1. The Capacity of Color Histogram Indexing Dong-Woei Lin 2003.3.6 NTUT CSIE

2. Outlines Preliminary Histogram and spatial information Effectiveness of histogram Histogram capacity M. Stricker, The capacity of color histogram indexing, ICCVPR, 1994 R. Brunelli, Histograms analysis for image retrieval, Pattern Recognition, 2001

3. Preliminary 1/4 Color histogram Incorporating spatial information Color coherence vector Correlogram (autocorrelogram) Proposed method Scale weighted (average distance of pixel pairs) Vector weighted (taking account of angle)

4. Preliminary 2/4 Performance evaluation (for CBIR) With relevant set through human subject: Precision: Recall: where A(q) and R(q) stands for answer set and relevant set for query image q respectively

5. Preliminary 3/4 Improving factor φ(for histogram-based) Histogram distance and similarity (based on vector norm or PDF)

6. Preliminary 4/4 Max.Min.MeanMean of top 10% 31.8%13.0%21.3%14.5% 45.7%15.2%26.0%17.0% 35.7%12.1%19.9%13.1% 40.6%14.7%24.6%15.9%

7. Histogram Space 1/2 For an image with N pixels, the histogram space ℌ is the subset of an n-dimensional vector space: ℌ For a given distance t : t-similar and t-different Identical (zero distance)

8. Histogram Space 2/2 Observations: The interval of reasonable values for t coincides with the first interval on the distance distribution increases very rapidly Indexing by color histograms works only if the histogram are sparse, i.e., most of the images contain only a fraction of the number of colors of the color space

9. The Capacity of Histogram Space 1/5 Definition of histogram capacity: C( ℌ, d, t), for a n-dimensional histogram space ℌ, a metric d, and a distance threshold t Assumption: uniform distribution across the color space

10. The Capacity of Histogram Space 2/5 Theorem: C( ℌ, d, t)  max w,l A(n, 2l, w) α =(wt/2N)  l  w  n, l  n/2 A(.) : the maximal number of codewords in any binary code of length n w : constant weight 2l : Hamming distance

11. The Capacity of Histogram Space 3/5 Using (1, 1, …, 0, 0, …, 1) to denote the histogram: a binary word of length n (number of bin) with exactly w 1 ’ s (non-zero bins) in it  each 1 represents the pixel number = N/w (w  n) 2l : the number of bins for two such histogram differ (l  w) n=64, w=62 N/62 11…..01….01..

12. The Capacity of Histogram Space 4/5 Distance of histogram H 1 and H 2 for d L1  t, solves l  wt/2N =  For any admissible w and l, the maximum of A(.) is still smaller than C

13. The Capacity of Histogram Space 5/5 Corollary for a computable lower bound: C( ℌ, d, t)  for L 1, l(w)=  wt/2N  q: smallest prime power such that q  n  = n

14. Histogram analysis for IR Revised notation of histogram capacity: Capacity curve C is defined as the density distribution of the dissimilarity through measure d between two elements of all possible histogram couples within a n- dimensional histogram space ℌ Capacity ℒ (t) =

15. Histogram analysis for IR Two major differences from Stricker(94) No distance function is defined Transforms difficult task “ maximal number ” into an empirical estimation by considering all image couples within the database The shape of C(t) Indicator of the distribution of histograms Induced by the selected dissimilarity measure The average value of dissimilarity represents the sparseness of histogram space ℌ

16. Histogram analysis for IR Indexing effectiveness ℰ = Can be used to assess several descriptor- dissimilarity combinations: Norm, distribution distance Chi-square, Kolmogorov-Smironv, Kuiper Hue, luminance, edgeness…

17. Histogram analysis for IR TestSet: 3500 images All 64 bins Rgb space: 4x4x4 Effectiveness: Hue=70 RGB=64

18. Experiments Establishments: RGB color space with 4x4x4 quantization Targets: Original image(uncompressed) DC image DC image with scalar-weighted Autocorrelogram of DC image Test sets: 47 320x240 JPEG images 150 192x128 JPEG images from Berkeley collections

19. Incorporating Spatial Info. Using mean dist. of all same-color pixel pairs as weight: Similarity measure: Mean value of DCT block For color j, image I 1 I 2 For intersection For Bhattacharyya * For compatible, the similarity will be transformed to dissimilarity * Intersection adopted only for comparison

20. Incorporating Spatial Info. Autocorrelogram of DC image: Color Dist. 0 0 … 0 1 1 … 1 …… 0 1 … d max 0 1 … d max …… Pair number p i,j : pair number of color i with distance j

21. Simulation Results I TypeCap. Original54.889 DC Image 56.105 DC w. SW64.172 Auto64.195 TestSet: nor.dat 47 320x240 images 1081 hist. Pairs

22. Simulation Results II TypeCap. Original58.028 DC Image 59.528 DC w. SW66.823 Auto68.184 TestSet: ber150.dat 150 192x128 images 11175 hist. Pairs

23. Semi-conclusion For histogram capacity: Autocorrelogram > scalar-weighted > DC image > original image The shape of autocorrelogram About the representation of curve

24. Spatial Histogram Capacity Spatial histogram (e.g. edgeness) Assessed features: E[dist] v.s. color # of pair v.s. pair dist

25. Simulation Result III TestsetCapacity nor4752.715 ber15031.834 nor47 ber150

26. Consistent of Last Exp. Considering the number of samples: Ber150(#)Capacity 5036.465 10031.447 15031.834

27. Future Works Types and properties of spatial histogram Study spatial descriptor Correlation of spatial and color features Sufficiency of definition of effectiveness