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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 1 The Fourier Transform Last week we talked about convolution as a technique for a variety of image processing applications (and we will see many more). Convolution works in the spatial domain, i.e., it uses spatial filters that are shifted across an input image. Sometimes it is useful to perform filtering in the frequency domain instead. To understand what this means and to perform such filtering, we need to take a look at the Fourier transform.

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 2 The Fourier Transform Jean Baptiste Joseph Fourier (and his Fourier-transformed image)

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 3 The Fourier Transform The Fourier transform receives a function as its input and outputs another function. The Fourier transform assumes that the input function is periodic, i.e., repetitive, and defined for all real numbers. It transforms this function into a weighted sum of sinusoidal terms. Let us first look at one-dimensional functions. Later, we will move on to two-dimensional functions such as images.

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 4 = 3 sin(x) A + 1 sin(3x) B A+B sin(5x) C A+B+C sin(7x)D A+B+C+D sin(x) A Adding Sine Waves

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 5 The Fourier Transform By adding a number of sine waves of different frequencies and amplitudes, we can approximate any given periodic function. However, if we limit ourselves to only sine waves with no offset, i.e., difference in phase at x = 0, we cannot perfectly reconstruct every function. For instance, the output of our function at x = 0 must always be 0. We could add a phase to each sine wave, but as Monsieur Fourier found out, it is sufficient to use both sine and cosine waves (which describe the same wave, but with a 90 phase difference) and keep all phases at 0.

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 6 The Fourier Transform So for any given function f, we could define a function F s that assigns to any frequency an amplitude (i.e., weight) that the sine wave of that frequency will receive in our sum of sine waves. Similarly, we could define a function F c that does the same for the cosine waves. In other words, these functions describe the contribution of different frequencies to the overall signal, our function f. While f describes our function in the spatial domain (i.e., what value does it have in what position x), F s and F c describe the same function in the frequency domain (i.e., the extent to which different frequencies are included in the function).

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 7 Refresher: Complex Numbers As you certainly remember, complex numbers consist of a real and an imaginary part, which we will use to represent the contribution of cosine and sine waves, respectively, to a given function. For example, 5 + 3i means Re = 5 and Im = 3. Importantly, the exponential function represents a rotation in the 2D space spanned by the real and imaginary axes: e i = cos + i sin cos 1 0 Im Re 0 sin

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 8 The Two-Dimensional Fourier Transform amplitude basis functions and phase of required basis functions Two-Dimensional Fourier Transform: Note: F(u,v) is complex, while f(x, y) is real. Two-Dimensional Inverse Fourier Transform:

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 9 2D Discrete Fourier Transform (u = 0,..., N-1; v = 0,…,M-1) (x = 0,..., N-1; y = 0,…,M-1)

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 10 2D Discrete Fourier Transform What do these equations mean? Well, if you look at the inverse transform, you see that each F(u, v) is multiplied by an exponential term for each pixel, and in sum they give us the original image. In other words, each exponential function represents one particular wave, and the sum of these waves, weighted by F, is our original image f. In the exponential term, you see an expression such as (ux + vy). This means that the variables u and v determine the “speed” with which the term increases when x and y, respectively, grow. Since this term determines the current angle of a wave, u and v determine the frequency of the waves in horizontal and vertical direction, respectively.

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 11

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 12 Magnitude And Phase Spectra Remember that we have both F s and F c to describe the sine and cosine contributions, respectively, to the spatial function. They can be used to compute the magnitude m: … and the phase :

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 13 Fourier Transform: Rectangle

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 14 Fourier Transform: Spatial Shifting

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 15 Fourier Transform: Rotation

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 16 Fourier Transform: Multiplicity

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 17 Fourier Transform: Real-World Images

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 18 Fourier Transform: Real-World Images

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 19 Fourier Transform: Real-World Images

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 20 Fourier Transform: Low-Pass Filtering

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 21 Fourier Transform: High-Pass Filtering

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September 30, 2014Computer Vision Lecture 7: The Fourier Transform 22 Fourier Transform: Noise Removal

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