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

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

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)

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

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

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%

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)

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

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

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

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..

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

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

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) =

Future Works Proceeds study of capacity How to cooperate with previous work?