /F8 1 Tf 12 0 0 12 237.7812 155 Tm 0 Tc 0 Tw (V)Tj /F10 1 Tf 9 0 0 9 246.5313 152 Tm (1)Tj /F3 1 Tf 12 0 0 12 252 155 Tm 0.0024 Tc 0.0239 Tw [( and )]TJ ET q 1 i 279 151 16 14 re W n BT /F8 1 Tf 12 0 0 12 279 151 Tm 0.0314 Tc 0 Tw ( )Tj ET Q BT /F8 1 Tf 12 0 0 12 279.7813 155 Tm 0 Tc 0 Tw (V)Tj /F10 1 Tf 9 0 0 9 289.25 152 Tm (2)Tj /F3 1 Tf 12 0 0 12 295 155 Tm 0.0078 Tc 0.0774 Tw [( used together include all constraints in the)]TJ -18.5833 -2 TD 0.0022 Tc 0 Tw (set )Tj /F8 1 Tf 1.5103 0 TD 0 Tc (V)Tj 10 0 0 10 97.967 129 Tm 0.0046 Tc ( )Tj /F3 1 Tf 12 0 0 12 102.459 131 Tm 0.0025 Tc 0.0249 Tw (due to )Tj /F6 1 Tf 12 0 2.64 12 141.5312 131 Tm 0 Tc 0 Tw ()Tj /F9 1 Tf 12 0 0 12 149.4687 131 Tm (D)Tj /F10 1 Tf 7 0 0 7 157.9062 128 Tm (1)Tj /F6 1 Tf 12 0 0 12 164.5625 131 Tm ()Tj 12 0 2.64 12 173.2812 131 Tm ()Tj /F9 1 Tf 12 0 0 12 180.4375 131 Tm (D)Tj /F10 1 Tf 7 0 0 7 189.4375 128 Tm (2)Tj /F3 1 Tf 12 0 0 12 194 131 Tm ( \()Tj /F6 1 Tf 12 0 2.64 12 202.5312 131 Tm ()Tj 12 0 0 12 213.6562 131 Tm ()Tj 12 0 2.64 12 222.375 131 Tm ()Tj 12 0 0 12 233.0625 131 Tm ()Tj /F10 1 Tf 0.7187 0 TD -0.0859 Tc [(1,)-203.1( )]TJ /F6 1 Tf 12 0 2.64 12 255.5937 131 Tm 0 Tc ()Tj /F10 1 Tf 12 0 0 12 264.0625 131 Tm (,)Tj /F6 1 Tf 12 0 2.64 12 267.75 131 Tm ()Tj 12 0 0 12 278.2188 131 Tm ()Tj /F10 1 Tf 0.7917 0 TD (0)Tj /F3 1 Tf 0.6068 0 TD 0.0045 Tc 0.045 Tw (\). The constraints in )Tj /F8 1 Tf 9.249 0 TD 0 Tc 0 Tw (I)Tj /F3 1 Tf 0.3918 0 TD 0.0037 Tc 0.0376 Tw [( can be expressed as a)]TJ -28.2242 -2 TD 0.0105 Tc 0.1046 Tw (series of inequalities in matrix form:)Tj ET q 1 i 108 81 42 12 re W n BT /F2 1 Tf 12 0 0 12 108 81 Tm 0 Tc 0 Tw ( )Tj ET Q BT /F2 1 Tf 12 0 0 12 109.8125 83 Tm 0 Tc 0 Tw (XM)Tj /F6 1 Tf 1.8984 0 TD ()Tj /F2 1 Tf 0.7995 0 TD (b)Tj /F3 1 Tf 31.9844 0 TD (\(6\))Tj ET endstream endobj 25 0 obj << /Type /Page /Parent 108 0 R /Resources 26 0 R /Contents 27 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 26 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R /F4 122 0 R /F6 96 0 R /F8 98 0 R /F9 97 0 R /F10 99 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 27 0 obj << /Length 4887 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (10)Tj 12 0 0 12 72 709 Tm 0.0057 Tc (Sets )Tj ET q 1 i 97 705 15 14 re W n BT /F8 1 Tf 12 0 0 12 97 705 Tm 0.0314 Tc ( )Tj ET Q BT /F8 1 Tf 12 0 0 12 97.7812 709 Tm 0 Tc (V)Tj /F10 1 Tf 9 0 0 9 106.5312 706 Tm (1)Tj /F3 1 Tf 12 0 0 12 112 709 Tm 0.0024 Tc 0.0239 Tw [( and )]TJ ET q 1 i 139 705 16 14 re W n BT /F8 1 Tf 12 0 0 12 139 705 Tm 0.0314 Tc 0 Tw ( )Tj ET Q BT /F8 1 Tf 12 0 0 12 139.7812 709 Tm 0 Tc 0 Tw (V)Tj /F10 1 Tf 9 0 0 9 149.25 706 Tm (2)Tj /F3 1 Tf 12 0 0 12 155 709 Tm 0.004 Tc 0.0402 Tw [( are constructed by assuming that if )]TJ /F9 1 Tf 16.8958 0 TD 0 Tc 0 Tw (D)Tj /F10 1 Tf 7 0 0 7 366.1875 706 Tm (1)Tj /F3 1 Tf 12 0 0 12 370 709 Tm 0.0024 Tc 0.0239 Tw [( and )]TJ /F9 1 Tf 2.3958 0 TD 0 Tc 0 Tw (D)Tj /F10 1 Tf 7 0 0 7 407.75 706 Tm (2)Tj /F3 1 Tf 12 0 0 12 413 709 Tm 0.0056 Tc 0.0561 Tw [( are applied then the)]TJ -28.4167 -2 TD 0.0075 Tc 0.0751 Tw (result is in )Tj /F8 1 Tf 5.1788 0 TD 0.0033 Tc 0 Tw (I )Tj /F3 1 Tf 0.6399 0 TD 0 Tc (:)Tj ET q 1 i 108 628 57 40 re W n BT /F2 1 Tf 12 0 0 12 108 628 Tm ( )Tj ET Q BT /F2 1 Tf 12 0 0 12 109.8125 657.4375 Tm (XD)Tj 9 0 0 9 127.875 654.4375 Tm (1)Tj 12 0 0 12 132.5625 657.4375 Tm (M)Tj /F6 1 Tf 1.2318 0 TD ()Tj /F2 1 Tf 0.7995 0 TD (b)Tj -3.9271 -2.0755 TD (XD)Tj 9 0 0 9 127.9688 629.5313 Tm (2)Tj 12 0 0 12 132.7812 632.5313 Tm (M)Tj /F6 1 Tf 1.2318 0 TD ()Tj /F2 1 Tf 0.7995 0 TD (b)Tj /F3 1 Tf 30.737 1.0391 TD (\(7\))Tj -37.8333 -3.0833 TD 0.014 Tc 0.14 Tw (This means that:)Tj ET q 1 i 108 551 71 52 re W n BT /F2 1 Tf 12 0 0 12 108 551 Tm 0 Tc 0 Tw ( )Tj ET Q BT /F6 1 Tf 12 0 2.64 12 109.5312 580.8437 Tm 0 Tc 0 Tw ()Tj /F2 1 Tf 12 0 0 12 117.4688 580.8437 Tm (XD)Tj /F10 1 Tf 9 0 0 9 134.875 577.8437 Tm (1)Tj /F2 1 Tf 12 0 0 12 139 580.8437 Tm (M)Tj /F6 1 Tf 1.2318 0 TD ()Tj 12 0 2.64 12 163.0937 580.8437 Tm ()Tj /F2 1 Tf 12 0 0 12 171.0313 580.8437 Tm (b)Tj /F6 1 Tf 12 0 2.64 12 109.2812 555.7188 Tm ()Tj /F2 1 Tf 12 0 0 12 116.4375 555.7188 Tm (XD)Tj /F10 1 Tf 9 0 0 9 134.5625 552.7187 Tm (2)Tj /F2 1 Tf 12 0 0 12 139.4063 555.7188 Tm (M)Tj /F6 1 Tf 1.2318 0 TD ()Tj 12 0 2.64 12 163.25 555.7188 Tm ()Tj /F2 1 Tf 12 0 0 12 170.4063 555.7188 Tm (b)Tj /F3 1 Tf 29.6328 1.5234 TD (\(8\))Tj -37.8333 -4.0833 TD 0.0071 Tc 0.0711 Tw (Adding these two equations, and insisting that )Tj /F6 1 Tf 12 0 2.64 12 334.5312 525 Tm 0 Tc 0 Tw ()Tj 12 0 0 12 345.6563 525 Tm ()Tj 12 0 2.64 12 354.375 525 Tm ()Tj 12 0 0 12 365.0625 525 Tm ()Tj /F10 1 Tf 0.7187 0 TD (1)Tj /F3 1 Tf 0.526 0 TD 0.0068 Tc 0.0684 Tw [( gives)]TJ ET q 1 i 108 489 107 17 re W n BT /F2 1 Tf 12 0 0 12 108 489 Tm 0 Tc 0 Tw ( )Tj ET Q BT /F2 1 Tf 12 0 0 12 109.8125 494 Tm 0 Tc 0 Tw (X)Tj /F6 1 Tf 12 0 2.64 12 122.0937 494 Tm ()Tj /F2 1 Tf 12 0 0 12 130.0313 494 Tm (D)Tj /F10 1 Tf 9 0 0 9 139.4375 491 Tm (1)Tj /F6 1 Tf 12 0 0 12 146.875 494 Tm ()Tj 12 0 2.64 12 155.5937 494 Tm ()Tj /F2 1 Tf 12 0 0 12 162.75 494 Tm (D)Tj /F10 1 Tf 9 0 0 9 172.875 491 Tm (2)Tj /F6 1 Tf 12 0 0 16.428 117.9687 493.125 Tm 4.7347 Tc ()Tj /F2 1 Tf 12 0 0 12 182.6875 494 Tm 0 Tc (M)Tj /F6 1 Tf 1.2318 0 TD ()Tj /F2 1 Tf 0.7995 0 TD (b)Tj /F3 1 Tf 26.5781 0 TD (\(9\))Tj -37.8333 -2.0833 TD 0.0069 Tc 0.069 Tw (which is the precisely condition imposed by using the mapping )Tj /F6 1 Tf 12 0 2.64 12 426.5312 469 Tm 0 Tc 0 Tw ()Tj /F9 1 Tf 12 0 0 12 434.4687 469 Tm (D)Tj /F10 1 Tf 7 0 0 7 442.9062 466 Tm (1)Tj /F6 1 Tf 12 0 0 12 449.5625 469 Tm ()Tj 12 0 2.64 12 458.2812 469 Tm ()Tj /F9 1 Tf 12 0 0 12 465.4375 469 Tm (D)Tj /F10 1 Tf 7 0 0 7 474.4375 466 Tm (2)Tj /F3 1 Tf 12 0 0 12 479 469 Tm (.)Tj -30.9167 -2 TD 0.0068 Tc 0.068 Tw (Despite the details, the additional constraint is very simple in that it says that)Tj -3 -2 TD 0.0067 Tc 0.0663 Tw (we have to be able to scale the illuminant by a certain amount and )Tj /F4 1 Tf 30.8373 0 TD 0.0145 Tc 0 Tw (still)Tj /F3 1 Tf 1.6285 0 TD 0.0072 Tc 0.0713 Tw [( satisfy the)]TJ -32.4658 -2 TD 0.009 Tc 0.09 Tw (illumination constraint. This constraint is realized by simply scaling the set of)Tj T* 0.0118 Tc 0.1184 Tw (illuminants by the inverse. As a simple example, consider the one-dimensional line)Tj T* 0.005 Tc 0.0499 Tw (segment [0,1]. If we have a condition on these points that when they are scaled by a)Tj T* 0.0072 Tc 0.0728 Tw [(factor of two the result must still be in that segment, then the set of points in our)]TJ 0 -1.8333 TD 0.0042 Tc 0.0425 Tw [(constrained set must be [0, )]TJ ET q 1 i 220 297 9 18 re W n BT 12 0 0 12 220 297 Tm 0 Tc 0 Tw ( )Tj ET Q BT 7 0 0 7 223 308.75 Tm 0 Tc 0 Tw (1)Tj -0.0134 -1.4152 TD (2)Tj ET q 1 i 220 297 9 18 re W n 0 G 0 J 0 j 0.5 w 10 M []0 d 222 306.094 m 227.156 306.094 l S Q BT 12 0 0 12 229 303 Tm 0.0043 Tc 0.0431 Tw (]. In other words, the set was scaled by the inverse of the)Tj -13.0833 -2.1667 TD 0.0032 Tc 0.0315 Tw (scale factor.)Tj ET endstream endobj 28 0 obj << /Type /Page /Parent 110 0 R /Resources 29 0 R /Contents 30 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 29 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 30 0 obj << /Length 2958 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (11)Tj /F2 1 Tf 12 0 0 12 72 710 Tm -0.0371 Tc (II.2)Tj 12 0 TD 0.0033 Tc 0.0328 Tw (Combining the Constraints)Tj /F3 1 Tf -9 -2 TD 0.0088 Tc 0.088 Tw [(Given the above formulation of the various constraints they can be easily)]TJ -3 -2 TD 0.0112 Tc 0.112 Tw [(combined into a forceful colour constancy algorithm. First the relative illumination)]TJ T* 0.0117 Tc 0.1163 Tw (field is used to remove the illumination variation from the image leaving an image)Tj T* 0.0065 Tc 0.0648 Tw [(which is of the scene with chromaticities as they would have appear if it had been)]TJ T* 0.0106 Tc 0.1056 Tw [(illuminated throughout by the illumination at the base point. Starting from this)]TJ T* 0.0107 Tc 0.1065 Tw (intermediate result the constraints sets on the possible illumination maps is derived)Tj T* 0.0099 Tc 0.0993 Tw [(for each of the surface chromaticities that is present. The illumination constraint)]TJ T* 0.008 Tc 0.0803 Tw (provided by the set of plausible illuminants is fixed by the initial measurement of)Tj T* 0.0101 Tc 0.1006 Tw (the various illuminants around the campus. Each hull point of the set of the)Tj T* 0.0135 Tc 0.1354 Tw (relative illumination field furnishes yet another constraint; namely, the)Tj T* 0.012 Tc 0.1197 Tw (illumination constraint multiplied by the appropriate diagonal transform. The)Tj T* 0.0138 Tc 0.1372 Tw [(illumination constraint and the transforms due to the relative illumination field)]TJ T* 0.0083 Tc 0.0837 Tw (are intersected, and the result is inverted. As mentioned above, this inverted set was)Tj T* 0.0824 Tw [(approximated well by its convex hull. The inverted set is then intersected with the)]TJ T* 0.0096 Tc 0.0959 Tw (intersection of all the surface constraints.)Tj 3 -2 TD 0.0074 Tc 0.0742 Tw (The final step of the algorithm is to chose a solution from the set of possible)Tj -3 -2 TD 0.0087 Tc 0.087 Tw (solutions. In [5, 1] the solution maximizes the volume of the mapped set, which is)Tj T* 0.0098 Tc 0.0975 Tw [(equivalent to maximizing the product of the components of the mapping. In this)]TJ T* 0.0092 Tc 0.0919 Tw (work, however, we use the centroid of the solution set, which is more natural. This)Tj T* 0.0077 Tc 0.0773 Tw [(choice can be shown to minimize the expected error if all solutions are equally likely)]TJ T* 0.0073 Tc 0.073 Tw (and error is measured by the distance from the choice to the actual solution.)Tj T* 0.0083 Tc 0.0833 Tw (Furthermore, in both synthesized and real images, the centroid was found to give)Tj T* 0.007 Tc 0.0703 Tw (better results.)Tj 3 -2 TD 0.01 Tc 0.1 Tw (The colour constancy algorithm that incorporates all the different constraints)Tj -3 -2 TD 0.0047 Tc 0.0478 Tw [(was tested first on generated data. One thousand sets containing 1, )47.7(2, )47.7(4, )47.8(8, and 16)]TJ T* 0.0093 Tc 0.0931 Tw (surfaces were randomly generated and used in conjunction with 1 of 5 illuminants,)Tj ET endstream endobj 31 0 obj << /Type /Page /Parent 110 0 R /Resources 32 0 R /Contents 33 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 32 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 33 0 obj << /Length 687 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (12)Tj 12 0 0 12 72 710 Tm 0.0097 Tc 0.0963 Tw (with 0 through 4 of the remaining lights playing the role of additional illuminants)Tj 0 -2 TD 0.0077 Tc 0.0775 Tw [(arising as a result of varying illumination. Table I gives the results which are)]TJ T* 0.0064 Tc 0.0643 Tw (exactly as wished. As either the number of surfaces or the number of extra lights)Tj T* 0.0093 Tc 0.0922 Tw [(increases, the answer consistently improves. Thus it was verified that varying)]TJ T* 0.0102 Tc 0.1019 Tw (illumination is a powerful constraint, and furthermore, it can be effectively)Tj T* 0.1024 Tw (integrated with the other constraints.)Tj ET endstream endobj 34 0 obj << /Type /Page /Parent 110 0 R /Resources 35 0 R /Contents 36 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 35 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 36 0 obj << /Length 8508 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (13)Tj 0 12 -12 0 88 550 Tm 0.023 Tc 0.2301 Tw (Solution Method)Tj 10 0 0 10 214 653 Tm 6.7 Tc 0 Tw [(1248)500(1)6700(6)]TJ ET 1 i 108 675 1 -1 re f 108 675 1 -1 re f 109 675 71 -1 re f 180 675 1 -1 re f 181 675 71 -1 re f 252 675 1 -1 re f 253 675 71 -1 re f 324 675 1 -1 re f 325 675 71 -1 re f 396 675 1 -1 re f 397 675 71 -1 re f 468 675 1 -1 re f 469 675 66 -1 re f 535 675 1 -1 re f 535 675 1 -1 re f 108 674 1 -24 re f 180 674 1 -24 re f 252 674 1 -24 re f 324 674 1 -24 re f 396 674 1 -24 re f 468 674 1 -24 re f 535 674 1 -24 re f BT 10 0 0 10 113 640 Tm 0.0041 Tc 0.0408 Tw [(B)-129.2(F)-5895.6(0.073 \(0.001\))-1395(0.073 \(0.001\))-1395(0.073 \(0.001\))-1395(0.073 \(0.001\))-1395(0.073 \(0.001\))]TJ ET 108 650 1 -1 re f 109 650 71 -1 re f 180 650 1 -1 re f 181 650 71 -1 re f 252 650 1 -1 re f 253 650 71 -1 re f 324 650 1 -1 re f 325 650 71 -1 re f 396 650 1 -1 re f 397 650 71 -1 re f 468 650 1 -1 re f 469 650 66 -1 re f 535 650 1 -1 re f 108 649 1 -12 re f 180 649 1 -12 re f 252 649 1 -12 re f 324 649 1 -12 re f 396 649 1 -12 re f 468 649 1 -12 re f 535 649 1 -12 re f BT 10 0 0 10 113 627 Tm 0.0067 Tc 0.0382 Tw [(B)-44.5(D)-44.5(T)-5092.9(0.116 \(0.002\))-1392.4(0.116 \(0.002\))-1392.4(0.116 )-47.3(\(0.002\))-1592.7(0.116 )-47.3(\(0.002\))-1592.7(0.116 )-47.3(\(0.002\))]TJ ET 108 637 1 -1 re f 109 637 71 -1 re f 180 637 1 -1 re f 181 637 71 -1 re f 252 637 1 -1 re f 253 637 71 -1 re f 324 637 1 -1 re f 325 637 71 -1 re f 396 637 1 -1 re f 397 637 71 -1 re f 468 637 1 -1 re f 469 637 66 -1 re f 535 637 1 -1 re f 108 636 1 -12 re f 180 636 1 -12 re f 252 636 1 -12 re f 324 636 1 -12 re f 396 636 1 -12 re f 468 636 1 -12 re f 535 636 1 -12 re f BT 10 0 0 10 113 614 Tm 0.007 Tc 0.0407 Tw [(G)-30.1(W)-5392.9(1.62 \(0.03\))-2392.9(1.01 \(0.01\))-2392.9(0.69 )-53.7(\(0.01\))-2592.8(0.513 )-44.5(\(0.004\))-1592.4(0.428 )-44.5(\(0.003\))]TJ ET 108 624 1 -1 re f 109 624 71 -1 re f 180 624 1 -1 re f 181 624 71 -1 re f 252 624 1 -1 re f 253 624 71 -1 re f 324 624 1 -1 re f 325 624 71 -1 re f 396 624 1 -1 re f 397 624 71 -1 re f 468 624 1 -1 re f 469 624 66 -1 re f 535 624 1 -1 re f 108 623 1 -12 re f 180 623 1 -12 re f 252 623 1 -12 re f 324 623 1 -12 re f 396 623 1 -12 re f 468 623 1 -12 re f 535 623 1 -12 re f BT 10 0 0 10 113 601 Tm 0.0042 Tc 0.0423 Tw [(RET)-5295.4(1.62 \(0.03\))-2395.7(1.10 \(0.01\))-2395.7(0.72 \(0.01\))-2395.7(0.478 \(0.004\))-1394.9(0.354 \(0.003\))]TJ ET 108 611 1 -1 re f 109 611 71 -1 re f 180 611 1 -1 re f 181 611 71 -1 re f 252 611 1 -1 re f 253 611 71 -1 re f 324 611 1 -1 re f 325 611 71 -1 re f 396 611 1 -1 re f 397 611 71 -1 re f 468 611 1 -1 re f 469 611 66 -1 re f 535 611 1 -1 re f 108 610 1 -12 re f 180 610 1 -12 re f 252 610 1 -12 re f 324 610 1 -12 re f 396 610 1 -12 re f 468 610 1 -12 re f 535 610 1 -12 re f BT 10 0 0 10 113 588 Tm 0.0049 Tc 0.0183 Tw [(S)-6670.1(12.4 )-26.1( )-26.1(\(0.3\))-2894.5(4.4 \(0.1\))-3194.8(1.65 )-24.5( )-24.5(\(0.05\))-2395(0.585 )-23( )-23(\(0.01\))-1894.7(0.285 )-69(\(0.003\))]TJ ET 108 598 1 -1 re f 109 598 71 -1 re f 180 598 1 -1 re f 181 598 71 -1 re f 252 598 1 -1 re f 253 598 71 -1 re f 324 598 1 -1 re f 325 598 71 -1 re f 396 598 1 -1 re f 397 598 71 -1 re f 468 598 1 -1 re f 469 598 66 -1 re f 535 598 1 -1 re f 108 597 1 -12 re f 180 597 1 -12 re f 252 597 1 -12 re f 324 597 1 -12 re f 396 597 1 -12 re f 468 597 1 -12 re f 535 597 1 -12 re f BT 10 0 0 10 113 575 Tm 0.0056 Tc 0.016 Tw [(S)-132.6(I)-6194.2(2.275 )-79.8(\(0.2\))-2594.2(1.677 )-24.6( )-24.6(\(0.02\))-1894(0.99 )-26.1( )-26.1(\(0.02\))-2394.3(0.480 \(0.01\))-1693.6(0.271 )-70.6(\(0.003\))]TJ ET 108 585 1 -1 re f 109 585 71 -1 re f 180 585 1 -1 re f 181 585 71 -1 re f 252 585 1 -1 re f 253 585 71 -1 re f 324 585 1 -1 re f 325 585 71 -1 re f 396 585 1 -1 re f 397 585 71 -1 re f 468 585 1 -1 re f 469 585 66 -1 re f 535 585 1 -1 re f 108 584 1 -12 re f 180 584 1 -12 re f 252 584 1 -12 re f 324 584 1 -12 re f 396 584 1 -12 re f 468 584 1 -12 re f 535 584 1 -12 re f BT 10 0 0 10 113 562 Tm 0.0721 Tc 0 Tw (SIV1)Tj 7.2 0 TD 0.0052 Tc 0.0421 Tw [(1.65 \(0.02\))-2394.7(1.26 \(0.01\))-2394.7(0.79 \(0.01\))-2394.7(0.420 \(0.01\))-1894.4(0.256 )-44.9(\(0.002\))]TJ ET 108 572 1 -1 re f 109 572 71 -1 re f 180 572 1 -1 re f 181 572 71 -1 re f 252 572 1 -1 re f 253 572 71 -1 re f 324 572 1 -1 re f 325 572 71 -1 re f 396 572 1 -1 re f 397 572 71 -1 re f 468 572 1 -1 re f 469 572 66 -1 re f 535 572 1 -1 re f 108 571 1 -12 re f 180 571 1 -12 re f 252 571 1 -12 re f 324 571 1 -12 re f 396 571 1 -12 re f 468 571 1 -12 re f 535 571 1 -12 re f BT 10 0 0 10 113 549 Tm 0.0721 Tc 0 Tw (SIV2)Tj 7.2 0 TD 0.0077 Tc 0.0867 Tw [(1.154 \(0.01\))-2091.7(0.896 \(0.01\))-2091.7(0.620 \(0.008\))-1591.7(0.351 )49.5( )49.5(\(0.004\))-1391.4(0.242 \(0.002\))]TJ ET 108 559 1 -1 re f 109 559 71 -1 re f 180 559 1 -1 re f 181 559 71 -1 re f 252 559 1 -1 re f 253 559 71 -1 re f 324 559 1 -1 re f 325 559 71 -1 re f 396 559 1 -1 re f 397 559 71 -1 re f 468 559 1 -1 re f 469 559 66 -1 re f 535 559 1 -1 re f 108 558 1 -12 re f 180 558 1 -12 re f 252 558 1 -12 re f 324 558 1 -12 re f 396 558 1 -12 re f 468 558 1 -12 re f 535 558 1 -12 re f BT 10 0 0 10 113 536 Tm 0.0721 Tc 0 Tw (SIV3)Tj 7.2 0 TD 0.0085 Tc 0.0845 Tw [(0.800 \(0.01\))-2090.9(0.656 \(0.008\))-1590.9(0.488 \(0.006\))-1590.9(0.311 \(0.004\))-1590.9(0.231 \(0.001\))]TJ ET 108 546 1 -1 re f 109 546 71 -1 re f 180 546 1 -1 re f 181 546 71 -1 re f 252 546 1 -1 re f 253 546 71 -1 re f 324 546 1 -1 re f 325 546 71 -1 re f 396 546 1 -1 re f 397 546 71 -1 re f 468 546 1 -1 re f 469 546 66 -1 re f 535 546 1 -1 re f 108 545 1 -12 re f 180 545 1 -12 re f 252 545 1 -12 re f 324 545 1 -12 re f 396 545 1 -12 re f 468 545 1 -12 re f 535 545 1 -12 re f BT 10 0 0 10 113 523 Tm 0.0721 Tc 0 Tw (SIV4)Tj 7.2 0 TD 0.0084 Tc 0.0838 Tw [(0.384 \(0.002\))-1591(0.359 \(0.002\))-1591(0.317 \(0.002\))-1591(0.274 \(0.002\))-1591(0.228 \(0.002\))]TJ ET 108 533 1 -1 re f 109 533 71 -1 re f 180 533 1 -1 re f 181 533 71 -1 re f 252 533 1 -1 re f 253 533 71 -1 re f 324 533 1 -1 re f 325 533 71 -1 re f 396 533 1 -1 re f 397 533 71 -1 re f 468 533 1 -1 re f 469 533 66 -1 re f 535 533 1 -1 re f 108 532 1 -12 re f 108 520 1 -1 re f 108 520 1 -1 re f 109 520 71 -1 re f 180 532 1 -12 re f 180 520 1 -1 re f 181 520 71 -1 re f 252 532 1 -12 re f 252 520 1 -1 re f 253 520 71 -1 re f 324 532 1 -12 re f 324 520 1 -1 re f 325 520 71 -1 re f 396 532 1 -12 re f 396 520 1 -1 re f 397 520 71 -1 re f 468 532 1 -12 re f 468 520 1 -1 re f 469 520 66 -1 re f 535 532 1 -12 re f 535 520 1 -1 re f 535 520 1 -1 re f BT 12 0 0 12 250 688 Tm 0.0115 Tc 0.1153 Tw (Number of Surfaces)Tj 10 0 0 10 72 462 Tm 0.0064 Tc 0.0636 Tw (Solution Method Key)Tj 0 -3.6 TD -0.0002 Tc 0 Tw [(B)-133.5(F)-5899.9(Error of best possible solution using full linear map)]TJ 0 -1.2 TD -0.0026 Tc [(B)-53.8(D)-53.8(T)-5102.2(Error of best possible solution using a diagonal map)]TJ T* 0.0088 Tc 0.088 Tw [(G)-28.3(W)-5391.1(Naive Grey World Algorithm \(scale each channel by average\))]TJ T* 0.0069 Tc 0.0699 Tw [(RET)-5292.7(Naive Retinex Algorithm \(scale each channel by maximum\))]TJ T* 0.0042 Tc 0.0424 Tw [(S)-6670.8(Surface constraints alone)]TJ T* 0.0056 Tc 0.0563 Tw [(S)-132.6(I)-6194.2(Surface and illumination constraints)]TJ T* 0.0039 Tc 0.0393 Tw [(S)-104.2(I)-104.2(V)-5395.9(Surface and illumination constraints with view under one extra illuminant)]TJ T* 0.0721 Tc 0 Tw (SIV2)Tj 7.2 0 TD 0.0044 Tc 0.0437 Tw (Surface and illumination constraints with view under two extra illuminants)Tj -7.2 -1.2 TD 0.0721 Tc 0 Tw (SIV3)Tj 7.2 0 TD 0.0058 Tc 0.0581 Tw (Surface and illumination constraints with view under three extra illuminants)Tj -7.2 -1.2 TD 0.0721 Tc 0 Tw (SIV4)Tj 7.2 0 TD 0.0039 Tc 0.0392 Tw (Surface and illumination constraints with view under four extra illuminants)Tj -7.2 -4.8 TD 0.0002 Tc -0.0004 Tw [(Table )-79.1(1)-3898.9(Results of color constancy experiments for 1000 sets of 1, 2, 4, 8, and 16 surfaces under all)]TJ 7.2 -1.2 TD 0.0063 Tc 0.0627 Tw (combinations of test lights and extra lights for varying illumination. The canonical)Tj T* 0.0075 Tc 0.0749 Tw (illuminant was a Philips CW fluorescent light. The values shown are the average)Tj T* 0.0057 Tc 0.0572 Tw (magnitude of the chromaticity vector difference between the estimate and the desired)Tj T* 0.0066 Tc 0.0664 Tw (answer, averaged over all results. The value in parenthesis is the error estimate of this)Tj T* 0.0395 Tc 0 Tw (average.)Tj ET endstream endobj 37 0 obj << /Type /Page /Parent 110 0 R /Resources 38 0 R /Contents 39 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 38 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 39 0 obj << /Length 2863 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (14)Tj /F2 1 Tf 14 0 0 14 72 708 Tm -0.0149 Tc [(I)32.7(I)32.7(I)-9514.6(Finding the Relative Illumination Field)]TJ /F3 1 Tf 12 0 0 12 108 660 Tm 0.0122 Tc 0.1225 Tw [(We now detail an algorithm for finding the relative illumination field)]TJ -3 -2 TD 0.0114 Tc 0.1138 Tw (describing the variation in the incident illumination. This algorithm can be divided)Tj T* 0.0072 Tc 0.0723 Tw [(into two parts. The first is a new technique for image segmentation appropriate for)]TJ T* 0.0105 Tc 0.1047 Tw (scenes with varying illumination. The second part uses the segmentation to)Tj T* 0.0151 Tc 0.1505 Tw (determine the illumination map robustly.)Tj 3 -2 TD 0.0093 Tc 0.0933 Tw (Unless the illumination is known to be constant, it is essential that a)Tj -3 -2 TD 0.0109 Tc 0.1095 Tw (segmentation method be able to accommodate varying illumination. In general, the)Tj T* 0.0101 Tc 0.1011 Tw (segmentation problem is quite difficult, especially with varying illumination, as in)Tj T* 0.0061 Tc 0.0609 Tw (this case an area of all one reflectance can exhibit a wide range of colour. Fortunately)Tj T* 0.0074 Tc 0.074 Tw (for our purposes, it is not critical if an occasional region is mistakenly divided into)Tj T* 0.0076 Tc 0.0762 Tw [(two pieces, nor if two regions which have almost the same colour are incorrectly)]TJ T* 0.008 Tc 0.0797 Tw (merged. This is because the goal at this point is simply to identify the illumination,)Tj T* 0.0096 Tc 0.096 Tw (not the surfaces. Nonetheless, the better the segmentation, the more reliable and)Tj T* 0.0065 Tc 0.0648 Tw (accurate the possible colour constancy.)Tj 3 -2 TD 0.0042 Tc 0.0428 Tw [(One approach to segmentation is that used by Retinex theory [6, 7, 8]. In)]TJ -3 -2 TD 0.0083 Tc 0.0834 Tw (Retinex small changes in pixel values at neighboring locations are assumed to be)Tj T* 0.0822 Tw [(due to changes in the illumination and large changes to changes in surface)]TJ T* 0.0066 Tc 0.0657 Tw [(reflectance. This idea can be used to segment an image into regions of constant)]TJ T* 0.0046 Tc 0.046 Tw (surface reflectance properties by growing regions by including pixels only if they are)Tj T* 0.0114 Tc 0.1141 Tw (less than some small threshold different from their neighbors. The threshold must)Tj T* 0.008 Tc 0.0802 Tw (be large enough to allow for both noise and the illumination changes and yet not)Tj T* (admit small changes in surface reflectance)Tj 10 0 0 10 304.282 156 Tm 0 Tc 0 Tw ()Tj 12 0 0 12 314.678 156 Tm 0.007 Tc 0.0702 Tw (a balance which is of course impossible)Tj -20.2232 -2 TD 0.008 Tc 0.0801 Tw (to establish.)Tj 3 -2 TD 0.0808 Tw (We use this method as part of our algorithm, but alone, it is not sufficient.)Tj -3 -2 TD 0.0098 Tc 0.0977 Tw [(The problem is that two dissimilar regions will eventually mistakenly merge if)]TJ ET endstream endobj 40 0 obj << /Type /Page /Parent 110 0 R /Resources 41 0 R /Contents 42 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 41 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 42 0 obj << /Length 2968 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (15)Tj 12 0 0 12 72 710 Tm 0.0074 Tc 0.0743 Tw (there exists a sequence of small jumps connecting them. This can occur if the edge is)Tj 0 -2 TD 0.0066 Tc 0.0657 Tw (gradual or because of noise. In essence, a threshold large enough to allow for noise)Tj T* 0.01 Tc 0.0999 Tw (\(and varying illumination\) allows for enough drift in the pixel values to include an)Tj T* 0.0102 Tc 0.1014 Tw [(entirely dissimilar region. Local information alone is insufficient, so we resolve the)]TJ T* 0.0115 Tc 0.1148 Tw (problem by adding a more global condition involving illumination constraints.)Tj 3 -2 TD 0.0074 Tc 0.0744 Tw (The new global segmentation condition is based on the idea that in order for)Tj -3 -2 TD 0.0057 Tc 0.0569 Tw (two pixelsno matter how far apart they are spatiallyto be considered part of the)Tj T* 0.0108 Tc 0.1082 Tw (same region, a plausible illumination change between them must exist. This)Tj T* 0.0102 Tc 0.1015 Tw (condition binding pixels within a single region based on plausible illumination)Tj T* 0.0082 Tc 0.0813 Tw (changes is called patch coherence. The patch coherence condition differs from the)Tj T* 0.0101 Tc 0.1006 Tw (Retinex condition in two important ways. First, the cumulative drift in pixel values)Tj T* 0.0058 Tc 0.0584 Tw (along a path is limited, as opposed to growing linearly with the pixel distance.)Tj T* 0.0081 Tc 0.0804 Tw (Second, the allowable drift is constrained more in certain directions due to the)Tj T* 0.0114 Tc 0.1138 Tw (nature of the set of common illuminants. For example, green illuminants are rare,)Tj T* 0.011 Tc 0.11 Tw (which means that the set of common illuminants is narrow in the green direction,)Tj T* 0.0069 Tc 0.0688 Tw (and thus overall, the drift towards or away from green is more restricted than that)Tj T* 0.0044 Tc 0.044 Tw (towards or away from blue.)Tj 3 -2 TD 0.0065 Tc 0.0647 Tw (It was found to be useful to retain the Retinex condition as well as the patch)Tj -3 -2 TD 0.0083 Tc 0.0827 Tw [(coherence method described above for two reasons. First, the Retinex condition is)]TJ T* 0.0048 Tc 0.0485 Tw (faster to compute, and thus can be use to reject pixels that do not need to be tested)Tj T* 0.0101 Tc 0.1007 Tw (further for inclusion. Second, if a comprehensive set of possible illuminants is used,)Tj T* 0.0083 Tc 0.0827 Tw [(then an occasional surface boundary change will also be a possible illumination)]TJ T* 0.0088 Tc 0.0874 Tw [(change. Since the Retinex method by itself works much of the time, these)]TJ T* 0.0102 Tc 0.102 Tw (exceptional cases in which a surface change mimics an illumination change)Tj T* 0.0075 Tc 0.0747 Tw (generally will be covered by the Retinex condition.)Tj 3 -2 TD 0.0089 Tc 0.0888 Tw (In detail our segmentation algorithm begins with an arbitrary initial starting)Tj -3 -2 TD 0.0093 Tc 0.0935 Tw (point in a region and assumes that the illumination at that point is constrained to)Tj ET endstream endobj 43 0 obj << /Type /Page /Parent 110 0 R /Resources 44 0 R /Contents 45 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 44 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R /F5 101 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 45 0 obj << /Length 3031 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (16)Tj 12 0 0 12 72 710 Tm 0.0078 Tc 0.0786 Tw [(be in the set of plausible illuminants. It is important to update the constraints on)]TJ 0 -2 TD 0.0082 Tc 0.082 Tw (the illumination at the starting point each time a new point is added to the region.)Tj T* 0.0102 Tc 0.1019 Tw (Each newly included point further constrains the possible illuminations at the)Tj T* 0.011 Tc 0.1103 Tw (starting point. Updating the constraints is similar to using the relative illumination)Tj T* 0.0071 Tc 0.0706 Tw (field to solve for colour constancy as described above. The element-wise ratio of the)Tj T* 0.0087 Tc 0.0865 Tw (chromaticities of the new point to that of the initial point induces a constraint set )Tj /F5 1 Tf 37.8908 0 TD 0 Tc 0 Tw (V)Tj /F3 1 Tf -37.8908 -2 TD 0.008 Tc 0.0797 Tw (defined by \(5\). Specifically, the illumination gamut is transformed by the inverse of)Tj T* 0.0078 Tc 0.0779 Tw (the ratio interpreted as a diagonal transform. This convex set is intersected with the)Tj T* 0.0084 Tc 0.0841 Tw (current constraint set. If the intersection is null, then the new point is excluded and)Tj T* 0.0086 Tc 0.0856 Tw (the constraint set is left unchanged. If it is not null, then the intersection becomes)Tj T* 0.006 Tc 0.06 Tw (the updated constraint set and the new point is added to the region.)Tj 3 -2 TD 0.0096 Tc 0.0955 Tw (Similar to the situation when solving for colour constancy, it is sufficient to)Tj -3 -2 TD 0.0087 Tc 0.087 Tw (perform the intersection only when the new transform to be applied to the)Tj T* 0.0113 Tc 0.1131 Tw (illumination gamut is outside the convex hull of the preceding transforms.)Tj T* 0.0103 Tc 0.1036 Tw (Although calculations are relative to the initial point, this procedure ensures that)Tj T* 0.0083 Tc 0.0825 Tw [(all points in the region can be assigned illuminants from the set of plausible)]TJ T* 0.015 Tc 0.1496 Tw (illuminants which are consistent with the illumination jumps between them.)Tj T* 0.0105 Tc 0.1046 Tw (Furthermore, the inclusion of any of the rejected points would violate this)Tj T* 0.0314 Tc 0 Tw (condition.)Tj 3 -2 TD 0.0101 Tc 0.1006 Tw (Given our segmentation we reduce the problem of finding the relative)Tj -3 -2 TD 0.0107 Tc 0.1078 Tw [(illumination field to that of finding the illumination at the center of each region)]TJ T* 0.0063 Tc 0.063 Tw (relative to that at the center of the base region. Since the center of a region, as)Tj T* 0.0082 Tc 0.083 Tw [(defined by the center of mass, need not be inside the region, the implementation)]TJ T* 0.0061 Tc 0.0605 Tw (uses the point in the region closest to the center of mass, preferably a few pixels from)Tj T* 0.009 Tc 0.0899 Tw (the boundary. The illumination at a point relative to that of the region center is)Tj T* 0.0075 Tc 0.0749 Tw (simply the ratio of its response to the response of the center point. This follows)Tj T* 0.0083 Tc 0.0829 Tw [(directly from the assumption that the pixels are from the same surface, given that)]TJ ET endstream endobj 46 0 obj << /Type /Page /Parent 110 0 R /Resources 47 0 R /Contents 48 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 47 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R /F6 96 0 R /F10 99 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 48 0 obj << /Length 6327 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (17)Tj 12 0 0 12 72 710 Tm 0.0075 Tc 0.0754 Tw (we accept a diagonal model for illumination change. Thus the map at an arbitrary)Tj 0 -2 TD 0.0077 Tc 0.0765 Tw (point is simply the map at the center, adjusted by this relative jump.)Tj 3 -2 TD 0.0092 Tc 0.0913 Tw (To determine the maps at the center points we make the assumption that)Tj -3 -2 TD 0.0103 Tc 0.1031 Tw (illumination does not change significantly at the region boundaries. Thus every)Tj T* 0.0076 Tc 0.0754 Tw (jump across a boundary gives a condition on the relative maps of the centers of the)Tj T* 0.0064 Tc 0.0643 Tw (two adjacent regions. More specifically, consider two regions A and B, with centers)Tj 0 -1.9167 TD 0 Tc 0 Tw (C)Tj 0.7343 -0.1667 TD (A)Tj 0.8057 0.1667 TD 0.004 Tc 0.0398 Tw [( and C)]TJ 3.0055 -0.1667 TD 0 Tc 0 Tw (B)Tj 0.6328 0.1667 TD 0.006 Tc 0.0602 Tw (, and boundary points B)Tj 11.049 -0.1667 TD 0 Tc 0 Tw (A)Tj 0.8057 0.1667 TD 0.0038 Tc 0.0384 Tw [( and B)]TJ 2.9038 -0.1667 TD 0 Tc 0 Tw (B)Tj 0.6327 0.1667 TD 0.0061 Tc 0.0611 Tw [( close to each other. Denote responses)]TJ -20.5694 -2.0833 TD 0.0051 Tc 0.0506 Tw (by R subscripted by the point label and denote the diagonal map relative to the grand)Tj 0 -2 TD 0.0057 Tc 0.0565 Tw (central point as D, also subscripted by the point label. Each channel or chromaticity)Tj T* 0.0089 Tc 0.0894 Tw (component is dealt with independently, so the quantities in the equations are)Tj T* 0.01 Tc 0.1 Tw (scalars. The assumption that the illumination does not change significantly at the)Tj T* 0.006 Tc 0.0602 Tw (boundary is simply:)Tj /F10 1 Tf 3.1484 -1.8333 TD 0 Tc 0 Tw (D)Tj 7 0 0 7 118.9062 421 Tm (B)Tj 5 0 0 5 124 419.25 Tm (A)Tj /F6 1 Tf 12 0 0 12 132.875 424 Tm ()Tj /F10 1 Tf 0.8099 0 TD (D)Tj 7 0 0 7 151.7187 421 Tm (B)Tj 5 0 0 5 156.7813 419.25 Tm (B)Tj /F3 1 Tf 12 0 0 12 520 424 Tm (\(10\))Tj -37.3333 -2.0833 TD 0.0088 Tc 0.0882 Tw (Since we are assuming a diagonal model of illumination change, and C)Tj 32.9295 -0.1667 TD 0.0042 Tc 0 Tw (A )Tj 1.0794 0.1667 TD 0.0358 Tc (is)Tj -34.0089 -2 TD 0.0058 Tc 0.0581 Tw (on the same surface as B)Tj 11.1944 -0.1667 TD 0 Tc 0 Tw (A)Tj 0.8091 0.1667 TD 0.0063 Tc 0.0633 Tw (, and similarly for the center and boundary of surface B,)Tj -12.0035 -2.0833 TD 0.0134 Tc 0.1343 Tw (we have:)Tj /F10 1 Tf 10 0 0 10 109.8125 315 Tm 0 Tc 0 Tw (D)Tj 7 0 0 7 117.5625 312.5 Tm (B)Tj 0.7232 -0.25 TD (A)Tj /F6 1 Tf 10 0 0 10 130.4687 315 Tm ()Tj /F10 1 Tf 0.6156 0 TD (D)Tj 7 0 0 7 144.2812 312.5 Tm (C)Tj 0.7634 -0.25 TD (A)Tj 10 0 0 10 162.4375 321.8125 Tm (R)Tj 7 0 0 7 169.8437 319.3125 Tm (B)Tj 0.7232 -0.25 TD (A)Tj 10 0 0 10 183.5625 310.7812 Tm (R)Tj 7 0 0 7 190.875 308.2812 Tm (C)Tj 0.7634 -0.25 TD (A)Tj ET 1 i 190.5 328.594 m 174.75 306.531 l 175 306.531 l 190.75 328.594 l f BT /F6 1 Tf 10 0 0 10 157.1875 320.0312 Tm ()Tj 0 -1.1281 TD ()Tj 4.6438 1.1281 TD ()Tj 0 -1.1281 TD ()Tj /F10 1 Tf 0.55 0.625 TD ( and D)Tj 7 0 0 7 261.3125 312.5 Tm (B)Tj 0.7187 -0.25 TD (B)Tj /F6 1 Tf 10 0 0 10 273.4062 315 Tm ()Tj /F10 1 Tf 0.6156 0 TD (D)Tj 7 0 0 7 287.2187 312.5 Tm (C)Tj 0.7589 -0.25 TD (B)Tj 10 0 0 10 304.5625 321.8125 Tm (R)Tj 7 0 0 7 311.9687 319.3125 Tm (B)Tj 0.7187 -0.25 TD (B)Tj 10 0 0 10 324.875 310.7812 Tm (R)Tj 7 0 0 7 332.1875 308.2812 Tm (C)Tj 0.7589 -0.25 TD (B)Tj ET 331.812 328.594 m 316.062 306.531 l 316.312 306.531 l 332.062 328.594 l f BT /F6 1 Tf 10 0 0 10 299.3125 320.0312 Tm ()Tj 0 -1.1281 TD ()Tj 4.4813 1.1281 TD ()Tj 0 -1.1281 TD ()Tj /F3 1 Tf 12 0 0 12 520 315 Tm (\(11\))Tj -37.3333 -4.0833 TD 0.0047 Tc 0.0477 Tw [(Combining \(10\) and \(11\) yields:)]TJ ET q 108 215 164 32 re W n BT 12 0 0 12 108 215 Tm 0 Tc 0 Tw ( )Tj ET Q BT 12 0 0 12 109.7187 227.9999 Tm 0 Tc 0 Tw (D)Tj 7 0 0 7 119.5937 224.8124 Tm (C)Tj 5 0 0 5 125.2187 222.9374 Tm (A)Tj 12 0 0 12 137.4375 236.0624 Tm (R)Tj 7 0 0 7 146.25 232.9687 Tm (B)Tj 5 0 0 5 151.2187 231.0937 Tm (A)Tj 12 0 0 12 159.0625 222.7812 Tm (R)Tj 7 0 0 7 167.9062 219.5937 Tm (C)Tj 5 0 0 5 173.5312 217.7187 Tm (A)Tj ET 167.375 244.375 m 148.375 217.719 l 148.875 217.719 l 167.875 244.375 l f BT /F6 1 Tf 12 0 0 12 131.5 234.3749 Tm ()Tj 0 -1.1797 TD ()Tj 0 0.5703 TD ()Tj 4.013 0.6094 TD ()Tj 0 -1.1797 TD ()Tj 0 0.5703 TD ()Tj 0.7292 0.0781 TD ()Tj /F3 1 Tf 0.8047 0 TD (D)Tj 7 0 0 7 207.9375 224.8124 Tm (C)Tj 5 0 0 5 213.5 222.9687 Tm (B)Tj 12 0 0 12 224.8437 236.0312 Tm (R)Tj 7 0 0 7 233.6562 232.9374 Tm (B)Tj 5 0 0 5 238.5625 231.0937 Tm (B)Tj 12 0 0 12 245.5 222.7812 Tm (R)Tj 7 0 0 7 254.3437 219.5937 Tm (C)Tj 5 0 0 5 259.9062 217.7499 Tm (B)Tj ET 253.812 244.344 m 234.844 217.75 l 235.344 217.75 l 254.312 244.344 l f BT /F6 1 Tf 12 0 0 12 218.9062 234.3437 Tm ()Tj 0 -1.1745 TD ()Tj 0 0.5703 TD ()Tj 3.8542 0.6042 TD ()Tj 0 -1.1745 TD ()Tj 0 0.5703 TD ()Tj /F3 1 Tf 21.237 0.0755 TD (\(12\))Tj -37.3333 -4.25 TD 0.0068 Tc 0.067 Tw [(Taking logarithms of both sides of \(12\), and rearranging terms gives:)]TJ ET q 108 135 291 17 re W n BT 12 0 0 12 108 135 Tm 0 Tc 0 Tw ( )Tj ET Q BT /F10 1 Tf 12 0 0 12 109.7812 142 Tm 0 Tc 0 Tw [(ln)-24.1(\()]TJ /F3 1 Tf 1.1641 0 TD (D)Tj 7 0 0 7 133.625 138.8125 Tm (C)Tj 5 0 0 5 139.25 136.9375 Tm (A)Tj /F10 1 Tf 12 0 0 12 145.6562 142 Tm (\))Tj /F6 1 Tf 0.5234 0 TD ()Tj /F10 1 Tf 0.7656 0 TD [(ln)-24.1(\()]TJ /F3 1 Tf 1.1641 0 TD (D)Tj 7 0 0 7 184.9687 138.8125 Tm (C)Tj 5 0 0 5 190.5312 136.9688 Tm (B)Tj /F10 1 Tf 12 0 0 12 196.0625 142 Tm (\))Tj /F6 1 Tf 0.5651 0 TD ()Tj /F10 1 Tf 0.8099 0 TD [(ln)-24.1(\()]TJ /F3 1 Tf 1.1641 0 TD (R)Tj 7 0 0 7 235.3437 138.9063 Tm (B)Tj 5 0 0 5 240.25 137.0625 Tm (B)Tj /F10 1 Tf 12 0 0 12 245.7812 142 Tm (\))Tj /F6 1 Tf 0.4401 0 TD ()Tj /F10 1 Tf 0.7656 0 TD [(ln)-24.1(\()]TJ /F3 1 Tf 1.1641 0 TD (R)Tj 7 0 0 7 283.0312 138.9063 Tm (B)Tj 5 0 0 5 288 137.0313 Tm (A)Tj /F10 1 Tf 12 0 0 12 294.4062 142 Tm (\))Tj /F6 1 Tf 0.526 0 TD ()Tj /F10 1 Tf 0.7682 0 TD [(ln)-24.1(\()]TJ /F3 1 Tf 1.1641 0 TD (R)Tj 7 0 0 7 332.7499 138.8125 Tm (C)Tj 5 0 0 5 338.3749 136.9375 Tm (A)Tj /F10 1 Tf 12 0 0 12 344.7812 142 Tm (\))Tj /F6 1 Tf 0.5234 0 TD ()Tj /F10 1 Tf 0.7656 0 TD [(ln)-24.1(\()]TJ /F3 1 Tf 1.1641 0 TD (R)Tj 7 0 0 7 383.0625 138.8125 Tm (C)Tj 5 0 0 5 388.625 136.9688 Tm (B)Tj /F10 1 Tf 12 0 0 12 394.1562 142 Tm (\))Tj /F3 1 Tf 10.487 0 TD (\(13\))Tj ET endstream endobj 49 0 obj << /Type /Page /Parent 110 0 R /Resources 50 0 R /Contents 51 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 50 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 51 0 obj << /Length 2923 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (18)Tj 12 0 0 12 72 710 Tm 0.0085 Tc 0.0846 Tw (This final equation is at the heart of the method. Here we have a condition on the)Tj 0 -2 TD 0.0076 Tc 0.0756 Tw (given component of the map for two of the regions. Other boundary point pairs)Tj T* 0.008 Tc 0.0801 Tw (produce additional equations. In order to have a robust method, one would like)Tj T* 0.0085 Tc 0.0848 Tw (long boundaries to have more weight in the process than short ones, since the latter)Tj T* 0.0064 Tc 0.0643 Tw (may due to a small region consisting entirely of noise. But this is exactly what we)Tj T* 0.0069 Tc 0.0688 Tw (will get if we enter one equation for each boundary pair and solve the resulting)Tj T* 0.0078 Tc 0.0784 Tw (system of equations in the least squares sense. Furthermore, some boundary pairs)Tj T* 0.0059 Tc 0.0594 Tw (can be identified as being more reliable and these are weighted even more by scaling)Tj T* 0.007 Tc 0.07 Tw (the equation by a number greater than one \(typically five\). In addition, some)Tj T* 0.0059 Tc 0.059 Tw (boundary pairs should contribute less, and their equations are scaled by a number)Tj T* 0.0106 Tc 0.1055 Tw (less than unity.)Tj 3 -2 TD 0.0072 Tc 0.0715 Tw (In order to have a solution to the set of equations, it must be insured that all)Tj -3 -2 TD 0.0081 Tc 0.0811 Tw (segments connect to each other through the boundary pairs. This might be)Tj T* 0.006 Tc 0.0597 Tw (accomplished simply by assigning a region to every point, and using each break in)Tj T* 0.0079 Tc 0.0795 Tw (either the horizontal or vertical directions to produce a boundary pair. This is)Tj T* 0.0068 Tc 0.0681 Tw (usually not an option because often some parts of the image should not be used; for)Tj T* 0.0091 Tc 0.0908 Tw (example, when an area is too dark. Therefore the likelihood of connectedness)Tj T* 0.0084 Tc 0.0842 Tw (between regions was increased in the following manner. Boundary pairs were)Tj T* 0.0067 Tc 0.0668 Tw (assigned at each horizontal and vertical change of region. If one of the regions was)Tj T* 0.0055 Tc 0.0554 Tw (to be ignored, a good region was sought in the same direction, taking as many pixels)Tj T* 0.0073 Tc 0.073 Tw (as required. The resulting equation was weighted inversely to the distance taken to)Tj T* 0.0072 Tc 0.0713 Tw (find a good region. Thus such a boundary would contribute little to the solution, but)Tj T* 0.0066 Tc 0.0655 Tw (connectivity was not a problem for reasonable images \(it is possible to construct an)Tj T* 0.0102 Tc 0.1019 Tw (image which will still lack connectivity\).)Tj 3 -2 TD 0.0093 Tc 0.0937 Tw (Several additional steps were taken to improve robustness. First small)Tj -3 -2 TD 0.0063 Tc 0.0634 Tw (regions were excluded from the computations. Second, it was found to be better to)Tj T* 0.0072 Tc 0.0715 Tw (use pixels one unit towards the insides of the respective regions, if these were)Tj ET endstream endobj 52 0 obj << /Type /Page /Parent 110 0 R /Resources 53 0 R /Contents 54 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 53 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 54 0 obj << /Length 2902 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (19)Tj 12 0 0 12 72 710 Tm 0.0075 Tc 0.0748 Tw (available. This way the pixels would tend to have contributions that were solely due)Tj 0 -2 TD 0.0052 Tc 0.0529 Tw [(to a single surface, as opposed to the possibility that they straddled more than one)]TJ T* 0.0046 Tc 0.0458 Tw (surface. These boundary pairs were weighted by a factor of five compared to ones)Tj T* 0.0044 Tc 0.0444 Tw (where it was necessary to use pixels exactly on the boundary.)Tj 3 -2 TD 0.0118 Tc 0.1188 Tw (The final step in determining the relative illumination field is to interpolate)Tj -3 -2 TD 0.0056 Tc 0.0562 Tw (over any excluded areas.)Tj /F2 1 Tf 14 0 0 14 72 542 Tm -0.0169 Tc 0 Tw [(I)149.8(V)-9302.3(The Complete Algorithm)]TJ /F3 1 Tf 12 0 0 12 108 504 Tm 0.0105 Tc 0.1052 Tw (Since the algorithm is fairly complex the implementation details of the)Tj -3 -2 TD 0.0119 Tc 0.1191 Tw (complete algorithm are summarized below.)Tj T* 0.0063 Tc 0.0631 Tw [(1)6.3(.)-743.7(Segment the image by region growing based on chromaticity, RGB, and patch)]TJ 1.5 -1.5 TD 0.0091 Tc 0.0906 Tw (coherence. Ignore regions smaller than some threshold \(20-50 pixels is)Tj T* 0.0058 Tc 0.0575 Tw [(reasonable\). Chose region centers and a base region.)]TJ -1.5 -2 TD 0.0039 Tc 0.0399 Tw [(2.)-746.1(Identify good boundary pairs \(described above\) and use these to form 3 systems of)]TJ 1.5 -1.5 TD 0.0087 Tc 0.0874 Tw [(equations \(one for each channel\) based on equation 13. Solve the equations in)]TJ T* 0.0081 Tc 0.0806 Tw (the least squares sense to get the ratios of the illumination between the region)Tj T* 0.0064 Tc 0.0638 Tw (centers and the base region center.)Tj -1.5 -2 TD 0.0112 Tc 0.1116 Tw [(3)11.2(.)-738.8(The segmentation determines the ratios within regions. Together with the ratios)]TJ 1.5 -1.5 TD 0.01 Tc 0.1004 Tw (from the previous step this gives the illumination field relative to the base)Tj T* 0.008 Tc 0.08 Tw (region. It is important to interpolate carefully the field over the image areas not)Tj T* 0.0053 Tc 0.053 Tw (yet accounted for due to being overly dark, saturated, or belonging to small)Tj T* 0.0207 Tc 0 Tw (regions.)Tj -1.5 -2 TD 0.01 Tc 0.1003 Tw [(4)10(.)-740(Take the convex hull of the set of illuminant chromaticities described in section)]TJ 1.5 -1.5 TD 0.0087 Tc 0.0863 Tw (II.1. Then take the element-wise reciprocal of a sampling of points on the hull)Tj T* 0.0061 Tc 0.0612 Tw (boundary \(we used 50 per edge segment\). Take the convex hull of the result and)Tj T* 0.0092 Tc 0.0915 Tw (scale by the chromaticity of the illuminant. The result is the constraint on the)Tj T* 0.0107 Tc 0.1075 Tw [(diagonal mapping of the chromaticity of a common illuminant to the)]TJ T* 0.0125 Tc 0.1254 Tw (chromaticity of the canonical illuminant \(referred to as the illumination)Tj T* 0.0097 Tc 0.0969 Tw (constraint in the text\).)Tj ET endstream endobj 55 0 obj << /Type /Page /Parent 110 0 R /Resources 56 0 R /Contents 57 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 56 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 118 0 R /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 57 0 obj << /Length 3090 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (20)Tj 12 0 0 12 72 710 Tm 0.0068 Tc 0.0684 Tw [(5)6.8(.)-743.2(Express the ratios found in step 3 as chromaticity ratios. Take the convex hull of)]TJ 1.5 -1.5 TD 0.0062 Tc 0.0618 Tw (the ratios. Scale the set found in step 4 by each of the hull points \(we implicitly)Tj T* 0.0055 Tc 0.0547 Tw (take the reciprocal twice to get the direct scaling\). Each resulting such hull is a)Tj T* 0.012 Tc 0.1197 Tw (constraint on the mappings due to varying illumination.)Tj -1.5 -2 TD 0.009 Tc 0.0902 Tw [(6)9(.)-741(Apply the illumination ratios to the original image. This is an image of the scene)]TJ 1.5 -1.5 TD 0.0103 Tc 0.1022 Tw [(as if it were taken under an unknown but constant illumination. Take the)]TJ T* 0.0078 Tc 0.0782 Tw (convex hull of the chromaticities \(r/b, g/b\) in this image. The convex hull of the)Tj T* 0.0045 Tc 0.0449 Tw (set of all possible \(r/b, g/b\) under the canonical illuminant is scaled by each of the)Tj T* 0.0086 Tc 0.0855 Tw (hull points. Each resulting such hull is a constraint on the mappings due to)Tj T* 0.0134 Tc 0.1343 Tw (image chromaticity.)Tj -1.5 -2 TD 0.0064 Tc 0.0643 Tw [(7)6.4(.)-743.6(Intersect the convex sets found in steps 4, 5, and 6 to obtain the constraint on the)]TJ 1.5 -1.5 TD 0.0116 Tc 0.1162 Tw (mapping of the base point illumination to the canonical illuminant. Chose the)Tj T* 0.01 Tc 0.0997 Tw (centroid of this region to obtain a concrete estimate. Use the illumination-ratio)Tj T* 0.0066 Tc 0.0661 Tw (map to calculate the estimate for rest of the image. Finally, apply the mapping)Tj T* 0.0069 Tc 0.0694 Tw (calculated to the original image to obtain an estimate of the appearance of the)Tj T* 0.015 Tc 0.1497 Tw (scene under the canonical illuminant.)Tj /F2 1 Tf 14 0 0 14 72 374 Tm -0.0167 Tc 0 Tw [(V)-13381.6(Results)]TJ /F3 1 Tf 12 0 0 12 108 336 Tm 0.0058 Tc 0.0584 Tw (The algorithm has been tested on a set of images of real scenes. In all the cases)Tj -3 -2 TD 0.0117 Tc 0.1165 Tw (the unknown illumination consists of light from an incandescent bulb coming)Tj T* 0.0071 Tc 0.0706 Tw (from one direction mixed with light from a fluorescent tube covered by a pale blue)Tj T* 0.0096 Tc 0.0956 Tw (filter coming from another. The latter is similar in colour temperature to sky light.)Tj T* 0.0103 Tc 0.1029 Tw [(Thus the scenes mimic a common real world situationan office with a window.)]TJ 3 -2 TD 0.0085 Tc 0.0849 Tw (Figure 1 shows the image of a three-dimensional Mondrian made by)Tj -3 -2 TD 0.006 Tc 0.0602 Tw (affixing coloured construction paper to a conical waste paper bin. The bin is lying on)Tj T* 0.0085 Tc 0.0849 Tw (its side with the incandescent light shining from the left and the blue light from the)Tj T* 0.0052 Tc 0.0516 Tw (right. The top has blue and green papers on it, in the middle is a grey patch, and near)Tj T* 0.0065 Tc 0.0649 Tw (the bottom are red and yellow papers. Comparing the image of the scene under the)Tj T* 0.0133 Tc 0.1337 Tw (unknown illumination to that under the canonical illumination, one can easily see)Tj ET endstream endobj 58 0 obj << /Type /Page /Parent 111 0 R /Resources 59 0 R /Contents 60 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 59 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 60 0 obj << /Length 2873 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (21)Tj 12 0 0 12 72 710 Tm 0.0069 Tc 0.0693 Tw (the variation from the reddish tinge at the top created by the incandescent light to)Tj 0 -2 TD 0.006 Tc 0.0598 Tw (the bluish tinge at the bottom created by the sky light. A Philips CW fluorescent)Tj T* 0.0174 Tc 0.1743 Tw (provides the canonical illumination.)Tj 3 -2 TD 0.0073 Tc 0.0729 Tw (Figure 1 also shows the colours the algorithm predicts will be seen under the)Tj -3 -2 TD 0.0114 Tc 0.1144 Tw (canonical illumination. Note that the algorithm does not predict the intensity)Tj T* 0.0095 Tc 0.0947 Tw (shading and cannot predict it since the location of the canonical illumination differs)Tj T* 0.01 Tc 0.0998 Tw [(from that of the unknown illumination. The printer is not calibrated so the images)]TJ T* 0.0073 Tc 0.0732 Tw [(gives only rough qualitative result. Quantitative results can be found in Table 2. In)]TJ T* 0.0065 Tc 0.0652 Tw (the recovered versus canonical images, the green, yellow, blue and grey patches are)Tj T* 0.0099 Tc 0.0993 Tw (much closer in colour than they are in the input versus the canonical.)Tj T* 0.1004 Tw (Quantitatively the RMS chromaticity difference between the input and the)Tj T* 0.0077 Tc 0.0769 Tw (canonical images is 0.96 while that between the recovered and canonical images is)Tj T* 0.0058 Tc 0.0573 Tw (only 0.26.)Tj 3 -2 TD 0.0106 Tc 0.1055 Tw (Figure 2 shows some intermediate steps in calculating the illumination field.)Tj -3 -2 TD 0.0077 Tc 0.077 Tw (The reader may notice that the bottom right region is in fact broken into three)Tj T* 0.0107 Tc 0.1072 Tw [(regions and includes many holes where their is insufficient information for robust)]TJ T* 0.0072 Tc 0.0726 Tw [(regions assignment. This is because of the darkness of the region, and the)]TJ T* 0.0093 Tc 0.0928 Tw [(conservative segmentation approach. Although it would be easy to tune the)]TJ T* 0.0943 Tw (segmentation for Mondrian type images, this example illustrates nicely how the)Tj T* 0.0078 Tc 0.078 Tw (next stage of the algorithm finds a good estimate of the illumination field regardless)Tj T* 0.0087 Tc 0.0865 Tw (of the segmentation error. Figure 2\(c\) illustrates the interpolation across areas where)Tj T* 0.0072 Tc 0.0725 Tw [(good regions are not available.)]TJ 3 -2 TD 0.0049 Tc 0.0493 Tw (Figure 3 is an image is of a person in front of a grey background with the)Tj -3 -2 TD 0.0096 Tc 0.0958 Tw (simulated blue sky illuminant on the left and the incandescent light on the right.)Tj T* 0.0069 Tc 0.069 Tw (Under this illumination the left side of the grey background appears quite blue, and)Tj T* 0.0073 Tc 0.0733 Tw (the flesh tones on the left are noticeably incorrect. The shift to the blue on the left is)Tj T* 0.0076 Tc 0.0756 Tw (slightly more noticeable than the shift to the red on the right because the camera)Tj ET endstream endobj 61 0 obj << /Type /Page /Parent 111 0 R /Resources 62 0 R /Contents 63 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 62 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 63 0 obj << /Length 2931 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (22)Tj 12 0 0 12 72 710 Tm 0.0099 Tc 0.0989 Tw (balance was set for indoor lightning. In the recovered image, the illumination)Tj 0 -2 TD 0.0073 Tc 0.0734 Tw [(variation has been removed. It is not possible to obtain a canonical image for)]TJ T* 0.0082 Tc 0.0816 Tw (comparison because people move too much in the time it takes to set up the new)Tj T* 0.0101 Tc 0.1011 Tw (illumination conditions. Figure 4 shows the constraint sets generated for the input)Tj T* 0.0079 Tc 0.0795 Tw (image in Figure 3.)Tj 3 -2 TD 0.0093 Tc 0.0929 Tw [(Table 2 provides numerical results for three other scenes not shown in the)]TJ -3 -2 TD 0.0083 Tc 0.0832 Tw (Figures. One is a simple two-dimensional Mondrian made by attaching eight)Tj T* 0.0062 Tc 0.0626 Tw [(sheets of coloured construction paper to the lab wall such that substantial parts of)]TJ T* 0.0635 Tw (the wall remained visible. The second is of a single piece of green poster board. The)Tj T* 0.0078 Tc 0.0783 Tw (third is a multi-coloured cloth ball. The cloth ball is interesting because the cloth has)Tj T* 0.0106 Tc 0.1056 Tw (more texture than the construction paper.)Tj 3 -2 TD 0.0098 Tc 0.0981 Tw (The numerical results in Table 2 reflect the RMS difference \(over the entire)Tj -3 -2 TD 0.0037 Tc 0.0366 Tw [(image\) between the [r/b, g/b] chromaticities at corresponding pixels in the)]TJ T* 0.0088 Tc 0.0887 Tw (recovered and canonical images. There are few colour constancy algorithms)Tj T* 0.0052 Tc 0.0515 Tw (designed to deal with scenes of the generality addressed here so it is difficult to make)Tj T* 0.0115 Tc 0.1146 Tw (comparisons with existing algorithms without violating their assumptions. As a)Tj T* 0.0058 Tc 0.0579 Tw (first measure, we compare the solution obtained by a straight least squares fit of the)Tj T* 0.0088 Tc 0.087 Tw [(input image to the canonical using a full linear model and then with a diagonal)]TJ T* 0.0142 Tc 0.1418 Tw [(model. Without accounting for the illumination variation no algorithm working)]TJ T* 0.0068 Tc 0.068 Tw (on the 3D Mondrian image can do better than the 0.78 error of the full linear case.)Tj T* 0.0073 Tc 0.0736 Tw (Note that the penalty for using a diagonal model instead of a full linear model is)Tj T* 0.0059 Tc 0.0591 Tw (very slight. leading to an error of 0.80. In contrast, by accounting for and utilizing the)Tj T* 0.0117 Tc 0.1168 Tw (illumination variation our new algorithm reduces the error to 0.26.)Tj 3 -1.8333 TD 0.0064 Tc 0.0638 Tw (Table 2 also shows the chromaticity error for the case of doing nothing at all.)Tj -3 -1.8333 TD 0.0077 Tc 0.0766 Tw (The grey world algorithm, which uses the average image chromaticity as an)Tj T* 0.0119 Tc 0.1188 Tw (illumination estimate, and the Retinex normalization strategy of taking the)Tj T* 0.011 Tc 0.1102 Tw (maximum response from each colour band as an illumination estimate are tried)Tj ET endstream endobj 64 0 obj << /Type /Page /Parent 111 0 R /Resources 65 0 R /Contents 66 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 65 0 obj << /ProcSet [ /PDF /Text ] /Font << /F3 120 0 R >> /ExtGState << /GS1 124 0 R >> >> endobj 66 0 obj << /Length 3132 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (23)Tj 12 0 0 12 72 710 Tm 0.0094 Tc 0.0938 Tw [(even though the comparison is somewhat unfair because they both assume the)]TJ 0 -1.8333 TD 0.0099 Tc 0.0992 Tw [(illumination to be constant. Similar tests are run using surface constraints alone)]TJ T* 0.008 Tc 0.0798 Tw (and surface constraints with the additional constraints on the set of plausible)Tj T* 0.0474 Tc 0 Tw (illuminants.)Tj 3 -1.8333 TD 0.0089 Tc 0.0892 Tw (To make the comparison fairer, we also include similar tests with these)Tj -3 -1.8333 TD 0.0105 Tc 0.1048 Tw (algorithms but applied after the illumination variation has been discounted. In)Tj T* 0.0082 Tc 0.0815 Tw (other words, we combined the first part of our algorithm \(removal of the)Tj T* 0.0112 Tc 0.1124 Tw [(illumination variation\) with each of the other algorithms. In this case, the other)]TJ T* 0.0101 Tc 0.1012 Tw (algorithms are applied to data which does not violate the constant illumination)Tj T* 0.0095 Tc 0.0947 Tw [(assumption, but they still do not exploit the information contained in the)]TJ T* 0.013 Tc 0.1298 Tw (illumination variation. The Retinex normalization applied in this way gives an)Tj T* 0.009 Tc 0.0899 Tw (algorithm which is close, in theory, to the original Retinex idea.)Tj 3 -1.8333 TD 0.0094 Tc 0.0941 Tw (The results show first that if the varying illumination is not accounted for,)Tj -3 -1.8333 TD 0.0084 Tc 0.0838 Tw (then all the colour constancy algorithms perform poorly. In all cases, the complete)Tj T* 0.0103 Tc 0.1034 Tw (new VIR-SIV algorithm did better than any algorithm which assumed the)Tj T* 0.0086 Tc 0.0856 Tw (chromaticity of the illumination to be constant. In fact, the performance is better)Tj T* 0.0073 Tc 0.0735 Tw (than that of the best diagonal and best linear fits. The complete algorithm also)Tj T* 0.0093 Tc 0.0923 Tw [(performed better than the others applied to the data with the varying illumination)]TJ T* 0.0077 Tc 0.0774 Tw [(removed, except when compared to applying the combination of surface and)]TJ T* 0.0119 Tc 0.1195 Tw (illumination constraints to the ball image. Most importantly, the algorithm)Tj T* 0.0081 Tc 0.081 Tw (performs better than the Retinex scaling applied to the data with the variation)Tj T* 0.0085 Tc 0.0853 Tw (removed. As mentioned above, this procedure is close to the spirit of the original)Tj T* 0.0116 Tc 0.1156 Tw (Retinex algorithm, which is unique as an alternative to our algorithm, even though)Tj T* 0.0107 Tc 0.1078 Tw [(its testing has been limited to scenes with more controlled illumination.)]TJ 3 -1.8333 TD 0.0072 Tc 0.0722 Tw [(The results for the green card are included to illustrate that the varying)]TJ -3 -1.8333 TD 0.0083 Tc 0.0828 Tw [(illumination constraint can be very useful in the case when there is a paucity of)]TJ T* 0.0095 Tc 0.095 Tw (other information. Most colour constancy algorithms require a good selection of)Tj T* 0.0068 Tc 0.0681 Tw (surfaces for reliable performance. The ability of this algorithm to go beyond that in)Tj T* 0.0107 Tc 0.1065 Tw (the case of varying illumination is encouraging.)Tj ET endstream endobj 67 0 obj << /Type /Page /Parent 111 0 R /Resources 68 0 R /Contents 69 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 68 0 obj << /ProcSet [ /PDF /Text /ImageC /ImageI ] /Font << /F3 120 0 R >> /XObject << /Im1 70 0 R /Im2 71 0 R /Im3 72 0 R /Im4 73 0 R /Im5 74 0 R /Im6 75 0 R /Im7 76 0 R /Im8 77 0 R >> /ExtGState << /GS1 124 0 R >> /ColorSpace << /Cs9 102 0 R >> >> endobj 69 0 obj << /Length 9138 >> stream BT /F3 1 Tf 10 0 0 10 566 747 Tm 0 g /GS1 gs 0 Tc 0 Tw (24)Tj ET q 143 0 0 176 72 544 cm /Im1 Do Q q 1 i 234 544 143 176 re W n q 143 0 0 176 234 544 cm /Im2 Do Q Q q 1 i 396 544 143 176 re W n q 143 0 0 176 396 544 cm /Im3 Do Q Q BT 10 0 0 10 72 532 Tm -0.0374 Tc [(\()-154.6(a)-154.6(\))-737( Scene under unknown)]TJ 2.1 -1.2 TD 0.0296 Tc (illumination)Tj 14.1 1.2 TD -0.0078 Tc [(\()-48.5(b)-48.5(\))-807.4(Scene under canonical)]TJ 2.1 -1.2 TD 0.0291 Tc (illuminant)Tj 14.1 1.2 TD -0.0043 Tc [(\()-49.6(c)-49.6(\))-903.7(Scene after colour constancy)]TJ 2.1 -1.2 TD -0.0156 Tc (processing)Tj -34.5 -1.6 TD 0.0034 Tc 0.0424 Tw [(Figure )49(1: )-753(Mondrian on conical waste basket illuminated \(a\) from the left by incandescent light and)]TJ 4.9 -1.2 TD 0.006 Tc 0.0603 Tw (from the right by simulated blue sky, \(b\) straight on with cool white fluorescent light \(the)Tj T* 0.0048 Tc 0.0485 Tw (canonical illuminant\), and \(c\) estimate of the appearance under an illuminant of the same)Tj T* 0.0067 Tc 0.0668 Tw (colour as the canonical illuminant but with the shading of the original illumination.)Tj ET 1 i 70 463 472 -1 re f q 86 306 117 150 re W n q 118 0 0 150 86 306 cm /Im4 Do Q Q 0.5 1 0 rg 110.99 351.495 m 110.99 350.115 110.094 348.995 108.99 348.995 c 107.886 348.995 106.99 350.115 106.99 351.495 c 106.99 352.875 107.886 353.995 108.99 353.995 c 110.094 353.995 110.99 352.875 110.99 351.495 c f 0.5 1 0 RG 0 J 0 j 1 w 10 M []0 d 110.505 351.495 m 110.505 350.391 109.833 349.495 109.005 349.495 c 108.177 349.495 107.505 350.391 107.505 351.495 c 107.505 352.599 108.177 353.495 109.005 353.495 c 109.833 353.495 110.505 352.599 110.505 351.495 c s 185.99 399.495 m 185.99 398.115 185.094 396.995 183.99 396.995 c 182.886 396.995 181.99 398.115 181.99 399.495 c 181.99 400.875 182.886 401.995 183.99 401.995 c 185.094 401.995 185.99 400.875 185.99 399.495 c f 185.505 399.495 m 185.505 398.391 184.833 397.495 184.005 397.495 c 183.177 397.495 182.505 398.391 182.505 399.495 c 182.505 400.599 183.177 401.495 184.005 401.495 c 184.833 401.495 185.505 400.599 185.505 399.495 c s 172.01 318.495 m 172.01 317.115 171.114 315.995 170.01 315.995 c 168.906 315.995 168.01 317.115 168.01 318.495 c 168.01 319.875 168.906 320.995 170.01 320.995 c 171.114 320.995 172.01 319.875 172.01 318.495 c f 171.495 318.495 m 171.495 317.391 170.823 316.495 169.995 316.495 c 169.167 316.495 168.495 317.391 168.495 318.495 c 168.495 319.599 169.167 320.495 169.995 320.495 c 170.823 320.495 171.495 319.599 171.495 318.495 c s 145.99 413.505 m 145.99 412.125 144.87 411.005 143.49 411.005 c 142.11 411.005 140.99 412.125 140.99 413.505 c 140.99 414.885 142.11 416.005 143.49 416.005 c 144.87 416.005 145.99 414.885 145.99 413.505 c f 145.49 413.505 m 145.49 412.401 144.594 411.505 143.49 411.505 c 142.386 411.505 141.49 412.401 141.49 413.505 c 141.49 414.609 142.386 415.505 143.49 415.505 c 144.594 415.505 145.49 414.609 145.49 413.505 c s 112.01 402.495 m 112.01 401.115 111.114 399.995 110.01 399.995 c 108.906 399.995 108.01 401.115 108.01 402.495 c 108.01 403.875 108.906 404.995 110.01 404.995 c 111.114 404.995 112.01 403.875 112.01 402.495 c f 111.495 402.495 m 111.495 401.391 110.823 400.495 109.995 400.495 c 109.167 400.495 108.495 401.391 108.495 402.495 c 108.495 403.599 109.167 404.495 109.995 404.495 c 110.823 404.495 111.495 403.599 111.495 402.495 c s 170 338.505 m 170 337.125 169.104 336.005 168 336.005 c 166.896 336.005 166 337.125 166 338.505 c 166 339.885 166.896 341.005 168 341.005 c 169.104 341.005 170 339.885 170 338.505 c f 169.5 338.505 m 169.5 337.401 168.828 336.505 168 336.505 c 167.172 336.505 166.5 337.401 166.5 338.505 c 166.5 339.609 167.172 340.505 168 340.505 c 168.828 340.505 169.5 339.609 169.5 338.505 c s 139 369.495 m 139 368.115 137.88 366.995 136.5 366.995 c 135.12 366.995 134 368.115 134 369.495 c 134 370.875 135.12 371.995 136.5 371.995 c 137.88 371.995 139 370.875 139 369.495 c f 138.5 369.495 m 138.5 368.391 137.604 367.495 136.5 367.495 c 135.396 367.495 134.5 368.391 134.5 369.495 c 134.5 370.599 135.396 371.495 136.5 371.495 c 137.604 371.495 138.5 370.599 138.5 369.495 c s 131.99 330.495 m 131.99 329.115 131.094 327.995 129.99 327.995 c 128.886 327.995 127.99 329.115 127.99 330.495 c 127.99 331.875 128.886 332.995 129.99 332.995 c 131.094 332.995 131.99 331.875 131.99 330.495 c f 131.505 330.495 m 131.505 329.391 130.833 328.495 130.005 328.495 c 129.177 328.495 128.505 329.391 128.505 330.495 c 128.505 331.599 129.177 332.495 130.005 332.495 c 130.833 332.495 131.505 331.599 131.505 330.495 c s q 248 306 123 148 re W n q 123 0 0 148 248 306 cm /Im5 Do Q Q 277 349.5 m 277 348.12 275.88 347 274.5 347 c 273.12 347 272 348.12 272 349.5 c 272 350.88 273.12 352 274.5 352 c 275.88 352 277 350.88 277 349.5 c f 276.5 349.5 m 276.5 348.396 275.604 347.5 274.5 347.5 c 273.396 347.5 272.5 348.396 272.5 349.5 c 272.5 350.604 273.396 351.5 274.5 351.5 c 275.604 351.5 276.5 350.604 276.5 349.5 c s 352 397.5 m 352 396.12 350.88 395 349.5 395 c 348.12 395 347 396.12 347 397.5 c 347 398.88 348.12 400 349.5 400 c 350.88 400 352 398.88 352 397.5 c f 351.5 397.5 m 351.5 396.396 350.604 395.5 349.5 395.5 c 348.396 395.5 347.5 396.396 347.5 397.5 c 347.5 398.604 348.396 399.5 349.5 399.5 c 350.604 399.5 351.5 398.604 351.5 397.5 c s 337.99 316.5 m 337.99 315.12 336.87 314 335.49 314 c 334.11 314 332.99 315.12 332.99 316.5 c 332.99 317.88 334.11 319 335.49 319 c 336.87 319 337.99 317.88 337.99 316.5 c f 337.49 316.5 m 337.49 315.396 336.594 314.5 335.49 314.5 c 334.386 314.5 333.49 315.396 333.49 316.5 c 333.49 317.604 334.386 318.5 335.49 318.5 c 336.594 318.5 337.49 317.604 337.49 316.5 c s 311 411.495 m 311 410.115 310.104 408.995 309 408.995 c 307.896 408.995 307 410.115 307 411.495 c 307 412.875 307.896 413.995 309 413.995 c 310.104 413.995 311 412.875 311 411.495 c f 310.5 411.495 m 310.5 410.391 309.828 409.495 309 409.495 c 308.172 409.495 307.5 410.391 307.5 411.495 c 307.5 412.599 308.172 413.495 309 413.495 c 309.828 413.495 310.5 412.599 310.5 411.495 c s 277.99 400.5 m 277.99 399.12 276.87 398 275.49 398 c 274.11 398 272.99 399.12 272.99 400.5 c 272.99 401.88 274.11 403 275.49 403 c 276.87 403 277.99 401.88 277.99 400.5 c f 277.49 400.5 m 277.49 399.396 276.594 398.5 275.49 398.5 c 274.386 398.5 273.49 399.396 273.49 400.5 c 273.49 401.604 274.386 402.5 275.49 402.5 c 276.594 402.5 277.49 401.604 277.49 400.5 c s 335.995 336.495 m 335.995 335.115 334.875 333.995 333.495 333.995 c 332.115 333.995 330.995 335.115 330.995 336.495 c 330.995 337.875 332.115 338.995 333.495 338.995 c 334.875 338.995 335.995 337.875 335.995 336.495 c f 335.495 336.495 m 335.495 335.391 334.599 334.495 333.495 334.495 c 332.391 334.495 331.495 335.391 331.495 336.495 c 331.495 337.599 332.391 338.495 333.495 338.495 c 334.599 338.495 335.495 337.599 335.495 336.495 c s 303.995 367.5 m 303.995 366.12 303.099 365 301.995 365 c 300.891 365 299.995 366.12 299.995 367.5 c 299.995 368.88 300.891 370 301.995 370 c 303.099 370 303.995 368.88 303.995 367.5 c f 303.495 367.5 m 303.495 366.396 302.823 365.5 301.995 365.5 c 301.167 365.5 300.495 366.396 300.495 367.5 c 300.495 368.604 301.167 369.5 301.995 369.5 c 302.823 369.5 303.495 368.604 303.495 367.5 c s 298 328.5 m 298 327.12 296.88 326 295.5 326 c 294.12 326 293 327.12 293 328.5 c 293 329.88 294.12 331 295.5 331 c 296.88 331 298 329.88 298 328.5 c f 297.5 328.5 m 297.5 327.396 296.604 326.5 295.5 326.5 c 294.396 326.5 293.5 327.396 293.5 328.5 c 293.5 329.604 294.396 330.5 295.5 330.5 c 296.604 330.5 297.5 329.604 297.5 328.5 c s q 405 306 120 149 re W n q 120 0 0 149 405 306 cm /Im6 Do Q Q BT 10 0 0 10 72 294 Tm 0 g 0.0028 Tc 0.0276 Tw [(\()-114.4(a)-114.4(\))-396.8(Segmentation with region centers)]TJ 1.8 -1.2 TD -0.0045 Tc 0 Tw (shown as green dots and pixels)Tj T* -0.0112 Tc (used as bottom or right member of)Tj T* -0.002 Tc (a boundary pair shown in red.)Tj 15.8 3.6 TD 0.0173 Tc 0.1732 Tw [(\()-23.4(b)-23.4(\))-482.3(Intermediate illumination)]TJ 1.8 -1.2 TD 0.0095 Tc 0.0947 Tw (field. The image is black)Tj T* 0.0948 Tw (where there is insufficient)Tj T* 0.0012 Tc 0.0116 Tw (information to assign a region.)Tj 14 3.6 TD 0.009 Tc 0.0898 Tw [(\()-36.3(c)-36.3(\))-990.4(Representation of the)]TJ 2.2 -1.2 TD 0.0188 Tc 0.1878 Tw (illumination field with)Tj T* 0.0121 Tc 0.1209 Tw (interpolation to fill in the)Tj T* 0.002 Tc 0.0201 Tw (unassigned areas.)Tj -35.6 -1.6 TD 0.0011 Tc 0.0194 Tw [(Figure )23.7(2: )-778.3(Intermediate processing steps for the image in Figure 1.)]TJ ET 70 237 472 -1 re f q 108 127 160 104 re W n q 160 0 0 104 108 127 cm /Im7 Do Q Q q 333 127 160 104 re W n q 160 0 0 104 333 127 cm /Im8 Do Q Q BT 10 0 0 10 108 115 Tm -0.0166 Tc 0 Tw [(\()-133.8(a)-133.8(\))-716.2( Scene under unknown illuminant)-5910.6(\()-57.3(b)-57.3(\))]TJ 24.8 0 TD 0.0005 Tc 0.0048 Tw (Scene after processing)Tj -28.4 -1.6 TD 0.0025 Tc 0.0321 Tw [(Figure )37.8(3: )-764.2(Person \(a\) illuminated from the left by simulated blue sky and on the right incandescent)]TJ 4.9 -1.2 TD 0.0008 Tc 0.0079 Tw (light, and \(b\) after colour constancy processing .)Tj ET endstream endobj 70 0 obj << /Type /XObject /Subtype /Image /Width 143 /Height 176 /BitsPerComponent 8 /ColorSpace /DeviceRGB /Interpolate true /Length 10488 /Filter /DCTDecode >> stream Adobe d C s !1AQa"q2B#R3b$r%C4Scs5D'6Tdt& EFVU(eufv7GWgw8HXhx)9IYiy*:JZjz ? zg- x|x/k DcLS]lW5?2elԦjf/,P0/Lԯlٸfo(2ͷWO>f\\٩_LO9T0#1iLeg'131Dg`' b+o^`˩㗸}~꺳zemą|۠N-omuvJ@>gv9MB>mJO] r-SY}G?70+q[o~ih }^Oi8 Λ/2#YMTaNˆpyB_bzD]9#pWfH@Cᓻ/4^ԯ