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Image Processing

Image Processing. Xuejin Chen xjchen99@ustc.edu.cn. Ref: http://fourier.eng.hmc.edu/e161/lectures/gradient/node10.html. Linear Filter. Smoothing Box, Bilinear, Gaussian. Linear Filter. Smoothing Box, Bilinear, Gaussian Edge Sobel. Linear Filter.

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Image Processing

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  1. Image Processing Xuejin Chen xjchen99@ustc.edu.cn Ref: http://fourier.eng.hmc.edu/e161/lectures/gradient/node10.html

  2. Linear Filter • Smoothing • Box, Bilinear, Gaussian

  3. Linear Filter • Smoothing • Box, Bilinear, Gaussian • Edge • Sobel

  4. Linear Filter • Smoothing • Box, Bilinear, Gaussian • Edge • Sobel • Corner

  5. Separable Filter • Convolution of K-size kernel requires K2 operations • Can be sped up to 2K operations by • First performing a 1D horizontal convolution • Followed by a 1D vertical convolution

  6. Separable Filter

  7. Fourier transformation

  8. Steerable Filters • Directional/Oriented filter • Sobel • Directional derivative A whole family of filters can be evaluated with very little cost by first convolving the image with (Gx, Gy)

  9. Steerable Filters • Second-order filter For directional Gaussian derivatives, it is possible to steer any order of derivative with a relatively small number of basis functions.

  10. Steerable Filters • Second-order filter Original image with oriented structures enhanced. Original image orientation map

  11. Steerable Filters • Fourth-order steerable filter test image containing bars (lines) and step edges at different orientations oriented energy as a function of angle average oriented energy dominant orientation (Freeman and Adelson 1991)

  12. Summed Area Table (Integral Image) • When an image needs to repeatedly convolved with different box filters (and especially filters of different sizes at different locations) • Precompute the summed area table (Crow1984)

  13. Summed Area Table (Integral Image) • Compute the sum of any rectangle area easily Recursive filtering

  14. Band-pass filters • Sobel, Corner • More sophisticated kernel: • Smooth image with a Gaussian filter • Take the first or second derivatives Laplacian Oriented Undirected

  15. Laplacian of Gaussian (LoG)

  16. LoG • Discrete convolution kernel • Can be any size • Sum_elements = Zero

  17. Laplacian of Gaussian (LoG)

  18. Difference of Gaussian (DoG) • Gaussian • DoG

  19. Difference of Gaussian (DoG)

  20. Difference of Gaussian (DoG)

  21. LoG and DoG

  22. LoG and DoG

  23. LoG and DoG LoG DoG

  24. Laplacian for Edge • Zero-crossing detection LoG DoG

  25. Pyramids • Change resolution • Upsampling (Interpolation) • Downsmapling (Decimation)

  26. Interpolation • Interpolation kernel h() with sampling rate r • Bilinear

  27. Interpolation • Interpolation kernel h() with sampling rate r • Bicubic interpolation

  28. Bicubic Interpolation • a specifies the derivative at x=1 • Usually a=-1, best matches the frequency characteristics of a sinc function • A small amount of sharpening • Ringing (does not linearly interpolate straight lines • Quadratic reproducing spline a=-0.5

  29. Bicubic Interpolation Bilinear Cubic a=-1 Cubic a=-0.5 windowed sinc

  30. Windowed sinc function • Best quality interpolator (Usually) • Both preserves details in the lower resolution image and avoids aliasing

  31. Windowed sinc function • Best quality interpolator (Usually) • Both preserves details in the lower resolution image and avoids aliasing • Ringing effect • Instead, repeatedly interpolate images by a small fractional amount

  32. Decimation (Downsampling) • Same kernel h(k,l) for both interpolation and decimation • Avoid aliasing • Convolve the image with a low-pass filter

  33. Decimation (Downsampling) • Linear • Binomial • Separating the high and low frequencies, • but leaves a fair amount of high-frequency detail, which can lead to aliasing after downsampling

  34. Decimation (Downsampling) • Linear • Binomial • Cubic • a=-1, • a=-0.5 • Windowed sinc • QMF-9 • Jpeg2000 Sample rate = 2

  35. Decimation (Downsampling) • Cubic a=-1 • Sharpest but ringing • QMF-9 and Jpeg2000 • Wavelet analysis filters • Useful for compression • More aliasing

  36. Multi-resolution Representations • Image pyramid • Accelerate coarse-to-fine search algorithms • Look for objects or patterns at different scales • Perform multi-resolution blending operations

  37. Multi-resolution Representations • Laplacian pyramid [Burt and Adelson’s (1983a)] • Best known and most widely used in computer vision

  38. Laplacian Pyramid • First: blur and subsample the original image with sample rate r=2 • Five-tap kernel Octave pyramid

  39. Laplacian pyramid • First: blur and subsample the original image by sample rate = 2 Gaussian pyramid: Repeated convolutions of the binomial kernel converge to a Gaussian

  40. Laplacian Pyramid

  41. Laplacian Pyramid • Actual computation of high-pass filter • Results in perfect reconstruction when Q=I Laplacian image Gaussian image

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