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An introduction to Generalized K-means with image processing applications

An introduction to Generalized K-means with image processing applications. Jeremy Watt and Reza Borhani. What we’ll discuss. The cutting edge – Dictionary Learning and large scale image compression From K-means to G- Kmeans (aka Dictionary Learning)

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An introduction to Generalized K-means with image processing applications

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  1. An introduction to Generalized K-means with image processing applications Jeremy Watt and Reza Borhani

  2. What we’ll discuss • The cutting edge – Dictionary Learning and large scale image compression • From K-means to G-Kmeans (aka Dictionary Learning) • A tried and true application of G-K Means in Image Processing

  3. Dictionary Learning and Joint Image Compression

  4. Image compression via G-Kmeans An 8x8 block

  5. Image compression via G-Kmeans

  6. Image compression via G-Kmeans • Say we have a robust matrix D that can represent image blocks from like images well as a sparse sum of its columns So we essentially have where # nonzero values in Standard basis

  7. Image compression via G-Kmeans Image adapted from Irina Rish’s Sparse Statistical Models Course

  8. Image compression via G-Kmeans • Can record far fewer coefficients than pixel values! • Since sender and receiver both have Dictionary, we just send a few coefficients – much cheaper!

  9. How do we find the right fit? subject to • Greed is Good - what is a greedy approach to solving this problem? • Repeat S times • Cycle through each column of D not yet used and find the single best fit • Subtract the best column’s influence from y

  10. Francis Bach et al. Sparse Coding and Dictionary Learning for Image Analysis Part III: Optimization for Sparse Coding and Dictionary Learning

  11. How do we find the right fit? subject to • Greed is Good - what is a greedy approach to solving this problem? for

  12. The era of Big Data 250 million images uploaded to Facebook everyday. There are currently   ~ 90 billion photos totalon Facebook. In 2007, almost 70 million CT-scans performed only in the U.S. bits Joint Compressor JPEG JPEG JPEG bits bits bits Joint Compressor bits DICOM DICOM DICOM bits bits bits

  13. Joint image compressor

  14. From K-means to G-Kmeans (aka Dictionary Learning)

  15. What is data clustering? • A method for understanding the structure of a given (training) data set via clustering. • A way of classifying points in a newly received (test) data set • K-means is the most popular method

  16. Begin Training Phase First – Initialize centroids

  17. Next – Assign data

  18. Recalculate centroids

  19. Re-assign data

  20. Re-compute centroids

  21. Re-assign data

  22. Pick K centroids • Repeat until convergence • Assign each (training) point to its closest centroid • Re compute centroid locations • Output final K centroids and assignments

  23. Begin Testing Phase ?

  24. Which centroid is closest?

  25. Shortcomings • Number of centroids? • Initialization? • Are yours the right ones? • non-glob clusters • complexity

  26. Classification • Not the best, but not bad right out of the box • MNIST dataset – 60,000 training, 10,000 test • Compeitors include SVMs, Logistic regression, etc.

  27. (Pseudo) Image Compression • Per pixel: instead of (R,G,B) store index of closest centroid • Also called vector quantization • Image taken from Bishop’s Pattern Recognition

  28. For K Centroids

  29. Y For P Points

  30. = = x Assignment vector

  31. = = X For P Assignments

  32. Notice • Columns of X – assignment of one data point • Rows of X – all assignments to a single centroid =

  33. K-Means algorithm: Notation • Centroid location matrix • Data matrix • Assignment matrix

  34. K-Means algorithm: Notation • Centroid matrix • Data matrix • Assignment matrix • the kth cluster • the cardinality of the kth cluster

  35. K-means algorithm Input: Y, initial D and X Output: Final D and X Until Convergence: for p (Update Assignments) for (Update Centroids)

  36. K-means algorithm Input: Y, initial D and X Output: Final D and X Until Convergence: for p (Update Assignments) for (Update Centroids)

  37. K-means algorithm Input: Y, initial D and X Output: Final D and X Until Convergence: for p (Update Assignments) for (Update Centroids)

  38. K-means algorithm Input: Y, initial D and X Output: Final D and X Until Convergence: for p (Update Assignments) for (Update Centroids)

  39. K-means algorithm Input: Y, initial D and X Output: Final D and X Until Convergence: for p (Update Assignments) for (Update Centroids) The kth row of X

  40. For example say we • 4 data points,,, • 2 centroids , Then records which points are assigned to the first centroid. And

  41. What value of is minimizes the squared distances?

  42. Not quite…

  43. Not quite…

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