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Entropy and some applications in image processing

Entropy and some applications in image processing. Neucimar J. Leite Institute of Computing neucimar@ic.unicamp.br. Outline. Introduction Intuitive understanding Entropy as global information Entropy as local information edge detection, texture analysis

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Entropy and some applications in image processing

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  1. Entropy and some applications in image processing Neucimar J. Leite Institute of Computing neucimar@ic.unicamp.br

  2. Outline • Introduction • Intuitive understanding • Entropy as global information • Entropy as local information • edge detection, texture analysis • Entropy as minimization/maximization constraints • global thresholding • deconvolution problem

  3. Information Entropy (Shannon´s entropy) An information theory concept closely related to the following question: • What is the minimum amount of data needed to represent an • information content? • For images (compression problems): • How few data are sufficient to completely describe an images • without (much) loss of information?

  4. Intuitive understanding: • relates the amount of uncertainty about an event with a given • probability distribution Event: randomly draw out a ball low uncertainty high uncertainty no uncertainty entropy = max min (uncertainty)

  5. Self-information: - Units of information used to represent an event E Example 1: Event: a coin flipping = { heads, tails } Probability: P(heads) = P(tails) = 1/2 0  heads 1  tails self-information: inversely related to the probability of E

  6. Example 2: amount of conveyed information of event E Entropy: average information

  7. coding the balls (3 bits/ball) 0 0 0 0 0 1 0 1 0 Degree of information compression: 0 1 1 equal length binary code 1 0 0 1 0 1 1 1 0 1 1 1 for independent data: Entropy: = 3 bits/ball

  8. code 0 0 0 1 medium uncertainty: 1 0 1 1 H= -( 5/8 log2(5/8) + 1/8 log2(1/8) + 1/8 log2(1/8)+ 1/8 log2(1/8) ) = 1.54 no uncertainty: H = -8log21 = 0

  9. 2 bits/ball > 1.54 bit/ball  code redundancy !!! and code 0 0 0 1 medium uncertainty: 1 0 1 1 H= -( 5/8 log2(5/8) + 1/8 log2(1/8) + 1/8 log2(1/8)+ 1/8 log2(1/8) ) = 1.54 22%  We need an encoding method for eliminating this code redundancy

  10. The Huffman encoding: Ball Probability Reduction 1 Reduction 2 red 5/8 black 1/8 1/8 blue 1/8 green

  11. Ball Probability Reduction 1 Reduction 2 5/8 red 5/8 2/8 black 1/8 1/8 1/8 blue 1/8 green

  12. Ball Probability Reduction 1 Reduction 2 5/8 5/8 red 5/8 3/8 2/8 black 1/8 1/8 1/8 blue 1/8 green

  13. Ball Probability Reduction 1 Reduction 2 (1) 5/8 5/8 red 5/8 (0) 3/8 2/8 black 1/8 1/8 1/8 blue 1/8 green

  14. Ball Probability Reduction 1 Reduction 2 (1) (1) 5/8 5/8 red 5/8 (01) (0) 3/8 2/8 black 1/8 1/8 1/8 blue (00) 1/8 green

  15. Ball Probability Reduction 1 Reduction 2 (1) (1) 5/8 5/8 (1) red 5/8 (01) (0) 3/8 2/8 (00) black 1/8 (011) 1/8 1/8 blue (00) 1/8 green (010) variable length code ball red 1 black 00 and blue 011 (18,6%) green 010

  16. 512 x 512 8-bit image: Entropy: 4.11 bits/pixel After Huffman encoding: Variable length coding does not take advantage of the high images pixel-to-pixel correlation:  a pixel can be predicted from the values of its neighbors  more redundancy  lower entropy (bits/pixel)

  17. Entropy: 7.45 After Huffman encoding: Entropy: 7.35 After Huffman encoding:

  18. Coding the interpixel difference  highlighting redundancies: Entropy: 4.73 instead of 7.45 After Huffman encoding: instead of 1.07 Entropy: 5.97 instead of 7.35 After Huffman encoding: instead of 1.08

  19. Entropy as a local information: the edge detection example

  20. Edge detection examples:

  21. Entropy-based edge detection • Low entropy values  low frequencies  uniform image regions • High entropy values  high frequencies  image edges

  22. Binary entropy function: 1.0 Entropy H p 0 0.5 1.0

  23. 1.0 Entropy H p 0 0.5 1.0

  24. 1.0 Entropy H p 0 0.5 1.0

  25. 1.0 Entropy H p 0 0.5 1.0

  26. 1.0 Entropy H p 0 0.5 1.0

  27. 1.0 Entropy H p 0 0.5 1.0

  28. Binary entropy function: Isotropic edge detection

  29. H in a 3x3 neighborhood:

  30. 5x5 neighborhood:

  31. 7x7 neighborhood:

  32. 9x9 neighborhood:

  33. Texture Analysis • Similarity grouping based on brightness, colors, slopes, sizes etc • The perceived patterns of lightness, directionality, coarseness, • regularity, etc can be used to describe and segment an image

  34. Texture description: statistical approach • Characterizes textures as smooth, coarse, periodic, etc - Based on the intensity histogram  prob. density function Descriptors examples: • Mean: a measure of average intensity p(zi) = the intensity histogram in a region zi = random variable denoting gray levels

  35. Other moments of different orders: - e.g., standard deviation: a measure of average contrast Entropy: a measure of randomness

  36. smooth coarse periodic

  37. Descriptors and segmentation: ?

  38. 0 1 2 3 4 0 0 0 0 0 0 1 0 4 2 1 0 2 0 3 3 2 0 3 0 1 2 3 0 4 0 0 0 0 0 Gray-level co-occurrence matrix: Haralick´s descriptors • Conveys information about the positions of pixels having • similar gray level values. Md(a,b) • 2 1 3 3 2 1 • 3 2 2 2 1 1 • 3 2 2 1 1 3 • 1 3 1 2 1 1 3 d=1

  39. Md = the probability that a pixel with gray level i will have a pixel with level j a distance of d pixels away in a given direction For the descriptor H: large empty spaces in M  little information content cluttered areas  large information content d = 2, horizontal direction

  40. Obviously, more complex texture analysis based on statistical descriptors should consider combination of information related to image scale, moments, contrast, homogeneity, directionality, etc

  41. Entropy as minimization/maximization constraints

  42. Global thresholding examples: histogram peaks mean

  43. For images with levels 0-255: The probability that a given pixel will have value less than or equal t is: Now considering: Class A: Class B:

  44. The optimal threshold is the value of t that maximizes where

  45. Examples:

  46. Entropy as a fuzziness measure In fuzzy set theory an element x belongs to a set S with a certain probability pxdefined by a membership function px(x) Example of a membership function for a given threshold t: px(x) gives the degree to which x belongs to the object or background with gray-level average and , respectively.

  47. How can the degree of fuzziness be measured? Example: t = 0 for a a binary image  fuzziness = 0

  48. Using the Shannon´s function (for two classes): the entropy of an entire fuzzy set of dimension MxN is and for segmentation purpose, the threshold t is such that E(t) is minimum  t minimizes fuzziness

  49. Segmentation examples

  50. Maximum Entropy Restoration: the deconvolution problem

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