W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 1998 Improvements to the color quantization process Rance David Necaise College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Computer Sciences Commons Recommended Citation Necaise, Rance David, "Improvements to the color quantization process" (1998). Dissertations, Theses, and Masters Projects. Paper 1539623933. https://dx.doi.org/doi:10.21220/s2-c72h-7024 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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IMPROVEMENTS TO THE COLOR QUANTIZATION PROCESS A Dissertation Presented to The Faculty of the Department of Computer Science The College of William & Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Ranee David Necaise 1998 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DM1 Number: 9920304 C o p y r ig h t 1999 b y N e c a is e, Ranee David All rights reserved. UMI Microform 9920304 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Ranee D. Necaise Approved, August 1998 Richard Prosl Thesis Advisor * * Stephen Park y l ( J —' ftX-it&' Weizhen Mao William Bynum' </Sidney Lawrence Department of Mathematics Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my parents . in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C ontents Acknowledgments ix List of Figures xiii Abstract xiv 1 Introduction 2 1.1 Background .......................................................................................................... 3 1.2 Quantization Process ........................................................................................... 4 1.3 Optimal Solution ................................................................................................. 5 1.4 Iterative Quantization Methods .......................................................................... 6 1.5 Color Image Quantization .................................................................................. 7 1.6 Motivation .......................................................................................................... 8 1.7 Thesis Overview ................................................................................................. 9 2 Quantitative Image Difference 11 2.1 Introduction .......................................................................................................... 11 2.2 Previous W ork .................................................................................................... 12 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.1 Metrics for Gray-Level Images ............................................................... 12 2.2.2 Metrics for Color Images........................................................................... 16 2.3 Problems with the Root-Mean-Square E rror ..................................................... 18 2.4 The Convolution Root-Mean-Square E rro r......................................................... 19 2.4.1 The Construction of the M etric ............................................................... 21 2.4.2 M o tiv a tio n ................................................................................................. 22 3 Heuristic Color Quantization Algorithms 24 3.1 Introduction ............................................................................................................. 24 3.1.1 Problem Formulation .............................................................................. 25 3.1.2 Heuristic Quantizer D esign .................................................................... 26 3.2 Previous W o r k ....................................................................................................... 28 3.2.1 Fixed Quantization Algorithms .............................................................. 28 3.2.2 Adaptive Quantization Algorithms ........................................................ 29 3.2.2.1 P o p u la rity .................................................................................. 30 3.2.2.2 Median Cut ............................................................................... 30 3.2.2.3 Center C u t .................................................................................. 32 3.2.2.4 Octree Quantization ................................................................... 33 3.2.2.5 Other A lgorithm s...................................................................... 35 3.3 The Color Cut Quantization Algorithm ............................................................ 36 3.3.1 Building The Histogram........................................................................... 38 3.3.2 Sorting and Splitting the L ist .................................................................. 39 3.3.3 Constructing the Colormap and Partition ............................................ 41 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.4 Color R e d a c tio n ........................................................................................ 43 3.4 Algorithm Analysis................................................................................................. 44 3.4.1 Worst Case A nalysis .................................................................................. 44 3.4.2 Execution A nalysis ..................................................................................... 48 3.5 Algorithm Comparisons........................................................................................ 49 3.5.1 The Color Cut A lg o rith m........................................................................ 49 3.5.2 Color Cut with Centroid M ap p in g ......................................................... 52 3.5.3 Color Cut with Color R e d u c tio n ............................................................ 53 4 Partitioning by Nearest-Neighbor 58 4.1 Introduction .............................................................................................................. 58 4.2 Locally Sorted S e a rc h ........................................................................................... 59 4.3 Adaptive Locally Sorted S earch ........................................................................... 60 4.3.1 Construction of the k-d Tree ................................................................... 62 4.3.2 Nearest-Neighbor Search L i s t................................................................... 64 4.3.3 Nearest-Neighbor Search............................................................................ 68 4.4 Algorithm Analysis.................................................................................................. 69 4.4.1 Worst Case A nalysis .................................................................................. 70 4.4.2 Empirical A nalysis .....................................................................................