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Sf!C subsequent amendments. 1970, &C-30, el ses amendements subsbquents Chromatic Aberration: A New Tool for Colour Constancy Jian Ho &Sc, Simon easer University, 1986 4 TI (ESIS SL'B\lI'J"TED I\ PARTIAL FULFII.LMENT OF THt REQL'IKEILIENTS FOR THE DEGREE OF Master of Science in the School 0 f Computing Science @ Jian Ilo 1988 SI\IOT FRXSER CNIVERSITY 4pn1 1988 411 rights reserved. This thesis may not be repr~ducedin b.hole or in part. by photocopy or ether means. krithout the permission of the author Permission has been granted L'autorisation a Qtb-accordhe to the National Library of h la ~ibliothbque' nationale Canada to microfilm this du Canada de microfilmer thesis and to lend or sell cette thhe et de pr8ter ou copies of the film. de vendre des exemplairee du film. The author (copyright owner) L'auteur (titulaire du droit has reserved other d'auteur) se rhserve lea publicatisn rights, and autres droits de pub1,ication; neither the thesis nor ni la th&ee ni de longs extensive extracts from it extrait.8 de celle-ci ne may be printed or otherwise doivent 6tre imprimbs ou reproduced without his/her autrement reprodu'its sans son written permission. autorisation &rite. ISBN 0-315-48753-4 Name: Jian Ho Degree' Master of Science 'I'itle of Thesis: Chromatic Aberration: A New Tool for Color Constancy Chairman: B;K. Bhattacharya . A - Hrlan V. Funt. Senior Supervisor Thomas W. Calvert.. Supervisory Committee Member Leigh H.%lmer. Supervisory Committee Member z~-N&. Li, External Committee Member -- Date of Approval PART IAL COPYR l GHT L I CENSE I hereby grant to Simn. Fraser Universlty She rlght to lend my thesis, proJect or exterlded essay (the title of which is shawn below) to users of the Simon Fraser Universlty Library, and to make partial or single copies only for such users or In response to a request from the library of any other university, or other educations; .Institution, on its own beha ?f or for one of Its users. I further agree that perrni ss ion for multiple copying of1 this work f r scholarly purposes may be gr2rnted by me or the Dean of Graduate Studies.S It is understood that copying or publication of this work for flnanclal gain shall not be allowed without my written permlsslon. Title of ThesislProject/Extended Essay Author: /J(signature) (date) Abstract Maloney proposed solving the colour constancy problem by using a fi&e-dimensional linear model. Zlis method depends on the number of sensor classes being larger than the number of basis functions needed to model the surface spectral reflectance. Since the surface spectral reflectances of most natural objects require at least three basis functions for accurate modeling. xcording to Maloney's paper we must have four sensor classes in order to achieve colour constancy. However, most experiments suggest that the human visual system is tri-chromatic. That is. there are only three classes of sensors in the human visual system. As a result, we must look for other spectral information to help us achieve colour constancy. We claim that chrom&c aberration can help here. , . f'rom the chromatic aberration which occurs at the edge between two coloured regions under unknown illumination. we derive the difference of the spectral power distributions of the lights reflected from these regions. Using finite-dimensional linear models of illumination and surface spectral reflectances as our basic assumption, we obtain a set of equations for the coefficients that describe the illumination and the surface spectral reflectance. In combination with the equations obtailred from the sensors inside each coloured reglon. we determine the surface spectral reflectances. b hence colours. of the regions and the spectral power distribution of the illumination. ~f the number of sensor classes used is not smaller than that of the dimensions of I he f inite-dimensional linear model for surface spectral reflectance. Since usiq degree three in the linear model approximates most of surfac; spectral reflectances very well, we only need to use three classes of sensors and chromatic aberration to recover the surface spectral reflectances. Without using chromatic aberration, Maloney had to use at least four sensors. ~histh; information provided by chromatic aberration is very valuable. Acknowledgements 1 would like to extend my greatest appreciatio"\ to Dr. Brian V. Funt for originally suggesting the topic of chromatic aberration, and is guidiance and encouragement throughout my thesis work. Many thanks also Dr. Leigh H. Palmer for his consultation and expertise in optics. In would like to thank Dr. Thomas W. Calvert, Dr. Ze-Nian Li, Mark Drew Tong for their helpful discussions and advice. This work is supported by Graduate Scholarship. Table of Contents Approval Abstract Acknowledgements Table of Con tents 1. Introduction 2. Previous Work 3. Motivation 4. Chromatic Aberration 4.1. Assumptions 1 4.2.. Background 4.2.1. Light Refraction 4.2.2. Index of Refraction 4.2.3. Thin Lens 4.2.4. Relationship between Wavelength and Focus 4.2.5. Diffraction 4.2.6. Spread Furiction 4.2.7. Cut-off Point for the Point Spread Function 4.3. Spectrum Information from Chromatic AberraLion 4.3.1. Point Source in One Dimension 4.3.2. Edges in One - ~imension 4.3.3. Edges In Two Dim~nsions 4.4. Discussion + 4.5. Depth from Chromatic Aberration 4.6. Implementation and Results 5. Colour Constancy 5.1. Assumptions 5.2. Finite-dimensional 31odels for Illumination and Surface Reflectance 5.3. Models of Illumination and Surface Spectral Reflectance 5.4. Finding the Colour of a Surface 5.4.1. Best Case 5.4.2. Worst Case 5.4.3. -4lgqrithm 5.4.4. Feasibility of the Solutions - 5.4.5. Implemontat ion 5.5. Implications Q 6. Separation of' Illumination and Surface Reflectance from Incoming & Light Spectrum 5 3 6.1. Exact models 6.2. Approximated Models 6.3. Theorems 6.4. Ihscussion 6.5. Implementation 6.6. Results 7. Physics and Anatomy of' the Human Eye 8. Conclusion Rcf crenccs List of Figures Figure 4-1: - Refractive effeA on a light ray passing from one ,medium into another Figure 4-2: Chromatic aberration effect of w-hite light passing through a prism Figure 4-3: Approximation of the refractive index of glass vs. wavelength Figure 4-4: Cro'ss-section of a circular double convex thin lens Figure 4-5: Circular double convex thin lens with surface radii K1 and K,.. Figure 4-6: This figure shows the case where the image of a point is in front of the image plane Figure 4-7: This figure shows the case where the image of a point is behind the image plane Figure 4-8: For uniform incoming flux, the flux arriving at the image plane will also be uniform Figure 4-9: This figure shows that at a long distance a spread function is too small to be measured Feure 4-10: Enlarged chromatic aberration effect of a thin lens Figure 4-1 1: Pure chromatic aberration effect of a point source Figure 4- 12: The image forming of an off-optical axis point. Figure 4-13: Formation of the image in the two-dimensiondl case Figure 4-14: Chromatic aberration in the two-dimensional case. The shaded areas are the areas of the An's. Figure 4-15: Figure for finding the depth of a coloured point or edge from its image of chromatic aberration Variation of the wavelength in focus on the retina as a function of accommodative stimulus. The thin line represents the mean results obtained by lvanoff (1953). while the bold line represents the results of Millodot and Sivak (1973). 7'hi.t figure is reproduced from (Sivak 781. Figure 4- 17: Intensity profile of the chromatic aberration effect of an edge Figure 6-1: Spectrum of 100UO•‹K daylight illuminating heather Figure 6-2: Reflectance curve of heather and the fitted curve. from our program. The reflectance has been magnified four times. Figure 6-3: Result of the illumination from our program and the actual 1000O•‹K daylight Chapter- Introduction Colour is becoming more and more useful -in computational vision today. Just as d black and white t6evision sets are gradually abandoned, we are moving from grey level computational vision to computational colour vision. Colour plays an important role in computational vision.