CS 691 Computational Photography Instructor: Gianfranco Doretto Frequency Domain

Overview Frequency domain analysis Sampling and reconstruction

Linear image transformations In analyzing images, it’s often useful to make a change of basis. Fourier transform, or Wavelet transform, or Steerable pyramid transform Vectorized image transformed image

Self-inverting transforms Same basis functions are used for the inverse transform U transpose and complex conjugate

Salvador Dali, “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976 Salvador Dali “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976

A nice set of basis This change of basis has a special name… Teases away fast vs. slow changes in the image.

Jean Baptiste Joseph Fourier ( ) had crazy idea (1807): Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies. Don’t believe it? –Neither did Lagrange, Laplace, Poisson and other big wigs –Not translated into English until 1878! But it’s (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 f(x) you want!

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

Frequency Spectra

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Fourier basis for image analysis Intensity Image Frequency Spectra

Signals can be composed += More:

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: Euler’s formula: Phase:

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

Fourier Transform pairs

Phase and Magnitude Fourier transform of a real function is complex –difficult to plot, visualize –instead, we can think of the phase and magnitude of the transform Phase is the phase of the complex transform Magnitude is the magnitude of the complex transform Curious fact – all natural images have about the same magnitude transform – hence, phase seems to matter, but magnitude largely doesn’t Demonstration – Take two pictures, swap the phase transforms, compute the inverse - what does the result look like?

This is the magnitude transform of the cheetah pic

This is the phase transform of the cheetah pic

This is the magnitude transform of the zebra pic

This is the phase transform of the zebra pic

Reconstructio n with zebra phase, cheetah magnitude

Reconstruction with cheetah phase, zebra magnitude

The Convolution Theorem The Fourier transform of the convolution of two functions is the product of their Fourier transforms The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse 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 * =

Filtering in frequency domain FFT Inverse FFT =

Play with FFT in Matlab Filtering with fft Displaying with fft im =... % “im” should be a gray-scale floating point image [imh, imw] = size(im); fftsize = 1024; % should be order of 2 (for speed) and include padding im_fft = fft2(im, fftsize, fftsize); % 1) fft im with padding hs = 50; % filter half-size fil = fspecial('gaussian', hs*2+1, 10); fil_fft = fft2(fil, fftsize, fftsize); % 2) fft fil, pad to same size as image im_fil_fft = im_fft.* fil_fft; % 3) multiply fft images im_fil = ifft2(im_fil_fft); % 4) inverse fft2 im_fil = im_fil(1+hs:size(im,1)+hs, 1+hs:size(im, 2)+hs); % 5) remove padding figure(1), imagesc(log(abs(fftshift(im_fft)))), axis image, colormap jet

Convolution versus FFT 1-d FFT: O(NlogN) computation time, where N is number of samples. 2-d FFT: 2N(NlogN), where N is number of pixels on a side Convolution: K N 2, where K is number of samples in kernel Say N=2 10, K= d FFT: , while convolution gives

Why is the Fourier domain particularly useful? It tells us the effect of linear convolutions. There is a fast algorithm for performing the DFT, allowing for efficient signal filtering. The Fourier domain offers an alternative domain for understanding and manipulating the image.

Analysis of our simple filters original 0 Pixel offset coefficient 1.0 Filtered (no change) constant

Analysis of our simple filters 0 Pixel offset coefficient original 1.0 shifted Constant magnitude, linearly shifted phase

Analysis of our simple filters 0 Pixel offset coefficient original 0.3 blurred Low-pass filter 0 1.0

Analysis of our simple filters original sharpened high-pass filter

Playing with the DFT of an image

Can change spectrum, then reconstruct

Low and High Pass filtering

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

Gaussian Filter

Box Filter

Overview Frequency domain analysis Sampling and reconstruction

Sampling and Reconstruction

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.14.14b / Wolberg]

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.14.14b / Wolberg]

1D Example: Audio lowhigh frequencies

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?

Sampling and Reconstruction Simple example: a sign wave

Undersampling What if we “missed” things between the samples? Simple example: undersampling a sine wave –unsurprising result: information is lost

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

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

Aliasing in video

Aliasing in images

What’s happening? Input signal: x = 0:.05:5; imagesc(sin((2.^x).*x)) Plot as image: Alias! Not enough samples

Fourier Transform of a Sampled Signal Sampling period T T 2T 3T … -F 0 F … Sampling frequency F = 1/T

Fourier Transform of a Sampled Signal Sampling period T T 2T 3T … -2F -F 0 F 2F … Sampling frequency F = 1/T

When sampling a signal at discrete intervals, the sampling frequency must be F > 2 x f max f max = max frequency of the input signal This will allow to reconstruct the original perfectly from the sampled version Do not know f max or cannot sample at that rate? Prefiltering! We lose something but still better than aliasing!!!!! good bad vvv Nyquist-Shannon Sampling Theorem

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

Image sub-sampling Throw away every other row and column to create a 1/2 size image - called image sub-sampling 1/4 1/8

Image sub-sampling 1/4 (2x zoom) 1/8 (4x zoom) Aliasing! What do we do? 1/2

Gaussian (lowpass) pre-filtering G 1/4 G 1/8 Gaussian 1/2 Solution: filter the image, then subsample Filter size should double for each ½ size reduction. Why?

Subsampling with Gaussian pre-filtering G 1/4G 1/8Gaussian 1/2

Compare with... 1/4 (2x zoom) 1/8 (4x zoom) 1/2

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);

Slide Credits This set of sides also contains contributions kindly made available by the following authors –Alexei Efros –Frédo Durand –Bill Freeman –Steve Seitz –Derek Hoiem –Steve Marschner