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SYDE 575: Introduction to Image Processing

Image Enhancement by Smoothing Textbook Sections 3.4 – 3.5, 5.3.1-5.3.2. SYDE 575: Introduction to Image Processing. Noise Reduction by Image Averaging. We want to be able to reduce the amount of noise in an image Assume that the noise process has a mean of zero

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SYDE 575: Introduction to Image Processing

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  1. Image Enhancement by Smoothing Textbook Sections 3.4 – 3.5, 5.3.1-5.3.2 SYDE 575: Introduction to Image Processing

  2. Noise Reduction by Image Averaging • We want to be able to reduce the amount of noise in an image • Assume that the noise process has a mean of zero • One approach is to take many images f(x,y) of the same scene and use these to generate a noise-free image g(x,y)

  3. Noise Reduction by Image Averaging • Assume noise is additive and Gaussian distributed with zero mean • Suppose we take the average of q number of noise samples at a point in the image

  4. Noise Reduction by Image Averaging • Now, apply concept to each pixel in the whole image • Given an infinite number of noise samples, the average approaches the mean of the distribution, which in this case is 0

  5. Example Source: Gonzalez and Woods

  6. Noise Reduction by Spatial Filtering • Image averaging takes advantage of information redundancy from the individual images to reduce noise • Not always possible to acquire so many images! • Images may not be perfectly registered so errors due to spatial misalignment • Alternative option: Take advantage of information redundancy from different pixels within the same image to reduce noise

  7. Spatial Filtering - Convolution • 2D discrete convolution output input point spread function (2d) or impulse response (1d) • Impulse response h[m,n] can be viewed as a spatial filter for an input image f[m,n] to produce output image g[m,n]

  8. Spatial Filtering • Let us represent w(x,y) as a 2D convolution mask Source: Gonzalez and Woods

  9. Spatial Filtering • Spatial filtering of an image f with 2D convolution using symmetrical mask w of size m x n can be expressed as output convolution mask; the “system” input

  10. Spatial Filtering Source: Gonzalez and Woods

  11. Local Average Filter for Smoothing • Instead of averaging between images, we can average neighboring pixels using the following point spread function (PSF) • Recall similar 1-d mask [1/3 1/3 1/3] • Why is the 1/9 required?

  12. Averaging Filter • Removes noise but also blurs Source: Gonzalez and Woods

  13. Properties of Averaging Filter • Linear? • Shift Invariant? • Memory? • Causal? • Stable? • Invertible?

  14. Weighted Average Filter • Problem: Simple averaging of neighboring pixels lead to over-smoothing • Possible solution: Instead of weighting all neighboring pixels equally, assign higher weights to pixels that are closer to the input pixel

  15. Weighted Averaging Filter: Example Source: Gonzalez and Woods

  16. Weighted Averaging Filter: Example Average Weighted Average Noisy

  17. Gaussian Smoothing • A weighted average can be produced using a Gaussian as a weight • In 1d • In 2d

  18. 1D Gaussian Source: Wikipedia

  19. 2d Gaussian Source: Wikipedia

  20. Characteristics of Local Averaging • h(m,n) >= 0 for all (m,n) • S h(m,n) = 1 (DC gain is 1) • Non-causal • Typically odd-dimensions (practical) • Even symmetry

  21. Order-Statistic Filters • Nonlinear spatial filters, i.e., a1f1 (m,n) + a2f2 (m,n) ≠ a1g1 (m,n) + a2g2 (m,n) • Best known example: median filter

  22. Median Filter • Provides good noise reduction for certain types of noise such as impulse noise • Considerably less blurring than weighted averaging filter • Forces a pixel to be like its neighbors • Steps • Order pixels within an area • Replace value of center pixel with median value (half of all pixels have intensities greater than or equal to the median value)

  23. Median Filter: Example 255 10 9 255 10 9 10 255 10 10 10 10 8 10 10 8 10 10 8 9 10 10 10 10 10 255 255 median=10

  24. Median Filter: Example Source: Gonzalez and Woods

  25. S-statistic Filter • Another method to avoid impact of outliers on smoothing is the S-statistic filter • Basically, only use those values within a region that are within a certain range relative to the mean • Typically, this is done by calculating the m and s of the local region and determining a local average based only on pixels within a certain number of s

  26. Recap: Spatial smoothing filters • All of the filters discussed so far are spatial smoothing filters • Weight of each pixel in the neighborhood covered by the filter depends on the proximity of the pixel to the center pixel being filtered • Typically, the closer the pixel is to the center pixel, the higher the weight

  27. Range Smoothing Filters • Problem: oversmoothing of edges and other fine image detail using ideal and Gaussian filters • Alternative solution: range smoothing filters • Weight of each pixel in the neighborhood covered by the filter depends on the similarity of the pixel's intensity value to that of the center pixel being filtered • The closer the pixel's intensity value is to that of the center pixel, the higher the weight

  28. Bilateral Filtering • Problem: range smoothing filter simply remaps intensity values • No notion of space • Poor noise reduction performance (in fact, not really useful by itself) • Idea: Combine spatial smoothing filters with range smoothing filters! • Good noise reduction (Spatial smoothing) • Good edge and detail preservation (Range smoothing)

  29. Bilateral Filtering • Resulting filter is non-linear Range smoothing filter Spatial smoothing filter

  30. Example • Suppose we use Gaussian models for the range and spatial smoothing filters Source: Tomasi et al. 1998

  31. Uses: Noise Suppression noisy Gaussian spatial filter Bilateral filter

  32. Uses: Special Effects • Bilateral filter reduces small image details while preserving large edge details • What would happen if we apply bilateral filtering multiple times? • More and more smaller details get smoothed out • Large edges remain well-preserved • Result: cartoon-like image

  33. Uses: Special Effects original 5 iterations

  34. Implementing Symmetrical 2-D Filters • Large images in use today (e.g., 1k x 1k, 5k x 5k, and even larger) require increased computational requirements • Symmetrical filters are commonly used in image processing systems • There is a faster way to implement a 2-d symmetrical mask using 2 1-d arrays

  35. Example: 2d as 2 x 1d * 1/9 Smoothing Mask – Note Symmetry “Image”

  36. Representing 2d Mask as 2 x 1d 1/9 = 1/3 * 1/3

  37. Top row and bottom row are equivalent 1/9 * = 1/9 1/3 * 1/3 * = same

  38. Computational Complexity • Assume M by N image and m by n filter mask • For 2d mask, number of multiplications is: M x N x m x n • For 2 1d masks, number of multiplication is M x N x (m + n) • So, for a 15 by 15 mask, 2d implementation is 225MN and 1d implementation is 30MN • Huge Savings! But note that 1d implementation requires storing an additional image in memory

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