slide credit fei fei li. image filtering image filtering: compute function of local neighborhood at...

Slide credit Fei Fei Li

Image filtering Image filtering: compute function of local neighborhood at each position Really important! Enhance images Denoise, resize, increase contrast, etc. Extract information from images Texture, edges, distinctive points, etc. Detect patterns Template matching

111 111 111 Slide credit: David Lowe (UBC) Example: box filter

0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 0 0000000000 0000000000 000 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 Credit: S. Seitz Image filtering 111 111 111

0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 010 0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 Image filtering 111 111 111 Credit: S. Seitz

0102030 0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 Image filtering 111 111 111 Credit: S. Seitz

0102030 0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 Image filtering 111 111 111 Credit: S. Seitz ?

0102030 50 0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 Image filtering 111 111 111 Credit: S. Seitz ?

0000000000 0000000000 00090 00 000 00 000 00 000 0 00 000 00 0000000000 00 0000000 0000000000 0102030 2010 0204060 4020 0306090 6030 0 5080 906030 0 5080 906030 0203050 604020 102030 2010 00000 Image filtering 111 111 111 Credit: S. Seitz

What does it do? Replaces each pixel with an average of its neighborhood Achieve smoothing effect (remove sharp features) 111 111 111 Slide credit: David Lowe (UBC) Box Filter

Smoothing with box filter

Practice with linear filters 000 010 000 Original ? Source: D. Lowe

Practice with linear filters 000 010 000 Original Filtered (no change) Source: D. Lowe

Practice with linear filters 000 100 000 Original ? Source: D. Lowe

Practice with linear filters 000 100 000 Original Shifted left By 1 pixel Source: D. Lowe

Practice with linear filters Original 111 111 111 000 020 000 - ? (Note that filter sums to 1) Source: D. Lowe

Practice with linear filters Original 111 111 111 000 020 000 - Sharpening filter - Accentuates differences with local average Source: D. Lowe

Sharpening Source: D. Lowe

Other filters 01 -202 01 Vertical Edge (absolute value) Sobel

Other filters -2 000 121 Horizontal Edge (absolute value) Sobel

Filtering vs. Convolution 2d filtering 2d convolution f=imageg=filter

Key properties of linear filters Linearity: filter(f 1 + f 2 ) = filter(f 1 ) + filter(f 2 ) Shift invariance: same behavior regardless of pixel location filter(shift(f)) = shift(filter(f)) Any linear, shift-invariant operator can be represented as a convolution Source: S. Lazebnik

More properties Commutative: a * b = b * a Conceptually no difference between filter and signal But particular filtering implementations might break this equality Associative: a * (b * c) = (a * b) * c Often apply several filters one after another: (((a * b 1 ) * b 2 ) * b 3 ) This is equivalent to applying one filter: a * (b 1 * b 2 * b 3 ) Distributes over addition: a * (b + c) = (a * b) + (a * c) Scalars factor out: ka * b = a * kb = k (a * b) Identity: unit impulse e = [0, 0, 1, 0, 0], a * e = a Source: S. Lazebnik

Weight contributions of neighboring pixels by nearness 0.003 0.013 0.022 0.013 0.003 0.013 0.059 0.097 0.059 0.013 0.022 0.097 0.159 0.097 0.022 0.013 0.059 0.097 0.059 0.013 0.003 0.013 0.022 0.013 0.003 5 x 5, = 1 Slide credit: Christopher Rasmussen Important filter: Gaussian

Smoothing with Gaussian filter

Smoothing with box filter

Gaussian filters Remove high-frequency components from the image (low-pass filter) Images become more smooth Convolution with self is another Gaussian So can smooth with small-width kernel, repeat, and get same result as larger-width kernel would have Convolving two times with Gaussian kernel of width is same as convolving once with kernel of width 2 Separable kernel Factors into product of two 1D Gaussians Source: K. Grauman

Separability of the Gaussian filter Source: D. Lowe

Separability example * * = = 2D convolution (center location only) Source: K. Grauman The filter factors into a product of 1D filters: Perform convolution along rows: Followed by convolution along the remaining column:

Separability Why is separability useful in practice?

Practical matters What about near the edge? the filter window falls off the edge of the image need to extrapolate methods: clip filter (black) wrap around copy edge reflect across edge Source: S. Marschner

Practical matters What is the size of the output? MATLAB: filter2(g, f, shape) shape = full: output size is sum of sizes of f and g shape = same: output size is same as f shape = valid: output size is difference of sizes of f and g f gg gg f gg g g f gg gg full samevalid Source: S. Lazebnik

2006 Steve Marschner 37 Median filters A Median Filter operates over a window by selecting the median intensity in the window. What advantage does a median filter have over a mean filter? Is a median filter a kind of convolution? Slide by Steve Seitz

2006 Steve Marschner 38 Comparison: salt and pepper noise Slide by Steve Seitz

Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts? GaussianBox filter

Hybrid Images A. Oliva, A. Torralba, P.G. Schyns, Hybrid Images, SIGGRAPH 2006 Hybrid Images,

Why do we get different, distance-dependent interpretations of hybrid images? ? Slide: Hoiem

Why does a lower resolution image still make sense to us? What do we lose? Image: http://www.flickr.com/photos/igorms/136916757/http://www.flickr.com/photos/igorms/136916757/Slide: Hoiem

Thinking in terms of frequency

Jean Baptiste Joseph Fourier (1768-1830) had crazy idea (1807): Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies. Dont believe it? Neither did Lagrange, Laplace, Poisson and other big wigs Not translated into English until 1878! But its (mostly) true! called Fourier Series there are some subtle restrictions...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. Laplace Lagrange Legendre

A sum of sines Our building block: Add enough of them to get any signal g(x) you want!

Frequency Spectra example : g(t) = sin(2f t) + (1/3)sin(2(3f) t) = + Slides: Efros

Frequency Spectra

Example: Music We think of music in terms of frequencies at different magnitudes Slide: Hoiem

Other signals We can also think of all kinds of other signals the same way xkcd.com

Fourier analysis in images Intensity Image Fourier Image http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering

Signals can be composed += http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering More: http://www.cs.unm.edu/~brayer/vision/fourier.html

Fourier Transform Fourier transform stores the magnitude and phase at each frequency Magnitude encodes how much signal there is at a particular frequency Phase encodes spatial information (indirectly) For mathematical convenience, this is often notated in terms of real and complex numbers Amplitude:Phase:

Computing the Fourier Transform Continuous Discrete k = -N/2..N/2 Fast Fourier Transform (FFT): NlogN

The Convolution Theorem The Fourier transform of the convolution of two functions is the product of their Fourier transforms Convolution in spatial domain is equivalent to multiplication in frequency domain!

Properties of Fourier Transforms Linearity Fourier transform of a real signal is symmetric about the origin The energy of the signal is the same as the energy of its Fourier transform See Szeliski Book (3.4)

Filtering in spatial domain 01 -202 01 * =

Filtering in frequency domain FFT Inverse FFT = Slide: Hoiem

Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts? GaussianBox filter Filtering

Gaussian

Box Filter

Why does a lower resolution image still make sense to us? What do we lose? Image: http://www.flickr.com/photos/igorms/136916757/http://www.flickr.com/photos/igorms/136916757/ Sampling

Throw away every other row and column to create a 1/2 size image Subsampling by a factor of 2

1D example (sinewave): Source: S. Marschner Aliasing problem

Source: S. Marschner 1D example (sinewave): Aliasing problem

Sub-sampling may be dangerous. Characteristic errors may appear: Wagon wheels rolling the wrong way in movies Checkerboards disintegrate in ray tracing Striped shirts look funny on color television Source: D. Forsyth Aliasing problem

Aliasing in video Slide by Steve Seitz

Source: A. Efros Aliasing in graphics

Sampling and aliasing

When sampling a signal at discrete intervals, the sampling frequency must be 2 f max f max = max frequency of the input signal This will allows to reconstruct the original perfectly from the sampled version good bad vvv Nyquist-Shannon Sampling Theorem

Anti-aliasing Solutions: Sample more often Get rid of all frequencies that are greater than half the new sampling frequency Will lose information But its better than aliasing Apply a smoothing filter

Algorithm for downsampling by factor of 2 1.Start with image(h, w) 2.Apply low-pass filter im_blur = imfilter(image, fspecial(gaussian, 7, 1)) 3.Sample every other pixel im_small = im_blur(1:2:end, 1:2:end);

Anti-aliasing Forsyth and Ponce 2002

Subsampling without pre-filtering 1/4 (2x zoom) 1/8 (4x zoom) 1/2 Slide by Steve Seitz

Subsampling with Gaussian pre-filtering G 1/4G 1/8Gaussian 1/2 Slide by Steve Seitz

Why does a lower resolution image still make sense to us? What do we lose? Image: http://www.flickr.com/photos/igorms/136916757/http://www.flickr.com/photos/igorms/136916757/

Why do we get different, distance-dependent interpretations of hybrid images? ?

Salvador Dali invented Hybrid Images? Salvador Dali Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln, 1976

Application: Hybrid Images A. Oliva, A. Torralba, P.G. Schyns, Hybrid Images, SIGGRAPH 2006 Hybrid Images,

Early processing in humans filters for various orientations and scales of frequency Perceptual cues in the mid-high frequencies dominate perception When we see an image from far away, we are effectively subsampling it Early Visual Processing: Multi-scale edge and blob filters Clues from Human Perception

Hybrid Image in FFT Hybrid ImageLow-passed ImageHigh-passed Image

Why do we get different, distance-dependent interpretations of hybrid images? ? Perception

Things to Remember Sometimes it makes sense to think of images and filtering in the frequency domain Fourier analysis Can be faster to filter using FFT for large images (N logN vs. N 2 for auto- correlation) Images are mostly smooth Basis for compression Remember to low-pass before sampling

Sharpening revisited What does blurring take away? original smoothed (5x5) detail = sharpened = Lets add it back: originaldetail +

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