Image Filtering and Gaussian Pyramids

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1 Image Filtering and Gaussian Pyramids CS94: Image Manipulation & Computational Photography Alexei Efros, UC Berkeley, Fall 27

2 Limitations of Point Processing Q: What happens if I reshuffle all pixels within the image? A: It s histogram won t change. No point processing will be affected

3 What is an image? We can think of an image as a function, f, from R 2 to R: f( x, y ) gives the intensity at position ( x, y ) Realistically, we expect the image only to be defined over a rectangle, with a finite range: f: [a,b]x[c,d] [,] A color image is just three functions pasted together. We can write this as a vector-valued function: rxy (, ) f( xy, ) = gxy (, ) bxy (, )

4 Images as functions

5 Sampling and Reconstruction

6 Sampled representations How to store and compute with continuous functions? Common scheme for representation: samples write down the function s values at many points [FvDFH fig.4.4b / Wolberg] 26 Steve Marschner 6

7 Reconstruction Making samples back into a continuous function for output (need realizable method) for analysis or processing (need mathematical method) amounts to guessing what the function did in between [FvDFH fig.4.4b / Wolberg] 26 Steve Marschner 7

8 D Example: Audio low frequencies high

9 Sampling in digital audio Recording: sound to analog to samples to disc Playback: disc to samples to analog to sound again how can we be sure we are filling in the gaps correctly? 26 Steve Marschner 9

10 Sampling and Reconstruction Simple example: a sign wave 26 Steve Marschner

11 Undersampling What if we missed things between the samples? Simple example: undersampling a sine wave unsurprising result: information is lost 26 Steve Marschner

12 Undersampling What if we missed things between the samples? Simple example: undersampling a sine wave unsurprising result: information is lost surprising result: indistinguishable from lower frequency 26 Steve Marschner 2

13 Undersampling What if we missed things between the samples? Simple example: undersampling a sine wave unsurprising result: information is lost surprising result: indistinguishable from lower frequency also, was always indistinguishable from higher frequencies aliasing: signals traveling in disguise as other frequencies 26 Steve Marschner 3

14 Aliasing in video Slide by Steve Seitz

15 Aliasing in images

16 What s happening? Input signal: Plot as image: x = :.5:5; imagesc(sin((2.^x).*x)) Alias! Not enough samples

17 Antialiasing What can we do about aliasing? Sample more often Join the Mega-Pixel craze of the photo industry But this can t go on forever Make the signal less wiggly Get rid of some high frequencies Will loose information But it s better than aliasing

18 Preventing aliasing Introduce lowpass filters: remove high frequencies leaving only safe, low frequencies choose lowest frequency in reconstruction (disambiguate) 26 Steve Marschner 8

19 Linear filtering: a key idea Transformations on signals; e.g.: bass/treble controls on stereo blurring/sharpening operations in image editing smoothing/noise reduction in tracking Key properties linearity: filter(f + g) = filter(f) + filter(g) shift invariance: behavior invariant to shifting the input delaying an audio signal sliding an image around Can be modeled mathematically by convolution 26 Steve Marschner 9

20 Moving Average basic idea: define a new function by averaging over a sliding window a simple example to start off: smoothing 26 Steve Marschner 2

21 Moving Average Can add weights to our moving average Weights [,,,,,,,, ] / 5 26 Steve Marschner 2

22 Cross-correlation Let be the image, be the kernel (of size 2k+ x 2k+), and be the output image This is called a cross-correlation operation: Can think of as a dot product between local neighborhood and kernel for each pixel

23 In 2D: box filter h[, ] Slide credit: David Lowe (UBC)

24 Image filtering h[, ] f [.,.] g[.,.] g[ m, n] = h[ k, l] k, l f [ m + k, n + l] Credit: S. Seitz

25 Image filtering h[, ] f [.,.] g[.,.] g[ m, n] = h[ k, l] k, l f [ m + k, n + l] Credit: S. Seitz

26 Image filtering h[, ] f [.,.] g[.,.] Credit: S. Seitz

27 Image filtering h[, ] f [.,.] g[.,.] Credit: S. Seitz

28 Image filtering h[, ] f [.,.] g[.,.] Credit: S. Seitz

29 Image filtering h[, ] f [.,.] g[.,.] ? 9 Credit: S. Seitz

30 Image filtering h[, ] f [.,.] g[.,.] ? 5 9 Credit: S. Seitz

31 Image filtering h[, ] f [.,.] g[.,.] g[ m, n] = h[ k, l] k, l f [ m + k, n + l] Credit: S. Seitz

32 Box Filter What does it do? h[, ] Replaces each pixel with an average of its neighborhood Achieve smoothing effect (remove sharp features) Slide credit: David Lowe (UBC)

33 Linear filters: examples = Original Blur (with a mean filter) Source: D. Lowe

34 Practice with linear filters? Original Source: D. Lowe

35 Practice with linear filters Original Filtered (no change) Source: D. Lowe

36 Practice with linear filters? Original Source: D. Lowe

37 Practice with linear filters Original Shifted left By pixel Source: D. Lowe

38 Other filters Sobel Vertical Edge (absolute value)

39 Other filters Q? Sobel Horizontal Edge (absolute value)

40 Back to the box filter

41 Moving Average Can add weights to our moving average Weights [,,,,,,,, ] / 5 26 Steve Marschner 43

42 Weighted Moving Average bell curve (gaussian-like) weights [,, 4, 6, 4,, ] 26 Steve Marschner 44

43 Moving Average In 2D What are the weights H? Steve Marschner Slide by 45 Steve Seitz

44 Gaussian filtering A Gaussian kernel gives less weight to pixels further from the center of the window This kernel is an approximation of a Gaussian function: 46 Slide by Steve Seitz

45 Mean vs. Gaussian filtering 47 Slide by Steve Seitz

46 Important filter: Gaussian Weight contributions of neighboring pixels by nearness x 5, σ = 48 Slide credit: Christopher Rasmussen

47 Gaussian Kernel σ = 2 with 3 x 3 kernel σ = 5 with 3 x 3 kernel Standard deviation σ: determines extent of smoothing 49 Source: K. Grauman

48 Gaussian filters = pixel = 5 pixels = pixels = 3 pixels

49 Choosing kernel width The Gaussian function has infinite support, but discrete filters use finite kernels 5 Source: K. Grauman

50 Practical matters How big should the filter be? Values at edges should be near zero Rule of thumb for Gaussian: set filter half-width to about 3 σ 52 Side by Derek Hoiem

51 Cross-correlation vs. Convolution cross-correlation: A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image: It is written: Convolution is commutative and associative Slide by Steve Seitz

52 Convolution Adapted from F. Durand

53 Convolution is nice! Notation: Convolution is a multiplication-like operation commutative associative distributes over addition scalars factor out identity: unit impulse e = [,,,,,, ] Conceptually no distinction between filter and signal Usefulness of associativity often apply several filters one after another: (((a * b ) * b 2 ) * b 3 ) this is equivalent to applying one filter: a * (b * b 2 * b 3 ) 26 Steve Marschner 55

54 Gaussian and convolution Removes high-frequency components from the image (low-pass filter) Convolution with self is another Gaussian * = Convolving twice with Gaussian kernel of width = convolving once with kernel of width Source: K. Grauman

55 Image half-sizing This image is too big to fit on the screen. How can we reduce it? How to generate a halfsized version?

56 Image sub-sampling /8 /4 Throw away every other row and column to create a /2 size image - called image sub-sampling Slide by Steve Seitz

57 Image sub-sampling /2 /4 (2x zoom) /8 (4x zoom) Aliasing! What do we do? Slide by Steve Seitz

58 Sampling an image Examples of GOOD sampling

59 Undersampling Examples of BAD sampling -> Aliasing

60 Gaussian (lowpass) pre-filtering G /8 G /4 Gaussian /2 Solution: filter the image, then subsample Filter size should double for each ½ size reduction. Why? Slide by Steve Seitz

61 Subsampling with Gaussian pre-filtering Gaussian /2 G /4 G /8 Slide by Steve Seitz

62 Compare with... /2 /4 (2x zoom) /8 (4x zoom) Slide by Steve Seitz

63 Gaussian (lowpass) pre-filtering G /8 G /4 Gaussian /2 Solution: filter the image, then subsample Filter size should double for each ½ size reduction. Why? How can we speed this up? Slide by Steve Seitz

64 Image Pyramids Known as a Gaussian Pyramid [Burt and Adelson, 983] In computer graphics, a mip map [Williams, 983] A precursor to wavelet transform Slide by Steve Seitz

65 A bar in the big images is a hair on the zebra s nose; in smaller images, a stripe; in the smallest, the animal s nose Figure from David Forsyth

66 The whole pyramid is only 4/3 the size of the original image! Slide by Steve Seitz Gaussian pyramid construction filter mask Repeat Filter Subsample Until minimum resolution reached can specify desired number of levels (e.g., 3-level pyramid)

67 What are they good for? Improve Search Search over translations Classic coarse-to-fine strategy Search over scale Template matching E.g. find a face at different scales

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