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Forty years of color quantization: a modern, algorithmic survey

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Abstract

Color quantization (cq), the reduction of the number of distinct colors in a given image with minimal distortion, is a common image processing operation with various applications in computer graphics, image processing/analysis, and computer vision. The first cq algorithm, median-cut, was proposed over 40 years ago. Since then, many clustering algorithms have been applied to the cq problem. In this paper, we present a comprehensive overview of the cq algorithms proposed in the literature. We first examine various aspects of cq, including the number of distinguishable colors, cq artifacts, types of cq, applications of cq, data structures, data reduction, color spaces and color difference equations, and color image fidelity assessment. We then provide an overview of image-independent cq algorithms, followed by a detailed survey of image-dependent ones. After presenting a brief discussion of pixel mapping, we conclude our survey with an outline of the open problems in cq.

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Notes

  1. Also known as color reduction (Bragg 1992; Kruger 1992).

  2. Also known as color-mapped, color-quantized, or palletized.

  3. Available at https://sipi.usc.edu/database/database.php?volume=misc.

  4. Also known as a color map or color (lookup) table.

  5. These visualizations were rendered using Color Space 1.1.1 (by Philippe Colantoni).

  6. Available at http://r0k.us/graphics/kodak/.

  7. Buades et al. (2011) demonstrate that the distribution of colors in most natural images can be modeled accurately by a 2d manifold rather than a 1d curve. In other words, reducing the color space dimensionality (D) from three to one often leads to a severe loss of information.

  8. For another application of space-filling curves in cq, see Subsection 1.7.

  9. The popularity of a cq algorithm appears to correlate well with the availability of its open-source implementations. However, practitioners should be aware of the fact that open-source implementations of popular cq algorithms such as median-cut (Heckbert 1982) and octree (Gervautz and Purgathofer 1988) have varying degrees of quality and faithfulness to the original algorithms.

  10. Heckbert (1980) was the first to use hashing in cq.

  11. Scanning an image of size \(N_p\) pixels in raster order with a step size of P yields a sample of size \(\left\lceil N_p / P \right\rceil\) pixels.

  12. In this paper, we assume that all color image data is encoded in the standard rgb (srgb) space (Anderson et al. 1996), which was standardized by the International Electrotechnical Commission (iec) in 1999. This means that the red, green, and blue components of the images are nonlinearly coded (i.e., gamma corrected). It is customary to denote such nonlinear components with primes (Poynton and Funt 2014) (i.e., \(r'\), \(g'\), and \(b'\) as opposed to r, g, and b, respectively). However, we omit the primes throughout the paper to avoid clutter.

  13. There is evidence that a ucs is not Euclidean, at least not in three dimensions (Urban et al. 2007).

  14. In 1976, the International Commission on Illumination (cie) recommended two approximately uniform color spaces, namely cielab and cieluv. Nearly half a century later, these spaces are still the cie recommendations, although cieluv has fallen out of favor (Fairchild and Johnson 2004). In fact, since its standardization, colorimetric research has mostly revolved around the cielab space and its cd equation (Luo 2002).

  15. Exceptions include a few cq studies (Sudha et al. 2003; Lo et al. 2003; Yu and Lo 2003; Nolle and Schaefer 2007; Schaefer 2014; Schaefer and Nolle 2015) that employ the s-cielab metric.

  16. There are, however, accelerated color space transformations with negligible loss of accuracy (Celebi et al. 2010).

  17. For example, cielab components are often represented in floating-point format to avoid loss of precision. Such a data representation, however, leads to slower computations than an integer representation.

  18. Ortiz-Jaramillo et al. (2019) compute two separate mse metrics in the cielab space, one using the cie76 equation and the other using the ciede2000 one. By contrast, Ramella (2021) computes psnr, and by extension mse, in the rgb space.

  19. wsnr is computed in the Fourier domain after filtering the two images with approximations of the human contrast sensitivity functions (Mannos and Sakrison 1974; Mitsa and Varkur 1993).

  20. msssim is a variant of ssim computed over a range of scales (Wang et al. 2003).

  21. vsnr is an hvs-based fidelity metric computed in the wavelet domain (Chandler and Hemami 2007).

  22. All three metrics are originally defined for grayscale images.

  23. By contrast, a rating experiment requires only \(N_i\) trials per observer (Mantiuk et al. 2012).

  24. There are also software for conducting pairwise comparison experiments (Vuong et al. 2018).

  25. This convention is adopted for computational simplicity. In the sq literature, it is more common to use the interval’s midpoint so as to minimize mse (Gersho and Gray 1992, p. 151).

  26. This scheme is inspired by the itu-r bt.601 luminance equation: \(y = 0.299 r + 0.587 g + 0.114 b\).

  27. In an asymptotically optimal quantization of a uniform 3d distribution, each quantization cell has the shape of a truncated octahedron rather than a cube (Barnes and Sloane 1983).

  28. This could be a \(4096 \times 4096\) rgb image with \(2^{24}\) distinct colors or the same image uniformly quantized using bit-cutting.

  29. We can generalize divisive algorithms to generate trees with greater branching factors than two (Makhoul et al. 1985) or even with variable branching factors (Schmidl et al. 1993).

  30. As Wu (1992a) notes, in a search tree, keeping the tree balanced is necessary to achieve logarithmic query time in the worst-case. However, in the palette design phase, distortion minimization is more important than achieving a balanced palette, while in the pixel mapping phase, we are concerned with amortized, rather than worst-case, time complexity.

  31. In the first iteration, we split the only cluster that contains the entire data set.

  32. For a formal definition of sse, see Eq. (12) in Subsection 5.3. Some cq researchers use the terms variance and sse synonymously even though, strictly speaking, the variance of a given cluster equals its mean squared error, that is, its sum of squared errors divided by its size.

  33. Other thresholding algorithms (Yang and Tsai 1998) can be used as well.

  34. A more elaborate approach would be to adjust all t (\(t \in \{1, \dotsc , K - 1\}\)) splitting planes simultaneously after split t using \((t + 1)\)-means clustering (Howard and Harris 1966).

  35. If the radius-weighted mean and centroid coincide, the splitting plane is taken orthogonal to the color axis with the greatest variance.

  36. Global optimality can be guaranteed only in the case of single linkage clustering. However, despite its theoretically appealing properties (Fisher and Van Ness 1971; Van Ness 1973; Ackerman et al. 2010; Carlsson and Memoli 2010), single linkage is generally not preferred due to its tendency to generate elongated clusters.

  37. For a formal definition of K-partition, see Eq. (7) in Sect. 5.

  38. The author reports that a sample of size \(S = 1,024\) pixels suffices for a \(512 \times 512\) input image.

  39. The radius r of \({\mathcal {X}}\) is given by \(r = \sqrt{\sum _{i = 1}^N \left\Vert {\textbf{x}}_i - {\bar{{\textbf{x}}}} \right\Vert ^2/N}\), where \({\bar{{\textbf{x}}}} = {\textbf{s}}/N\) is the centroid of \({\mathcal {X}}\). Given \({\textsc {cf}}({\mathcal {X}})\), the radius can be computed as \(r = \sqrt{ss / N - \left\Vert {\bar{{\textbf{x}}}} \right\Vert ^2}\).

  40. The tree building phase requires only \({\mathcal {O}}(N)\) time.

  41. Less commonly known as the populosity algorithm (Velho et al. 1997).

  42. In the context of divisive cq algorithms, the primary purpose of bit-cutting is to reduce time and memory requirements. For the popularity algorithm and its variants, however, bit-cutting is necessary to detect the dominant peaks of the color histogram effectively.

  43. Courtesy of Larry Ewing (lewing@isc.tamu.edu) and the gimp

  44. For color image data, one option is the most frequent color (Houle and Dubois 1986).

  45. For conciseness, we omit the handling of exceptional cases such as data points equidistant to multiple centers.

  46. For faster implementations of maximin, refer to Geraci et al. (2006), Zhao et al. (2013), and Borgelt and Yarikova (2020).

  47. Yuan and Goldberg (1988) proposed a similar algorithm in the context of vq.

  48. In the clustering literature, k -means may refer to an objective function to be minimized or the best-known algorithm for minimizing this objective.

  49. In the early cq literature, the hardness of the km problem was incorrectly attributed to various authors, including Hyafil and Rivest (1976), Brucker (1978), and Garey et al. (1982).

  50. Also known as the Lloyd’s algorithm (Lloyd 1982), generalized Lloyd algorithm (gla) (Gray and Karnin 1982), or Linde–Buzo–Gray algorithm (lbg) (Linde et al. 1980).

  51. As before, we omit the handling of exceptional cases such as empty clusters or data points equidistant to multiple centers.

  52. Various cq researchers incorrectly cite Shafer and Kanade (1987) or Celenk (1990) in the context of bkm.

  53. For example, Wu and Witten (1985) report execution times ranging from 12 to 34 hours for \(256 \times 256\) images (\(K = 256\)).

  54. In this paper, an exact accelerated algorithm refers to an accelerated algorithm that generates the same output as the corresponding naive algorithm when started from identical initial conditions. On the other hand, an approximate accelerated algorithm refers to an accelerated algorithm that generates approximately the same output as the corresponding naive algorithm when started from identical initial conditions. We restrict ourselves to sequential acceleration techniques, as opposed to parallel ones, as the former techniques are more appropriate for and common in cq applications. Note that a sequential acceleration algorithm does not necessarily have a lower time complexity than its naive counterpart.

  55. This algorithm was proposed for vq earlier by Chang et al. (1992).

  56. An exact version of this algorithm was proposed earlier by Kaukoranta et al. (2000).

  57. We can arbitrarily assign all data points to \({\mathcal {P}}_1\) before the first iteration.

  58. In general, the exact number of iterations depends on the number, dimensionality, and distribution of the data points, the number of clusters sought, and the initial centers.

  59. Thompson et al. (2020) show that a single-pass variant of bkm is faster than okm, but performs very poorly.

  60. For a discussion of the alternative soft competitive learning (or winner-take-most learning) paradigm, see Subsection 5.5.

  61. For conciseness, we omit the handling of exceptional cases such as empty clusters or data points coincident with centers.

  62. In practice (Celebi 2009), pim is only slightly faster than fcm because the former is based on \(\ell _2\) distances, whereas the latter is based on \(\ell _2^2\) distances.

  63. Recall that fcm with \(M = 1\) is identical to bkm.

  64. According to Pei and Lo (1998), in the context of cq, a 2d som performs slightly worse than a 1d one.

  65. For computational efficiency, Dekker adopts the \(\ell _1\) metric rather than the \(\ell _2^2\) distance.

  66. By contrast, conventional partitional algorithms discussed in Sect. 5 are all local optimization algorithms.

  67. Sometimes referred to as inverse color map computation (Heckbert 1982)

  68. Dekker (1994) also employs a mos-like accelerated pm algorithm.

  69. Available at https://data.mendeley.com/datasets/vw5ys9hfxw/2.

References

  • Abernathy AD, Celebi ME (2022) The incremental online \(k\)-means clustering algorithm and its application to color quantization. Expert Syst Appl 207(117):927

    Google Scholar 

  • Ackerman M, Ben-David S, Loker D (2010) Towards property-based classification of clustering paradigms. In: Advances in neural information processing systems, pp 10–18

  • Ahalt SC, Krishnamurthy AK, Chen P et al (1990) Competitive learning algorithms for vector quantization. Neural Netw 3(3):277–290

    Google Scholar 

  • Akarun L, Yardimci Y, Cetin AE (1997) Adaptive methods for dithering color images. IEEE Trans Image Process 6(7):950–955

    Google Scholar 

  • Akinduko AA, Mirkes EM, Gorban AN (2016) SOM: stochastic initialization versus principal components. Inf Sci 364–365:213–221

    Google Scholar 

  • Alexandrov VV, Gorsky ND, Mysko SN (1984) Recursive pyramids and their use for image coding. Pattern Recogn Lett 2(5):301–310

    Google Scholar 

  • Aloise D, Deshpande A, Hansen P et al (2009) NP-hardness of euclidean sum-of-squares clustering. Mach Learn 75(2):245–248

    MATH  Google Scholar 

  • Anderson M, Motta R, Chandrasekar S, et al (1996) Proposal for a standard default color space for the internet—sRGB. In: Proceedings of the color and imaging conference, pp 238–245

  • Andersson P, Nilsson J, Akenine-Möller T et al (2020) FLIP: a difference evaluator for alternating images. Proc ACM Comput Graph Interact Tech 3(2):15-1-15:23

    Google Scholar 

  • Arthur D, Vassilvitskii S (2007) \(K\)-means++: the advantages of careful seeding. In: Proceedings of the 18th annual ACM–SIAM symposium on discrete algorithms, pp 1027–1035

  • Atsalakis A, Papamarkos N (2006) Color reduction and estimation of the number of dominant colors by using a self-growing and self-organized neural gas. Eng Appl Artif Intell 19(7):769–786

    Google Scholar 

  • Atsalakis A, Papamarkos N, Andreadis I (2002) On estimation of the number of image principal colors and color reduction through self-organized neural networks. Int J Imaging Syst Technol 12(3):117–127

    Google Scholar 

  • Atsalakis A, Papamarkos N, Kroupis N et al (2004) Colour quantisation technique based on image decomposition and its embedded system implementation. IEE Proc 151(6):511–524

    Google Scholar 

  • Avanaki A, Espig K, Kimpe T, et al (2014) Perceptual Uniformity of commonly used color spaces. In: Proceedings of the medical imaging 2014: digital pathology conference, pp 90410V-1–90410V-6

  • Awasthi P, Charikar M, Krishnaswamy R, et al (2015) The hardness of approximation of euclidean \(K\)-means. In: Proceedings of the 31st international symposium on computational geometry, pp 754–767

  • Baarsch J, Celebi ME (2012) Investigation of internal validity measures for \(K\)-means clustering. In: Proceedings of the international multiconference of engineers and computer scientists, pp 471–476

  • Bader M (2013) Space-filling curves: an introduction with applications in scientific computing. Springer, Berlin

    MATH  Google Scholar 

  • Balasubramanian R, Allebach JP (1991a) A new approach to palette selection for color images. J Imaging Technol 17(6):284–290

    Google Scholar 

  • Balasubramanian R, Allebach JP (1991b) A new approach to palette selection for color images. In: Proceedings of the SPIE electronic imaging symposium, pp 58–69

  • Balasubramanian R, Allebach JP, Bouman CA (1994a) Color-image quantization with use of a fast binary splitting technique. J Opt Soc Am A 11(11):2777–2786

    Google Scholar 

  • Balasubramanian R, Bouman CA, Allebach JP (1994b) Sequential scalar quantization of color images. J Electron Imaging 3(1):45–59

    Google Scholar 

  • Banerjee A, Merugu S, Dhillon IS et al (2005) Clustering with bregman divergences. J Mach Learn Res 6:1705–1749

    MathSciNet  MATH  Google Scholar 

  • Baqai FA, Lee JH, Agar AU et al (2005) Digital color halftoning. IEEE Signal Process Mag 22(1):87–96

    Google Scholar 

  • Barata C, Celebi ME, Marques JS et al (2016) Clinically inspired analysis of dermoscopy images using a generative model. Comput Vis Image Underst 151:124–137

    Google Scholar 

  • Barnes ES, Sloane NJA (1983) The optimal lattice quantizer in three dimensions. SIAM J Algebr Discret Methods 4(1):30–41

    MathSciNet  MATH  Google Scholar 

  • Bartholdi JJ III, Platzman LK (1988) Heuristics based on spacefilling curves for combinatorial problems in euclidean space. Manag Sci 34(3):291–305

    MathSciNet  MATH  Google Scholar 

  • Bei CD, Gray RM (1985) An improvement of the minimum distortion encoding algorithm for vector quantization. IEEE Trans Commun 33(10):1132–1133

    Google Scholar 

  • Bentley JL (1975) Multidimensional binary search trees used for associative searching. Commun ACM 18(9):509–517

    MATH  Google Scholar 

  • Berman D, Treibitz T, Avidan S (2016) Non-local image Dehazing. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 1674–1682

  • Bermejo S, Cabestany J (2002) The effect of finite sample size on online \(K\)-Means. Neurocomputing 48(1–4):511–539

    MATH  Google Scholar 

  • Bernard Y, Hueber N, Girau B (2020) A fast algorithm to find best matching units in self-organizing maps. In: Proceedings of the international conference on artificial neural networks, pp 825–837

  • Bezdek JC (1981) Pattern recognition with fuzzy objective function algorithms. Springer, Berlin

    MATH  Google Scholar 

  • Bezdek JC, Keller J, Krisnapuram R et al (1999) Fuzzy models and algorithms for pattern recognition and image processing. Kluwer Academic Publishers, Boston

    MATH  Google Scholar 

  • Bhagavathy S, Llach J, Zhai J (2009) Multiscale probabilistic dithering for suppressing contour artifacts in digital images. IEEE Trans Image Process 18(9):1936–1945

    MathSciNet  MATH  Google Scholar 

  • Bing Z, Junyi S, Qinke P (2004) An adjustable algorithm for color quantization. Pattern Recogn Lett 25(16):1787–1797

    Google Scholar 

  • Blum C, Roli A (2003) Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput Surv 35(3):268–308

    Google Scholar 

  • Borgelt C, Yarikova O (2020) Initializing \(K\)-means clustering. In: Proceedings of the 9th international conference on data science, technology and applications, pp 260–267

  • Bottisti D, Mendez L, Dechev D (2012) CuNeuQuant: a CUDA implementation of the NeuQuant ImageQuantization algorithm. In: Proceedings of the international conference on image processing, computer vision, and pattern recognition

  • Bottou L (1998) Online learning and stochastic approximations. In: Saad D (ed) On-line learning in neural networks. Cambridge University Press, Cambridge, pp 9–42

    MATH  Google Scholar 

  • Bragg D (1992) A simple color reduction filter. In: Kirk D (ed) Graphics gems III. Morgan Kaufmann, pp 20–22

    Google Scholar 

  • Braquelaire JP, Brun L (1997) Comparison and optimization of methods of color image quantization. IEEE Trans Image Process 6(7):1048–1052

    Google Scholar 

  • Braudaway GW (1987) Procedure for optimum choice of a small number of colors from a large color palette for color imaging. In: Proceedings of the electronic imaging conference, pp 71–75

  • Bregman LM (1967) The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput Math Math Phys 7(3):200–217

    MathSciNet  MATH  Google Scholar 

  • Brucker P (1978) On the complexity of clustering problems. In: Henn R, Korte B, Oettli W (eds) Optimization and operations research. Springer, Berlin, pp 45–54

    Google Scholar 

  • Brun L, Mokhtari M (2000) Two high speed color quantization algorithms. In: Proceedings of the 1st international conference on color in graphics and image processing, pp 116–121

  • Brun L, Secroun C (1998) A fast algorithm for inverse colormap computation. Comput Graph Forum 17(4):263–271

    Google Scholar 

  • Brun L, Trémeau A (2003) Color quantization. In: Sharma G (ed) Digital color imaging handbook. CRC Press, Boca Raton, pp 589–638

    Google Scholar 

  • Buades A, Lisani JL, Morel JM (2011) Dimensionality of color space in natural images. J Opt Soc Am A 28(2):203–209

    Google Scholar 

  • Budrikis ZL (1972) Visual fidelity criterion and modeling. Proc IEEE 60(7):771–779

    Google Scholar 

  • Buzo A, Gray, Jr. RMA. H.and Gray, Markel JD (1980) Speech coding based upon vector quantization. IEEE Trans Acoust Speech Signal Process 28(5):562–574

  • Cannon RL, Dave JV, Bezdek JC (1986) Efficient implementation of the Fuzzy \(C\)-means clustering algorithms. IEEE Trans Pattern Anal Mach Intell 8(2):248–255

    MATH  Google Scholar 

  • Carlsson G, Memoli F (2010) Characterization, stability and convergence of hierarchical clustering methods. J Mach Learn Res 11:1425–1470

    MathSciNet  MATH  Google Scholar 

  • Cattelan M (2012) Models for paired comparison data: a review with emphasis on dependent data. Stat Sci 27(3):412–433

    MathSciNet  MATH  Google Scholar 

  • Çak S, Dizdar EN, Ersak A (1998) A Fuzzy colour quantizer for renderers. Displays 19(2):61–65

    Google Scholar 

  • Celebi ME (2009) Fast color quantization using weighted sort-means clustering. J Opt Soc Am A 26(11):2434–2443

    Google Scholar 

  • Celebi ME (2011) Improving the performance of \(K\)-means for color quantization. Image Vis Comput 29(4):260–271

    Google Scholar 

  • Celebi ME (ed) (2015) Partitional clustering algorithms. Springer

  • Celebi ME, Aydin K (eds) (2016) Unsupervised learning algorithms. Springer

  • Celebi ME, Kingravi H (2012) Deterministic initialization of the \(K\)-means algorithm using hierarchical clustering. Int J Pattern Recognit Artif Intell 26(7):1250,018

    MathSciNet  Google Scholar 

  • Celebi ME, Kingravi HA (2015) Linear, deterministic, and order-invariant initialization methods for the \(K\)-means clustering algorithm. In: Celebi ME (ed) Partitional clustering algorithms. Springer, Berlin, pp 79–98

    Google Scholar 

  • Celebi ME, Zornberg A (2014) Automated quantification of clinically significant colors in dermoscopy images and its application to skin Lesion classification. IEEE Syst J 8(3):980–984

    Google Scholar 

  • Celebi ME, Kingravi H, Celiker F (2010) Fast colour space transformations using minimax approximations. IET Image Proc 4(2):70–80

    MathSciNet  Google Scholar 

  • Celebi ME, Celiker F, Kingravi HA (2011) On Euclidean norm approximations. Pattern Recogn 44(2):278–283

    MATH  Google Scholar 

  • Celebi ME, Kingravi HA, Celiker F (2012a) Comments on ‘on approximating Euclidean metrics by weighted \(t\)-cost distances in arbitrary dimension’. Pattern Recogn Lett 33(10):1422–1425

    Google Scholar 

  • Celebi ME, Wen Q, Hwang S, et al (2012b) Color quantization of dermoscopy images using the \(K\)-means clustering algorithm. In: Celebi ME, Schaefer G (eds) Color medical image analysis. Springer, Berlin, pp 87–107

  • Celebi ME, Kingravi H, Vela PA (2013) A comparative study of efficient initialization methods for the \(K\)-means clustering algorithm. Expert Syst Appl 40(1):200–210

    Google Scholar 

  • Celebi ME, Hwang S, Wen Q (2014) Colour quantisation using the adaptive distributing units algorithm. Imaging Sci J 62(2):80–91

    Google Scholar 

  • Celebi ME, Wen Q, Hwang S (2015) An effective real-time color quantization method based on divisive hierarchical clustering. J Real-Time Image Proc 10(2):329–344

    Google Scholar 

  • Celenk M (1990) A color clustering technique for image segmentation. Comput Vis Graph Image Process 52(2):145–170

    Google Scholar 

  • Chan YH, Fung YH (2005) A regularized constrained iterative restoration algorithm for restoring color-quantized images. Signal Process 85(7):1375–1387

    MATH  Google Scholar 

  • Chandler DM, Hemami SS (2007) VSNR: a wavelet-based visual signal-to-noise ratio for natural images. IEEE Trans Image Process 16(9):2284–2298

    MathSciNet  Google Scholar 

  • Chang RF, Chen WT, Wang JS (1992) A fast finite-state algorithm for vector quantizer design. IEEE Trans Signal Process 40(1):221–225

    Google Scholar 

  • Chang CH, Xu P, Xiao R et al (2005) New adaptive color quantization method based on self-organizing maps. IEEE Trans Neural Networks 16(1):237–249

    Google Scholar 

  • Chang CH, Shibu M, Xiao R (2006) Self organizing feature map for color quantization on FPGA. In: Omondi AR, Rajapakse JC (eds) FPGA implementations of neural networks. Springer, Berlin, pp 225–245

    Google Scholar 

  • Chao CKT, Singh K, Gingold Y (2021) PosterChild: blend-aware artistic posterization. Comput Graph Forum 40(4):87–99

    Google Scholar 

  • Chaudhuri D, Murthy CA, Chaudhuri BB (1992) A modified metric to compute distance. Pattern Recogn 25(7):667–677

    Google Scholar 

  • Chen SH, Pan JS (1989) Fast search algorithm for VQ-based recognition of isolated words. IEE Proc I 136(6):391–396

    Google Scholar 

  • Chen MS, Wang SW (1999) Fuzzy clustering analysis for optimizing Fuzzy membership functions. Fuzzy Sets Syst 103(2):239–254

    Google Scholar 

  • Cheng H, Bouman CA (2001) Document compression using rate-distortion optimized segmentation. J Electron Imaging 10(2):460–474

    Google Scholar 

  • Cheng SC, Yang CK (2001) Fast and novel technique for color quantization using reduction of color space dimensionality. Pattern Recogn Lett 22(8):845–856

    MATH  Google Scholar 

  • Cheng DY, Gersho A, Ramamurthi B, et al (1984) Fast search algorithms for vector quantization and pattern matching. In: Proceedings of the IEEE international conference on acoustics, speech, and signal processing, pp 372–375

  • Cheng TW, Goldgof DB, Hall LO (1998) Fast Fuzzy clustering. Fuzzy Sets Syst 93(1):49–56

    MATH  Google Scholar 

  • Cheng SS, Xiong Z, Wu X (2002) Fast trellis-coded color quantization of images. Real-Time Imaging 8(4):265–275

    MATH  Google Scholar 

  • Cheng MM, Mitra NJ, Huang X et al (2015) Global contrast based salient region detection. IEEE Trans Pattern Anal Mach Intell 37(3):569–582

    Google Scholar 

  • Chuang YY, Curless B, Salesin DH, et al (2001) A Bayesian approach to digital matting. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 264–271

  • Chung KL, Huang YH, Wang JP et al (2012) Speedup of color palette indexing in self-organization of Kohonen feature map. Expert Syst Appl 39(3):2427–2432

    Google Scholar 

  • Cohen-Addad V, Kanade V, Mallmann-Trenn F et al (2019) Hierarchical clustering: objective functions and algorithms. J ACM 66(4):1–42

    MathSciNet  MATH  Google Scholar 

  • Cottrell M, Olteanu M, Rossi F, et al (2016) Theoretical and applied aspects of the self-organizing maps. In: Proceedings of the 11th international workshop WSOM 2016. Springer, Berlin, pp 3–26

  • Crevier D (1993) Computing statistical properties of Hue distributions for color image analysis. In: Proceedings of the SPIE intelligent robots and computer vision XII conference, pp 613–623

  • Daly SJ, Feng X (2004) Decontouring: prevention and removal of false contour artifacts. In: Proceedings of the SPIE electronic imaging symposium, pp 130–149

  • Darken C, Moody J (1990) Fast adaptive \(K\)-means clustering: some empirical results. In: Proceedings of the 1990 international joint conference on neural networks, pp 233–238

  • Dasgupta S (2016) A cost function for similarity-based hierarchical clustering. In: Proceedings of the 48th annual ACM symposium on theory of computing, pp 118–127

  • Dasgupta S, Freund Y (2009) Random projection trees for vector quantization. IEEE Trans Inf Theory 55(7):3229–3242

    MathSciNet  MATH  Google Scholar 

  • Dekker A (1994) Kohonen neural networks for optimal colour quantization. Netw Comput Neural Syst 5(3):351–367

    MathSciNet  MATH  Google Scholar 

  • Delon J, Desolneux A, Lisani JL et al (2007) Automatic color palette. Inverse Problems and Imaging 1(2):265–287

    MathSciNet  MATH  Google Scholar 

  • Dembélé D, Kastner P (2003) Fuzzy \(C\)-means method for clustering microarray data. Bioinformatics 19(8):973–980

    Google Scholar 

  • DeSieno D (1988) Adding a conscience to competitive learning. In: Proceedings of the IEEE 1988 international conference on neural networks, pp 117–124

  • Dixit SS (1991) Quantization of color images for display/printing on limited color output devices. Comput Graph 15(4):561–567

    Google Scholar 

  • Domański M, Bartkowiak M (1998) Compression. In: Sangwine SJ, Horne REN (eds) The colour image processing handbook. Chapman & Hall, Boca Raton, pp 242–304

    Google Scholar 

  • Dosselmann R, Yang XD (2011) A comprehensive assessment of the structural similarity index. SIViP 5(1):81–91

    Google Scholar 

  • Dyer ME, Frieze AM (1985) A simple heuristic for the \(P\)-centre problem. Oper Res Lett 3(6):285–288

    MathSciNet  MATH  Google Scholar 

  • Eiben AE, Hinterding R, Michalewicz Z (1999) Parameter control in evolutionary algorithms. IEEE Trans Evolut Comput 3(2):124–141

    Google Scholar 

  • Elkan C (2003) Using the triangle inequality to accelerate \(K\)-Means. In: Proceedings of the 20th international conference on machine learning, pp 147–153

  • Equitz WH (1984) Fast algorithms for vector quantization picture coding. Master’s thesis, Massachusetts Institute of Technology

  • Equitz WH (1989) A new vector quantization clustering algorithm. IEEE Trans Acoust Speech Signal Process 37(10):1568–1575

    Google Scholar 

  • Eschrich S, Ke J, Hall LO et al (2003) Fast accurate Fuzzy clustering through data reduction. IEEE Trans Fuzzy Syst 11(2):262–270

    Google Scholar 

  • Fairchild MD, Johnson GM (2004) iCAM framework for image appearance, differences, and quality. J Electron Imaging 13(1):126–138

    Google Scholar 

  • Fan Z, Jiang T, Huang T (2017) Active sampling exploiting reliable informativeness for subjective image quality assessment based on pairwise comparison. IEEE Trans Multimedia 19(12):2720–2735

    Google Scholar 

  • Farber B, Zeger K (2006) Quantization of multiple sources using nonnegative integer bit allocation. IEEE Trans Inf Theory 52(11):4945–4964

    MathSciNet  MATH  Google Scholar 

  • Feder T, Greene DH (1988) Optimal algorithms for approximate clustering. In: Proceedings of the 20th annual ACM symposium on theory of computing, pp 434–444

  • Feldman D (2020) Core-sets: updated survey. In: Ros F, Guillaume S (eds) Sampling techniques for supervised or unsupervised tasks. Springer, Berlin, pp 23–44

    Google Scholar 

  • Fisher L, Van Ness JW (1971) Admissible clustering procedures. Biometrika 58(1):91–104

    MathSciNet  MATH  Google Scholar 

  • Fiume E, Ouellette M (1989) On distributed, probabilistic algorithms for computer graphics. In: Proceedings of the graphics interface ’89, pp 211–218

  • Fletcher P (1991) A SIMD parallel colour quantization algorithm. Comput Graph 15(3):365–373

    Google Scholar 

  • Flinkman M, Laamanen H, Vahimaa P et al (2012) Number of colors generated by smooth nonfluorescent reflectance spectra. J Opt Soc Am A 29(12):2566–2575

    Google Scholar 

  • Fowler RJ, Paterson MS, Tanimoto SL (1981) Optimal packing and covering in the plane are NP-complete. Inf Process Lett 12(3):133–137

    MathSciNet  MATH  Google Scholar 

  • Fox B (1966) Discrete optimization via marginal analysis. Manag Sci 13(3):210–216

    MATH  Google Scholar 

  • Freisleben B, Schrader A (1997) An evolutionary approach to color image quantization. In: Proceedings of the 1997 IEEE international conference on evolutionary computation, pp 459–464

  • Fung YH, Chan YH (2004) POCS-based algorithm for restoring colour-quantised images. IEE Proc Vis Image Signal Process 151(2):119–127

    Google Scholar 

  • Fung YH, Chan YH (2006) A POCS-based restoration algorithm for restoring halftoned color-quantized images. IEEE Trans Image Process 15(7):1985–1992

    Google Scholar 

  • Fung YH, Chan YH (2006) A simulated annealing restoration algorithm for restoring halftoned color-quantized images. Signal Process 21(4):280–292

    Google Scholar 

  • Garey MR, Johnson D, Witsenhausen HS (1982) The complexity of the generalized Lloyd-Max problem. IEEE Trans Inf Theory 28(2):255–256

    MathSciNet  MATH  Google Scholar 

  • Gentile RS, Allebach JP, Walowit E (1990) Quantization of color images based on uniform color spaces. J Imaging Technol 16(1):11–21

    Google Scholar 

  • Geraci F, Pellegrini M, Maggini M, et al (2006) Cluster generation and cluster labelling for web snippets. In: Proceedings of the 13th international conference on string processing and information retrieval, pp 25–36

  • Gersho A, Gray RM (1992) Vector quantization and signal compression. Kluwer Academic Publishers, Boston

    MATH  Google Scholar 

  • Gervautz M, Purgathofer W (1988) A simple method for color quantization: octree quantization. In: Magnenat-Thalmann N, Thalmann D (eds) New trends in computer graphics. Springer, Berlin, pp 219–231

    Google Scholar 

  • Ghanbarian A, Kabir E, Charkari N (2007) Color reduction based on ant colony. Pattern Recogn Lett 28(12):1383–1390

    Google Scholar 

  • Goldberg N (1991) Colour image quantization for high resolution graphics display. Image Vis Comput 9(5):303–312

    Google Scholar 

  • Gonzalez TF (1985) Clustering to minimize the maximum intercluster distance. Theoret Comput Sci 38(2–3):293–306

    MathSciNet  MATH  Google Scholar 

  • González AI, Graña M, Albizuri FX et al (2000) A near real-time evolution-based adaptation strategy for dynamic color quantization of image sequences. Inf Sci 122(2–4):161–183

    MATH  Google Scholar 

  • Gotsman C, Lindenbaum M (1996) On the metric properties of discrete space-filling curves. IEEE Trans Image Process 5(5):794–797

    Google Scholar 

  • Gray RM, Karnin ED (1982) Multiple local optima in vector quantizers. IEEE Trans Inf Theory 28(2):256–261

    MathSciNet  MATH  Google Scholar 

  • Großwendt A, Røglin H, Schmidt M (2019) Analysis of Ward’s method. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 2939–2957

  • Grossberg S (1987) Competitive learning: from interactive activation to adaptive resonance. Cogn Sci 11(1):23–63

    Google Scholar 

  • Hadizadeh H, Bajic IV, Saeedi P, et al (2011) Good-looking green images. In: Proceedings of the 18th IEEE international conference on image processing, pp 3177–3180

  • Hains C, Wang SG, Knox K (2003) Digital color halftones. In: Sharma G (ed) Digital color imaging handbook. CRC Press, Boca Raton, pp 385–490

    Google Scholar 

  • Hamerly G, Drake J (2015) Accelerating Lloyd’s algorithm for \(K\)-means clustering. In: Celebi ME (ed) Partitional clustering algorithms. Springer, Berlin, pp 41–78

    Google Scholar 

  • Hanbury A (2003) Circular statistics applied to colour images. Proc Comput Vis Winter Workshop 2003:55–60

    Google Scholar 

  • Hanbury A (2008) Constructing cylindrical coordinate colour spaces. Pattern Recogn Lett 29(4):494–500

    Google Scholar 

  • Hansen P, Lazić J, Mladenović N (2007) Variable neighbourhood search for colour image quantization. IMA J Manag Math 18(2):207–221

    MathSciNet  MATH  Google Scholar 

  • Hardeberg JY, Bando E, Pedersen M (2008) Evaluating colour image difference metrics for Gamut-mapped images. Color Technol 124(4):243–253

    Google Scholar 

  • Harding EF (1967) The number of partitions of a set of \(N\) points in \(K\) dimensions induced by hyperplanes. Proc Edinb Math Soc (Ser 2) 15(4):285–289

    MathSciNet  MATH  Google Scholar 

  • Har-Peled S, Sadri B (2005) How fast is the \(K\)-means method? Algorithmica 41(3):185–202

    MathSciNet  MATH  Google Scholar 

  • Hasegawa S, Imai H, Inaba M et al (1993) Efficient algorithms for variance-based \(K\)-clustering. In: Shin SY, Kunii TL (eds) Computer graphics and applications. World Scientific Publishing Co., Singapore, pp 75–88

    Google Scholar 

  • Hatam M, Masnadi-Shirazi MA (2015) Optimum nonnegative integer bit allocation for wavelet based signal compression and coding. Inf Sci 297:332–344

    MathSciNet  MATH  Google Scholar 

  • Hathaway RJ, Bezdek JC (2006) Extending Fuzzy and probabilistic clustering to very large data sets. Comput Stat Data Anal 51(1):215–234

    MathSciNet  MATH  Google Scholar 

  • Hathaway RJ, Hu Y (2009) Density-weighted Fuzzy \(C\)-means clustering. IEEE Trans Fuzzy Syst 17(1):243–252

    Google Scholar 

  • Hathaway RJ, Bezdek JC, Huband JM (2006) Maximin initialization for cluster analysis. In: Proceedings of the 11th iberoamerican congress in pattern recognition, pp 14–26

  • Hatzinger R, Dittrich R (2012) prefmod: an R package for modeling preferences based on paired comparisons, rankings, or ratings. J Stat Softw 48(10):1–31

    Google Scholar 

  • Heckbert PS (1980) Color image quantization for frame buffer display. Bachelor’s thesis, Massachusetts Institute of Technology

  • Heckbert P (1982) Color image quantization for frame buffer display. ACM SIGGRAPH Comput Graph 16(3):297–307

    Google Scholar 

  • Heckel R, Shah NB, Ramchandran K et al (2019) Active ranking from pairwise comparisons and when parametric assumptions do not help. Ann Stat 47(6):3099–3126

    MathSciNet  MATH  Google Scholar 

  • Hoare CAR (1971) Proof of a program: find. Commun ACM 14(1):39–45

    MATH  Google Scholar 

  • Höppner F (2002) Speeding up Fuzzy \(C\)-means: using a hierarchical data organisation to control the precision of membership calculation. Fuzzy Sets Syst 128(3):365–376

    MathSciNet  MATH  Google Scholar 

  • Hore A, Ziou D (2010) Image quality metrics: PSNR vs. SSIM. In: Proceedings of the 2010 international conference on pattern recognition, pp 2366–2369

  • Hou Y, Zheng L, Gould S (2020) Learning to structure an image with few colors. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 10,116–10,125

  • Houle G, Dubois E (1986) Quantization of color images for display on graphics terminals. In: Proceedings of the IEEE global telecommunications conference, pp 1138–1142

  • Howard N, Harris B (1966) A hierarchical grouping routine (IBM 360/65 Fortran IV Program). Tech. rep., University of Pennsylvania Computer Center

  • Hsieh IS, Fan KC (2000) An adaptive clustering algorithm for color quantization. Pattern Recogn Lett 21(4):337–346

    Google Scholar 

  • Hsu WL, Nemhauser GL (1979) Easy and hard bottleneck location problems. Discret Appl Math 1(3):209–215

    MathSciNet  MATH  Google Scholar 

  • Hu YC, Lee MG (2007) \(K\)-means based color palette design scheme with the use of stable flags. J Electron Imaging 16(3):033,003

    Google Scholar 

  • Hu YC, Su BH (2008a) Accelerated \(K\)-means clustering algorithm for colour image quantization. Imaging Sci J 56(1):29–40

    MathSciNet  Google Scholar 

  • Hu YC, Su BH (2008b) Accelerated pixel mapping scheme for colour image quantisation. Imaging Sci J 56(2):68–78

    Google Scholar 

  • Hu YC, Lee MG, Tsai P (2009) Colour palette generation schemes for colour image quantization. Imaging Sci J 57(1):46–59

    Google Scholar 

  • Huang SC (2021) An efficient palette generation method for color image quantization. Appl Sci 11(3):1043

    Google Scholar 

  • Huang YL, Chang RF (2004) A fast finite-state algorithm for generating RGB palettes of color quantized images. J Inf Sci Eng 20(4):771–782

    Google Scholar 

  • Huang J, Schultheiss P (1963) Block quantization of correlated gaussian random variables. IEEE Trans Commun Syst 11(3):289–296

    Google Scholar 

  • Huang TS, Schreiber WF, Tretiak OJ (1971) Image processing. Proc IEEE 59(11):1586–1609

  • Huang M, Xia Z, Wang H et al (2012) The range of the value for the Fuzzifier of the Fuzzy \(C\)-means algorithm. Pattern Recogn Lett 33(16):2280–2284

    Google Scholar 

  • Huang HZ, Xu K, Martin RR et al (2016) Efficient, edge-aware, combined color quantization and dithering. IEEE Trans Image Process 25(3):1152–1162

    MathSciNet  MATH  Google Scholar 

  • Huang Q, Kim HY, Tsai WJ et al (2018) Understanding and removal of false contour in HEVC compressed images. IEEE Trans Circuits Syst Video Technol 28(2):378–391

    Google Scholar 

  • Hung KL, Chang CC (2002) An improvement of a technique for color quantization using reduction of color space dimensionality. Informatica 26(1):11–16

    MATH  Google Scholar 

  • Hwang KF, Chang CC (2002) A fast pixel mapping algorithm using principal component analysis. Pattern Recogn Lett 23(14):1747–1753

    MATH  Google Scholar 

  • Hyafil L, Rivest RL (1976) Constructing optimal binary decision trees is NP-complete. Inf Process Lett 5(1):15–17

    MathSciNet  MATH  Google Scholar 

  • Ienne P, Thiran P, Vassilas N (1997) Modified self-organizing feature map algorithms for efficient digital hardware implementation. IEEE Trans Neural Networks 8(2):315–330

    Google Scholar 

  • Inaba M, Katoh N, Imai H (1994) Applications of weighted voronoi diagrams and randomization to variance-based \(K\)-clustering. In: Proceedings of the 10th annual symposium on computational seometry, pp 332–339

  • Jackins CL, Tanimoto SL (1980) Oct-trees and their use in representing three-dimensional objects. Comput Graphics Image Process 14(3):249–270

    Google Scholar 

  • Jain AK, Pratt WK (1972) color image quantization. In: Proceedings of the 1972 national telecommunications conference, p 34

  • Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. ACM Comput Surv 31(3):264–323

    Google Scholar 

  • Jamieson KG, Nowak RD (2011) Active ranking using pairwise comparisons. In: Proceedings of the 25th annual conference on neural information processing systems, pp 2240–2248

  • Johnson GM, Fairchild MD (2003) A top down description of S-CIELAB and CIEDE2000. Color Res Appl 28(6):425–435

    Google Scholar 

  • Johnson GM, Song X, Montag ED et al (2010) Derivation of a color space for image color difference measurement. Color Res Appl 35(6):387–400

    Google Scholar 

  • Joy G, Xiang Z (1993) Center-cut for color image quantization. Visual Comput 10(1):62–66

    Google Scholar 

  • Joy G, Xiang Z (1996) Reducing false contours in quantized color images. Comput Graph 20(2):231–242

    Google Scholar 

  • Kanungo T, Mount D, Netanyahu N et al (2002) An efficient \(K\)-means clustering algorithm: analysis and implementation. IEEE Trans Pattern Anal Mach Intell 24(7):881–892

    Google Scholar 

  • Kasuga H, Yamamoto H, Okamoto M (2000) Color quantization using the fast \(K\)-means algorithm. Syst Comput Japan 31(8):33–40

    Google Scholar 

  • Kaukoranta T, Fränti P, Nevalainen O (2000) A fast exact GLA based on code vector activity detection. IEEE Trans Image Process 9(8):1337–1342

    Google Scholar 

  • Keysers D, Lampert CH, Breuel TM (2006) Color image dequantization by constrained diffusion. In: Proceedings of the SPIE/IS &T electronic imaging symposium, pp 605,803–1–605,803–10

  • Khalifa KB, Blaiech AG, Abadi M et al (2020) New hardware architecture for self-organizing map used for color vector quantization. J Circuits Syst Comput 29(1):2050,002

    Google Scholar 

  • Kim N, Kehtarnavaz N (2005) DWT-based scene-adaptive color quantization. Real-Time Imaging 11(5–6):443–453

    Google Scholar 

  • Kim TH, Ahn J, Choi MG (2007) Image dequantization: restoration of quantized colors. Comput Graph Forum 26(3):619–626

    Google Scholar 

  • Kohonen T (1982) Self-organized formation of topologically correct feature maps. Biol Cybern 43:59–69

    MathSciNet  MATH  Google Scholar 

  • Kohonen T (1993) Things you haven’t heard about the self-organizing map. In: Proceedings of the IEEE international conference on neural networks, pp 1147–1156

  • Kohonen T (2013) Essentials of the self-organizing map. Neural Netw 37:52–65

    Google Scholar 

  • Koikkalainen P, Oja E (1990) Self-organizing hierarchical feature maps. In: Proceedings of the 1990 IJCNN international joint conference on neural networks, pp 279–284

  • Kok CW, Chan SC, Leung SH (1993) Color quantization by Fuzzy quantizer. In: Proceedings of the IS &T/SPIE’s symposium on electronic imaging: science and technology, pp 235–242

  • Kolen JF, Hutcheson T (2002) Reducing the time complexity of the Fuzzy \(C\)-means algorithm. IEEE Trans Fuzzy Syst 10(2):263–267

    Google Scholar 

  • Kruger A (1992) Reduction of computer-generated images. PhD thesis, University of Iowa

  • Kuehni RG (2003) Color space and its divisions: color order from antiquity to the present. Wiley, New York

    Google Scholar 

  • Kuehni RG (2016) How many object colors can we distinguish? Color Res Appl 41(5):439–444

    Google Scholar 

  • Kuhn GR, Oliveira MM, Fernandes LA (2008) An improved contrast enhancing approach for color-to-grayscale mappings. Vis Comput 24(7):505–514

    Google Scholar 

  • Kurdthongmee W (2008) A novel hardware-oriented Kohonen SOM image compression algorithm and its FPGA implementation. J Syst Architect 54(10):983–994

    Google Scholar 

  • Kurdthongmee W (2011) Utilization of a rational-based representation to improve the image quality of a hardware-based \(K\)-SOM quantizer. J Real-Time Image Proc 6(3):199–211

    Google Scholar 

  • Kurdthongmee W (2016) A hardware centric algorithm for the best matching unit searching stage of the SOM-based quantizer and its FPGA implementation. J Real-Time Image Proc 12(1):71–80

    Google Scholar 

  • Lai JZC, Liaw YC (2008) Improvement of the \(K\)-means clustering filtering algorithm. Pattern Recogn 41(12):3677–3681

    MATH  Google Scholar 

  • Lampinen J, Oja E (1990) Fast computation of Kohonen self-organization. In: Soulié FF, Hérault J (eds) Neurocomputing: algorithms, architectures and applications. Springer, p 65–74

  • Lawrence RD, Almasi GS, Rushmeier HE (1999) A scalable parallel algorithm for self-organizing maps with applications to sparse data mining problems. Data Min Knowl Disc 3(2):171–195

    Google Scholar 

  • Lee E, Schmidt M, Wright J (2017) Improved and simplified inapproximability for \(K\)-means. Inf Process Lett 120:40–43

    MathSciNet  MATH  Google Scholar 

  • Lempel A, Ziv J (1986) Compression of two-dimensional data. IEEE Trans Inf Theory 32(1):2–8

    Google Scholar 

  • Leung CS, Ho TY, Xiao Y (2010) GPU color quantization. In: Engel W (ed) GPU Pro: advanced rendering techniques. A K Peters, p 3–13

  • Levkowitz H, Herman GT (1993) GLHS: a generalized lightness, Hue, and saturation color model. CVGIP 55(4):271–285

    Google Scholar 

  • Li J, Mantiuk RK, Wang J, et al (2018) Hybrid-MST: a hybrid active sampling strategy for pairwise preference aggregation. In: Proceedings of the 32nd international conference on neural information processing systems, pp 3479–3489

  • Linde Y, Buzo A, Gray RM (1980) An algorithm for vector quantizer design. IEEE Trans Commun 28(1):84–95

    Google Scholar 

  • Linhares JMM, Pinto PD et al (2008) The number of discernible colors in natural scenes. J Opt Soc Am A 25(12):2918–2924

    Google Scholar 

  • Liu TS, Chang LW (1995) Fast color image quantization with error diffusion and morphological operations. Signal Process 43(3):293–303

    MATH  Google Scholar 

  • Liu Q, Crispino M, Scheel I, et al (2019) Model-based learning from preference data. Annu Rev Stat Appl pp 329–354

  • Liu Y, Sun J, Yao Q, et al (2018) A scalable heterogeneous parallel SOM based on MPI/CUDA. In: Proceedings of the 10th Asian conference on machine learning, pp 264–279

  • Lloyd S (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28(2):129–136

    MathSciNet  MATH  Google Scholar 

  • Lo KC, Chan YH, Yu M (2003) Colour quantization by three-dimensional frequency diffusion. Pattern Recogn Lett 24(14):2325–2334

    MATH  Google Scholar 

  • Lucic M, Bachem O, Krause A (2016) Strong coresets for hard and soft bregman clustering with applications to exponential family mixtures. In: Proceedings of the 19th international conference on artificial intelligence and statistics, pp 1–9

  • Luo MR (2002) Development of colour-difference formulae. Rev Prog Color Relat Top 32(1):28–39

    Google Scholar 

  • Luzardo G, Aelterman J, Luong H, et al (2017) Real-time false-contours removal for inverse tone mapped HDR content. In: Proceedings of the 25th ACM international conference on multimedia, pp 1472–1479

  • MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the 5th Berkeley symposium on mathematical statistics and probability, pp 281–297

  • Mahajan M, Nimbhorkar P, Varadarajan K (2009) The planar \(K\)-means problem is NP-hard. In: Proceedings of the 3rd international workshop on algorithms and computation, pp 274–285

  • Makhoul J, Roucos S, Gish H (1985) Vector quantization in speech coding. Proc IEEE 73(11):1551–1588

    Google Scholar 

  • Mannos JL, Sakrison DJ (1974) The effects of a visual fidelity criterion of the encoding of images. IEEE Trans Inf Theory 20(4):525–536

    MATH  Google Scholar 

  • Mantiuk RK, Tomaszewska A, Mantiuk R (2012) Comparison of four subjective methods for image quality assessment. Comput Graph Forum 31(8):2478–2491

    Google Scholar 

  • Martínez-Verdú F, Perales E, Chorro E et al (2007) Computation and visualization of the MacAdam limits for any lightness, Hue angle, and light source. J Opt Soc Am A 24(6):1501–1515

    Google Scholar 

  • Masaoka K, Berns RS, Fairchild MD et al (2013) Number of discernible object colors is a conundrum. J Opt Soc Am A 30(2):264–277

    Google Scholar 

  • Masuyama S, Ibaraki T, Hasegawa T (1981) The computational complexity of the \(M\)-center problems on the plane. Trans IEICE Jpn 64(2):E-57-64

    Google Scholar 

  • Matsumoto M, Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans Model Comput Simul 8(1):3–30

    MATH  Google Scholar 

  • Max J (1960) Quantizing for minimum distortion. IRE Trans Inf Theory 6(1):7–12

    MathSciNet  Google Scholar 

  • Maystre L, Grossglauser M (2017) Just sort it! A simple and effective approach to active preference learning. In: Proceedings of the 34th international conference on machine learning, pp 2344–2353

  • Megiddo N, Supowit KJ (1984) On the complexity of some common geometric location problems. SIAM J Comput 13(1):182–196

    MathSciNet  MATH  Google Scholar 

  • Melgosa M, Huertas R (2004) Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas. J Opt Soc Am A 21(12):2269–2275

    Google Scholar 

  • Mignotte M (2008) Segmentation by fusion of histogram-based \(K\)-means clusters in different color spaces. IEEE Trans Image Process 17(5):780–787

    MathSciNet  Google Scholar 

  • Mikhailiuk A, Wilmot C, Pérez-Ortiz M, et al (2021) Active sampling for pairwise comparisons via approximate message passing and information gain maximization. In: Proceedings of the 25th international conference on pattern recognition, pp 2559–2566

  • Milvang O (1987) An adaptive algorithm for color image quantization. In: Proceedings of the 5th scandinavian conference on image analysis, pp 43–47

  • Mitsa T, Varkur KL (1993) Evaluation of contrast sensitivity functions for the formulation of quality measures incorporated in halftoning algorithms. In: Proceedings of the 1993 IEEE international conference on acoustics, speech, and signal processing, pp 301–304

  • Mohr AE (2002) Bit allocation in sub-linear time and the multiple-choice knapsack problem. In: Proceedings of the data compression conference, pp 352–361

  • Mojsilović A, Soljanin E (2001) Color quantization and processing by fibonacci lattices. IEEE Trans Image Process 10(11):1712–1725

    MATH  Google Scholar 

  • Mojsilović A, Hu J, Soljanin E (2002) Extraction of perceptually important colors and similarity measurement for image matching, retrieval, and analysis. IEEE Trans Image Process 11(11):1238–1248

    MathSciNet  Google Scholar 

  • Monga V, Damera-Venkata N, Evans BL (2006) Color image halftoning. In: Lukac R, Plataniotis KN (eds) Color image processing: methods and applications. CRC Press, Boca Raton, pp 157–183

  • Montagne C, Lelandais S, Smolarz A et al (2006) Adaptive color quantization using the “Baker’s transformation’’. J Electron Imaging 15(2):023,015

    Google Scholar 

  • Moon B, Jagadish HV, Faloutsos C et al (2001) Analysis of the clustering properties of the Hilbert space-filling curve. IEEE Trans Knowl Data Eng 13(1):124–141

    Google Scholar 

  • Morovic J, Cheung V, Morovic P (2012) Why we don’t know how many colors there are? In: Proceedings of the conference on colour in graphics, imaging, and vision, pp 49–53

  • Mulier FM, Cherkassky VS (1995) Statistical analysis of self-organization. Neural Netw 8(5):717–727

    Google Scholar 

  • Müllner D (2013) fastcluster: fast hierarchical, agglomerative clustering routines for R and Python. J Stat Softw 53(9):1–18

    Google Scholar 

  • Murtagh F, Legendre P (2014) Ward’s hierarchical agglomerative clustering method: which algorithms implement Ward’s criterion? J Classif 31(3):274–295

    MathSciNet  MATH  Google Scholar 

  • Murthy SK, Kasif S, Salzberg S (1994) A system for induction of oblique decision trees. J Artif Intell Res 2(1):1–32

    MATH  Google Scholar 

  • Necaise RD (1998) Improvements to the color quantization process. PhD thesis, College of William & Mary

  • Nieves JL, Gomez-Robledo L, Chen YJ et al (2020) Computing the relevant colors that describe the color palette of paintings. Appl Opt 59(6):1732–1740

    Google Scholar 

  • Nikolaou N, Papamarkos N (2009) Color reduction for complex document images. Int J Imaging Syst Technol 19(1):14–26

    Google Scholar 

  • Nolle L, Schaefer G (2007) Colour map design through optimization. Eng Optim 39(3):327–343

    Google Scholar 

  • Ohta YI, Kanade T, Sakai T (1980) Color information for region segmentation. Comput Graphics Image Process 13(3):222–241

    Google Scholar 

  • Omran MG, Engelbrecht AP, Salman A (2005) A color image quantization algorithm based on particle swarm optimization. Informatica 29(3):261–269

    MATH  Google Scholar 

  • Oppenheim AV, Weinstein CJ (1972) Effects of finite register length in digital filtering and the fast Fourier transform. Proc IEEE 60(8):957–976

    Google Scholar 

  • Orchard M, Bouman C (1991) Color quantization of images. IEEE Trans Signal Process 39(12):2677–2690

    Google Scholar 

  • Ortiz-Jaramillo B, Kumcu A, Platisa L et al (2019) Evaluation of color differences in natural scene color images. Signal Process 71:128–137

    Google Scholar 

  • Otsu N (1979) A threshold selection method from gray-level histograms. IEEE Trans Syst Man Cybern 9(1):62–66

    Google Scholar 

  • Özdemir D, Akarun L (2002) Fuzzy algorithm for color quantization of images. Pattern Recogn 35(8):1785–1791

    MATH  Google Scholar 

  • Ozkan I, Turksen IB (2007) Upper and lower values for the level of fuzziness in FCM. Inf Sci 177(23):5143–5152

    MATH  Google Scholar 

  • Paeth AW (1990) Mapping RGB triples onto four bits. In: Glassner A (ed) Graphics gems I. Morgan Kaufmann, pp 233–245

    Google Scholar 

  • Paeth AW (1991) Mapping RGB triples onto 16 distinct values. In: Arvo J (ed) Graphics gems II. Morgan Kaufmann, pp 143–146

    Google Scholar 

  • Pal NR, Bezdek JC (2002) Complexity reduction for “large image’’ processing. IEEE Trans Syst Man Cybern 32(5):598–611

    Google Scholar 

  • Palomo EJ, Domínguez E (2014) Hierarchical color quantization based on self-organization. J Math Imaging Vis 49(1):1–19

    Google Scholar 

  • Papamarkos N, Atsalakis AE, Strouthopoulos CP (2002) Adaptive color reduction. IEEE Trans Syst Man Cybern 32(1):44–56

    MATH  Google Scholar 

  • Park JH, Kim SH, Lee JC, et al (2022) Scalable color quantization for task-centric image compression. ACM Trans Multimed Comput Commun Appl

  • Parker JK, Hall LO (2014) Accelerating Fuzzy-\(C\) means using an estimated subsample size. IEEE Trans Fuzzy Syst 22(5):1229–1244

    Google Scholar 

  • Pedersen M (2015) Evaluation of 60 full-reference image quality metrics on the CID:IQ. In: Proceedings of the 2015 IEEE international conference on image processing, pp 1588–1592

  • Pedersen M, Hardeberg JY (2012) Full-reference image quality metrics: classification and evaluation. Found Trends Comput Graph Vis 7(1):1–80

    MATH  Google Scholar 

  • Pei SC, Cheng CM (1995) Dependent scalar quantization of color images. IEEE Trans Circuits Syst Video Technol 5(2):124–139

    Google Scholar 

  • Pei SC, Lo YS (1998) Color image compression and limited display using self-organization Kohonen map. IEEE Trans Circuits Syst Video Technol 8(2):191–205

    Google Scholar 

  • Pei SC, Chuang YT, Chuang WH (2006) Effective palette indexing for image compression using self-organization of Kohonen feature map. IEEE Trans Image Process 15(9):2493–2498

    Google Scholar 

  • Pérez-Delgado ML (2015) Colour quantization with ant-tree. Appl Soft Comput 36:656–669

    Google Scholar 

  • Pérez-Delgado ML (2018) Artificial ants and fireflies can perform colour quantisation. Appl Soft Comput 73:153–177

    Google Scholar 

  • Pérez-Delgado ML (2019) The color quantization problem solved by swarm-based operations. Appl Intell 49:2482–2514

    Google Scholar 

  • Pérez-Delgado ML (2020) Color quantization with particle swarm optimization and artificial ants. Soft Comput 24:4545–4573

    Google Scholar 

  • Pérez-Delgado ML (2021) Revisiting the iterative ant-tree for color quantization algorithm. J Vis Commun Image Represent 78(103):180

    Google Scholar 

  • Pérez-Delgado ML, Gallego JAR (2019) A hybrid color quantization algorithm that combines the greedy orthogonal Bi-partitioning method with artificial ants. IEEE Access 7:128,714-128,734

    Google Scholar 

  • Pérez-Delgado ML, Gallego JAR (2020) A two-stage method to improve the quality of quantized images. J Real-Time Image Proc 17(3):581–605

    Google Scholar 

  • Pérez-Ortiz M, Mantiuk RK (2017) A practical guide and software for analysing pairwise comparison experiments. https://arxiv.org/abs/1712.03686

  • Pérez-Ortiz M, Mikhailiuk A, Zerman E et al (2019) From pairwise comparisons and rating to a unified quality scale. IEEE Trans Image Process 29:1139–1151

    MathSciNet  MATH  Google Scholar 

  • Phung SL, Bouzerdoum A, Chai D (2005) Skin segmentation using color pixel classification: analysis and comparison. IEEE Trans Pattern Anal Mach Intell 27(1):148–154

    Google Scholar 

  • Pointer MR, Attridge GG (1998) The number of discernible colours. Color Res Appl 23(1):52–54

    Google Scholar 

  • Ponomarenko N, Jin L, Ieremeiev O et al (2015) Image database TID2013: peculiarities, results and perspectives. Signal Process 30:57–77

    Google Scholar 

  • Poynton C, Funt B (2014) Perceptual uniformity in digital image representation and display. Color Res Appl 39(1):6–15

    Google Scholar 

  • Pratt WK (1970) Spatial transform coding of color images. IEEE Trans Commun Technol 19(6):980–992

    Google Scholar 

  • Press WH, Teukolsky SA, Vetterling WT et al (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge, Cambridge University Press

    MATH  Google Scholar 

  • Puzicha J, Held M, Ketterer J et al (2000) On spatial quantization of color images. IEEE Trans Image Process 9(4):666–682

    Google Scholar 

  • Ra SW, Kim JK (1993) A fast mean-distance-ordered partial codebook search algorithm for image vector quantization. IEEE Trans Circuits Syst II 40(9):576–579

    Google Scholar 

  • Ramanath R, Snyder WE, Yoo Y et al (2005) Color image processing pipeline. IEEE Signal Process Mag 22(1):34–43

    Google Scholar 

  • Ramella G (2021) Evaluation of quality measures for color quantization. Multimed Tools Appl 80:32,975-33,009

    Google Scholar 

  • Ramella G, di Baja GS (2013) A new technique for color quantization based on histogram analysis and clustering. Int J Pattern Recognit Artif Intell 27(3):1360,006

    MathSciNet  Google Scholar 

  • Ramstad TA (1982) Sub-band coder with a simple adaptive bit-allocation algorithm: a possible candidate for digital mobile telephony? In: Proceedings of the IEEE international conference on acoustics, speech, and signal processing, pp 203–207

  • Reitan PJ (1998) 3D visualization of color image histograms. Comput Netw ISDN Syst 30(20–21):2025–2035

    Google Scholar 

  • Reitan PJ (1999) Hybrid approaches to color image quantization. PhD thesis, University of Maryland, Baltimore County

  • Ren M, Wang Z, Jiang J (2019) A self-adaptive FCM for the optimal Fuzzy weighting exponent. Int J Comput Intell Appl 18(2):1950,008

    Google Scholar 

  • Ritter HJ, Schulten K (1988) Kohonen’s self-organizing maps: exploring their computational capabilities. In: Proceedings of the IEEE 1988 international conference on neural networks, pp 109–116

  • Robbins H, Monro S (1951) A stochastic approximation method. Ann Math Stat 22(3):400–407

    MathSciNet  MATH  Google Scholar 

  • Robertson PK (1988) Visualizing color Gamuts: a user interface for the effective use of perceptual color spaces in data displays. IEEE Comput Graph Appl 8(5):50–64

    Google Scholar 

  • Rumelhart D, Zipser D (1985) Feature discovery by competitive learning. Cogn Sci 9(1):75–112

    Google Scholar 

  • Ruspini EH (1970) Numerical methods for Fuzzy clustering. Inf Sci 2(3):319–350

    MATH  Google Scholar 

  • Sabin MJ, Gray RM (1984) Product code vector quantizers for waveform and voice coding. IEEE Trans Acoust Speech Signal Process 32(3):474–488

    Google Scholar 

  • Safavian SR, Landgrebe D (1991) A survey of decision tree classifier methodology. IEEE Trans Syst Man Cybern 21(3):660–674

    MathSciNet  Google Scholar 

  • Sakrison DJ, Algazi VR (1971) Comparison of line-by-line and two-dimensional encoding of random images. IEEE Trans Inf Theory 17(4):386–398

    MathSciNet  MATH  Google Scholar 

  • Schaefer G (2014) Soft computing-based colour quantisation. EURASIP J Image Video Process 2014(1):1–9

    Google Scholar 

  • Schaefer G, Nolle L (2015) A hybrid color quantization algorithm incorporating a human visual perception model. Comput Intell 31(4):684–698

    MathSciNet  Google Scholar 

  • Schaefer G, Zhou H (2009) Fuzzy clustering for colour reduction in images. Telecommun Syst 40(1):17–25

    Google Scholar 

  • Schaefer G, Zhou H, Celebi ME et al (2011) Rough colour quantisation. Int J Hybrid Int Syst 8(1):25–30

    Google Scholar 

  • Schaefer G, Hu Q, Zhou H et al (2012) Rough \(C\)-means and Fuzzy rough \(C\)-means for colour quantisation. Fund Inform 119(1):113–120

    MathSciNet  Google Scholar 

  • Scheunders P (1997) A genetic \(C\)-means clustering algorithm applied to color image quantization. Pattern Recogn 30(6):859–866

    Google Scholar 

  • Schmidl TM, Cosman PC, Gray RM (1993) Unbalanced non-binary tree-structured vector quantizers. In: Proceedings of the 27th asilomar conference on signals, systems and computers, pp 1519–1523

  • Schmitz BE, Stevenson RL (1995) Color palette restoration. Graph Models Image Process 57(5):409–419

    Google Scholar 

  • Schreiber T (1991) A voronoi diagram based adaptive \(K\)-means-type clustering algorithm for multidimensional weighted data. In: Proceedings of the international workshop on computational geometry, pp 265–275

  • Schwämmle V, Jensen ON (2010) A simple and fast method to determine the parameters for Fuzzy \(C\)-means cluster analysis. Bioinformatics 26(22):2841–2848

    Google Scholar 

  • Serrano C, Lazo M, Serrano A et al (2022) Clinically inspired skin lesion classification through the detection of dermoscopic criteria for basal cell carcinoma. J Imaging 8(7):197

    Google Scholar 

  • Shafer SA, Kanade T (1987) Color Vision. In: Shapiro SC (ed) Encyclopedia of artificial intelligence, vol 1. Wiley, New York, pp 124–131

    Google Scholar 

  • Sharma G, Vrhel MJ, Trussell HJ (1998) Color imaging for multimedia. Proc IEEE 86(6):1088–1108

    Google Scholar 

  • Sharma G, Wu W, Dalal EN (2005) The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations. Color Res Appl 30(1):21–30

    Google Scholar 

  • Sheikh HR, Sabir MF, Bovik AC (2006) A statistical evaluation of recent full reference image quality assessment algorithms. IEEE Trans Image Process 15(11):3440–3451

    Google Scholar 

  • Sheikholeslami G, Chatterjee S, Zhang A (1998) WaveCluster: a multi-resolution clustering approach for very large spatial databases. In: Proceedings of the 24th international conference on very large data bases, pp 428–439

  • Shufelt JA (1997) Texture analysis for enhanced color image quantization. Graph Models Image Process 59(3):149–163

    Google Scholar 

  • Silverstein DA, Farrell JE (2001) Efficient method for paired comparison. J Electron Imaging 10(2):394–398

    Google Scholar 

  • Sobol’ IM, Asotsky D, Kreinin A et al (2011) Construction and comparison of high-dimensional Sobol’ generators. Wilmott 56:64–79

    Google Scholar 

  • Soljanin E (2002) Writing sequences on the plane. IEEE Trans Inf Theory 48(6):1344–1354

    MathSciNet  MATH  Google Scholar 

  • Song Q, Su GM, Cosman PC (2020) Efficient debanding filtering for inverse tone mapped high dynamic range videos. IEEE Trans Circuits Syst Video Technol 30(8):2575–2589

    Google Scholar 

  • Sproull RF (1991) Refinements to nearest-neighbor searching in \(K\)-dimensional trees. Algorithmica 6(4):579–589

    MathSciNet  MATH  Google Scholar 

  • Stevens RJ, Lehar AF, Preston FH (1983) Manipulation and presentation of multidimensional image data using the Peano scan. IEEE Trans Pattern Anal Mach Intell 5(5):520–526

    Google Scholar 

  • Stockham TG Jr (1972) Image processing in the context of a visual model. Proc IEEE 60(7):828–842

    Google Scholar 

  • Stokes M, Fairchild MD, Berns RS (1992) Precision requirements for digital color reproduction. ACM Trans Graph 11(4):406–422

    MATH  Google Scholar 

  • Streijl RC, Winkler S, Hands DS (2016) Mean opinion score (MOS) revisited: methods and applications. Multimed Syst 22(2):213–227

    Google Scholar 

  • Su MC, Chang HT (2000) Fast self-organizing feature map algorithm. IEEE Trans Neural Netw 11(3):721–733

    Google Scholar 

  • Su T, Dy JG (2007) In search of deterministic methods for initializing \(K\)-means and Gaussian mixture clustering. Intell Data Anal 11(4):319–338

    Google Scholar 

  • Sudha N, Srikanthan T, Mailachalam B (2003) A VLSI architecture for 3-D self-organizing map based color quantization and its FPGA implementation. J Syst Architect 48(11–12):337–352

    Google Scholar 

  • Szilágyi L, Benyó Z, Szilágyi SM, et al (2003) MR brain image segmentation using an enhanced Fuzzy \(C\)-means algorithm. In: Proceedings of the 25th annual international conference of the IEEE engineering in medicine and biology society, pp 724–726

  • Szilágyi L, Dénesi G, Kovács L, et al (2014) Comparison of various improved-partition Fuzzy \(C\)-means clustering algorithms in fast color reduction. In: Proceedings of the 2014 IEEE 12th international symposium on intelligent systems and informatics, pp 197–202

  • Szilágyi L, Dénesi G, Enăchescu C (2016) Fast color quantization via Fuzzy clustering. In: Proceedings of the international conference on neural information processing, pp 95–103

  • Taşdizen T, Akarun L, Ersoy C (1998) Color quantization with genetic algorithms. Signal Process 12(1):49–57

    Google Scholar 

  • Thomas SW (1991) Efficient inverse color map computation. In: Arvo J (ed) Graphics gems II. Academic Press, Cambridge, pp 116–125

    Google Scholar 

  • Thompson S, Celebi ME, Buck KH (2020) Fast color quantization using MacQueen’s \(K\)-means algorithm. J Real-Time Image Proc 17(5):1609–1624

    Google Scholar 

  • Trapp M, Pasewaldt S, Döllner J (2019) Techniques for GPU-based color quantization. In: Proceedings of the 27th international conference in central Europe on computer graphics, visualization and computer vision, pp 81–87

  • Tseng HW, Ding WB (2012) Reversible data hiding scheme for colour images based on pixel clustering and histogram shifting. Imaging Sci J 60(1):47–53

    Google Scholar 

  • Tsukida K, Gupta MR (2011) How to analyze paired comparison data. Tech. Rep. UWEETR-2011-0004, University of Washington

  • Tu Z, Lin J, Wang Y et al (2020) Adaptive debanding filter. IEEE Signal Process Lett 27:1715–1719

    Google Scholar 

  • Turnbull D, Elkan C (2005) Fast recognition of musical genres using RBF networks. IEEE Trans Knowl Data Eng 17(4):580–584

    Google Scholar 

  • Turner H, Firth D (2012) Bradley-Terry models in R: the BradleyTerry2 package. J Stat Softw 48(9):1–21

    Google Scholar 

  • Uchiyama T, Arbib MA (1994a) An algorithm for competitive learning in clustering problems. Pattern Recogn 27(10):1415–1421

    Google Scholar 

  • Uchiyama T, Arbib MA (1994b) Color image segmentation using competitive learning. IEEE Trans Pattern Anal Mach Intell 16(12):1197–1206

    Google Scholar 

  • Ueda Y, Koga T, Suetake N et al (2017) Color quantization method based on principal component analysis and linear discriminant analysis for Palette-based image generation. Opt Rev 24(6):741–756

    Google Scholar 

  • Urban P, Rosen MR, Berns RS et al (2007) Embedding non-euclidean color spaces into euclidean color spaces with minimal isometric disagreement. J Opt Soc Am A 24(6):1516–1528

    Google Scholar 

  • Valenzuela G, Celebi ME, Schaefer G (2018) Color quantization using coreset sampling. In: Proceedings of the 2018 IEEE international conference on systems, man, and cybernetics, pp 2096–2101

  • Van Hulle MM (2012) Self-organizing maps. In: Rozenberg G, Bäck T, Kok JN (eds) Handbook of natural computing. Springer, Berlin, pp 585–622

    Google Scholar 

  • Van Ness JW (1973) Admissible clustering procedures. Biometrika 60(2):422–424

    MathSciNet  Google Scholar 

  • Vattani A (2009) \(K\)-means requires exponentially many iterations even in the plane. In: Proceedings of the 25th annual symposium on computational geometry, pp 324–332

  • Velho L, Gomez J, Sobreiro MVR (1997) Color image quantization by pairwise clustering. In: Proceedings of the 10th Brazilian symposium on computer graphics and image processing, pp 203–210

  • Verevka O, Buchanan JW (1995) Local \(K\)-means algorithm for colour image quantization. In: Proceedings of the graphics/vision interface conference, pp 128–135

  • Vuong J, Kaur S, Heinrich J et al (2018) Versus—a tool for evaluating visualizations and image quality using a 2AFC methodology. Visual Inf 2(4):225–234

    Google Scholar 

  • Wan X, Kuo CCJ (1998) A new approach to image retrieval with hierarchical color clustering. IEEE Trans Circuits Syst Video Technol 8(5):628–643

    Google Scholar 

  • Wan SJ, Wong SKM, Prusinkiewicz P (1988) An algorithm for multidimensional data clustering. ACM Trans Math Softw 14(2):153–162

    Google Scholar 

  • Wan SJ, Prusinkiewicz P, Wong SKM (1990) Variance-based color image quantization for frame buffer display. Color Res Appl 15:52–58

    Google Scholar 

  • Wang Z, Bovik AC (2009) Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process Mag 26(1):98–117

    Google Scholar 

  • Wang Z, Simoncelli EP, Bovik AC (2003) Multiscale structural similarity for image quality assessment. In: Proceedings of the 37th Asilomar conference on signals, systems and computers, pp 1398–1402

  • Wang Z, Bovik AC, Sheikh HR et al (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612

    Google Scholar 

  • Wang CH, Lee CN, Hsieh CH (2007) Sample-size adaptive self-organization map for color images quantization. Pattern Recogn Lett 28(13):1616–1629

    Google Scholar 

  • Wang Y, Pan Z, Li R (2018) Performance re-evaluation on “codewords distribution-based optimal combination of equal-average equal-variance equal-norm nearest neighbor fast search algorithm for vector quantization encoding. IEEE Trans Image Process 27(2):718–720

  • Wang Y, Huang H, Wang C, et al (2019) GIF2Video: color dequantization and temporal interpolation of GIF images. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 1419–1428

  • Wang S, Sun Y, Bao Z (2020) On the efficiency of \(K\)-means clustering: evaluation, optimization, and algorithm selection. Proc VLDB Endow 14(2):163–175

    Google Scholar 

  • Ward J (1963) Hierarchical grouping to optimize an objective function. J Am Stat Assoc 58(301):236–244

    MathSciNet  Google Scholar 

  • Watanabe T (1988) A fast algorithm for color image quantization using only 256 colors. Syst Comput Jpn 19(3):64–72

    MathSciNet  Google Scholar 

  • Wen Q, Celebi ME (2011) Hard vs. Fuzzy \(C\)-means clustering for color quantization. EURASIP J Adv Signal Process 1:118–129

    Google Scholar 

  • Wickelmaier F, Schmid C (2004) A Matlab function to estimate choice model parameters from paired-comparison data. Behav Res Methods Instrum Comput 36(1):29–40

    Google Scholar 

  • Williams WT (1971) Principles of clustering. Annu Rev Ecol Syst 2:303–326

    Google Scholar 

  • Wittek P, Gao SC, Lim IS et al (2017) somoclu: an efficient parallel library for self-organizing maps. J Stat Softw 78(9):1–21

    Google Scholar 

  • Wu X (1991a) Efficient statistical computations for optimal color quantization. In: Arvo J (ed) Graphics gems II. Academic Press, Cambridge, pp 126–133

    Google Scholar 

  • Wu X (1991b) Optimal quantization by matrix searching. J Algorithms 12(4):663–673

    MathSciNet  MATH  Google Scholar 

  • Wu X (1992a) Color quantization by dynamic programming and principal analysis. ACM Trans Graph 11(4):348–372

    MATH  Google Scholar 

  • Wu X (1992b) Statistical colour quantization for minimum distortion. In: Falcidieno B, Herman I, Pienovi C (eds) Computer graphics and mathematics. Springer, Berlin, pp 189–202

    Google Scholar 

  • Wu KL (2012) Analysis of parameter selections for Fuzzy \(C\)-means. Pattern Recogn 45(1):407–415

    MATH  Google Scholar 

  • Wu X, Witten IH (1985) A fast \(K\)-means type clustering algorithm. Tech. Rep. 85/197/10, University of Calgary

  • Wu KL, Yang MS (2006) Alternative learning vector quantization. Pattern Recogn 39(3):351–362

    MATH  Google Scholar 

  • Wu X, Zhang K (1993) Quantizer monotonicities and globally optimal scalar quantizer design. IEEE Trans Inf Theory 39(3):1049–1053

    MATH  Google Scholar 

  • Wu X, Kumar V, Quinlan JR et al (2008) Top 10 algorithms in data mining. Knowl Inf Syst 14(1):1–37

    Google Scholar 

  • Xiang Z (1997) Color image quantization by minimizing the maximum intercluster distance. ACM Trans Graph 16(3):260–276

    Google Scholar 

  • Xiang Z (2018) Color quantization. In: Gonzalez TF (ed) Handbook of approximation algorithms and metaheuristics, 2nd edn. CRC Press, Boca Raton, pp 691–709

    Google Scholar 

  • Xiang Z, Joy G (1994) Color image quantization by agglomerative clustering. IEEE Comput Graph Appl 14(3):44–48

    Google Scholar 

  • Xiao Y, Feng RB, Han ZF et al (2015) GPU accelerated self-organizing map for high dimensional data. Neural Process Lett 41(3):341–355

    Google Scholar 

  • Xie YF, Liu JH, Zhang CF et al (2016) Codewords distribution-based optimal combination of equal-average equal-variance equal-norm nearest neighbor fast search algorithm for vector quantization encoding. IEEE Trans Image Process 25(12):5806–5813

    MathSciNet  MATH  Google Scholar 

  • Xu Q, Jiang T, Yao Y, et al (2011) Random partial paired comparison for subjective video quality assessment via HodgeRank. In: Proceedings of the 19th ACM international conference on multimedia, pp 393–402

  • Yager RR, Filev DP (1994) Approximate clustering via the mountain method. IEEE Trans Syst Man Cybern 24(8):1279–1284

    Google Scholar 

  • Yair E, Zeger K, Gersho A (1992) Competitive learning and soft competition for vector quantizer design. IEEE Trans Signal Process 40(2):294–309

    Google Scholar 

  • Yang CY, Lin JC (1996) RWM-Cut for color image quantization. Comput Graph 20(4):577–588

    Google Scholar 

  • Yang CK, Tsai WH (1998) Color image compression using quantization, thresholding, and edge detection techniques all based on the moment-preserving principle. Pattern Recogn Lett 19(2):205–215

    MathSciNet  Google Scholar 

  • Yang MS, Wu KL, Hsieh JN et al (2008) Alpha-cut implemented Fuzzy clustering algorithms and switching regressions. IEEE Trans Syst Man Cybern Part B (Cybernetics) 38(3):588–603

    Google Scholar 

  • Ye P, Doermann D (2014) Active sampling for subjective image quality assessment. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 4249–4256

  • Yoo I, Luo X, Wang Y, et al (2020) GIFnets: differentiable GIF encoding framework. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 14473–14482

  • Yu CH, Chen SY (2006) Universal colour quantisation for different colour spaces. IEE Proc Vis Image Signal Process 153(4):445–455

    Google Scholar 

  • Yu MP, Lo KK (2003) Contextual algorithm for color quantization. J Electron Imaging 12(3):442–447

    Google Scholar 

  • Yu J, Yang MS (2005) Optimality test for generalized FCM and its application to parameter selection. IEEE Trans Fuzzy Syst 13(1):164–176

    Google Scholar 

  • Yu J, Cheng Q, Huang H (2004) Analysis of the weighting exponent in the FCM. IEEE Trans Syst Man Cybern Part B (Cybernetics) 34(1):634–639

    Google Scholar 

  • Yuan G, Goldberg M (1988) A sequential initialization technique for vector quantizer design. Pattern Recogn Lett 7(3):157–161

    Google Scholar 

  • Zhang X, Wandell BA (1997) A spatial extension of CIELAB for digital color-image reproduction. J Soc Inform Display 5(1):61–63

    Google Scholar 

  • Zhang X, Wandell BA (1998) Color image fidelity metrics evaluated using image distortion maps. Signal Process 70(3):201–214

    MATH  Google Scholar 

  • Zhang T, Ramakrishnan R, Livny M (1997) BIRCH: a new data clustering algorithm and its applications. Data Min Knowl Disc 1(2):141–182

    Google Scholar 

  • Zhao Y, Sheong FK, Sun J et al (2013) A fast parallel clustering algorithm for molecular simulation trajectories. J Comput Chem 34(2):95–104

    Google Scholar 

  • Zhou K, Yang S (2019) Fuzzifier selection in Fuzzy \(C\)-means from cluster size distribution perspective. Informatica 30(3):613–628

    MathSciNet  Google Scholar 

  • Zhou H, Schaefer G, Sadka A et al (2009) Anisotropic mean shift based Fuzzy \(C\)-means segmentation of dermoscopy images. IEEE J Select Top Signal Proces 3(1):26–34

    Google Scholar 

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This material is based upon work supported by the National Science Foundation under Award No. oia-1946391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Celebi, M.E. Forty years of color quantization: a modern, algorithmic survey. Artif Intell Rev 56, 13953–14034 (2023). https://doi.org/10.1007/s10462-023-10406-6

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