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Fourier Transform – Chapter 13

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Image space Cameras (regardless of wave lengths) create images in the spatial domain Pixels represent features (intensity, reflectance, heat, etc.) from the “real 3D world”

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Image space All operations we’ve looked at so far are applied in the spatial domain –Histogram (statistical) operations –Point operations –Filter (convolution) operations –Edge operations –Corner (feature) operations –Line (curve) detection –Morphological operations –Region operations –Color space

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Frequency domain As it turns out, all the spatial domain “signals” can be represented in the frequency domain And, some of the previously mentioned operations can be performed more efficiently in the frequency domain –Convolution –Filtering –Compression

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The Fourier Transform The Fourier Transform provides the means from moving between the spatial and frequency domains Developed in the area of sound processing –Decomposition of sound waves into elementary harmonic functions

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Preliminaries

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Sine and Cosine functions You’ve seen these many times before

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Sine and Cosine functions Frequency – cycles on horizontal axis

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Sine and Cosine functions Angular frequency and “common” frequency (f)

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Sine and Cosine functions Amplitude – increasing on vertical axis

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Sine and Cosine functions Phase – shifting on horizontal axis

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Sine and Cosine functions Orthogonality –We can combine sine and cosine waveforms with varying frequency, amplitude, and phase parameters to create other sine and cosine waveforms

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Sine and Cosine functions Orthogonality example

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Sine and Cosine functions Vector representation Complex number representation Euler notation (complex numbers on the unit circle) vector length ≡ amplitude

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Euler notation This brings us to the “complex-valued sinusoid” Since it’s on a unit circle the amplitude is And the phase is That is, multiplying by a real value alters the amplitude, multiplying by a complex value alters the phase

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Fourier Series Not only can sinusoidal functions [of varying frequency, amplitude, and phase] be combined to create other sinusoidal functions but… They can be combined to create almost and periodic function

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Fourier Series Frequencies kω 0 x are harmonics (multiples) of the fundamental A k and B k are derived via Fourier Analysis

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Fourier Integral But that wasn’t enough…Fourier wanted to cover non-periodic functions too Requires more than just integer multiples (harmonics) of the fundamental frequency Requires infinitely many frequencies

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Fourier Integral To solve for the amplitudes A ω and B ω we need the following integrals A ω and B ω form continuous functions of coefficients (corresponding to infinitely many, densely spaced frequencies) A ω and B ω form the Fourier Spectrum

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Fourier Transform Apply the Fourier Series to complex- valued functions using Euler’s notation to get the Fourier Transform And the inverse Fourier Transform

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Fourier Transform In general –A real-valued function yields a complex- valued Fourier Transform –A complex-valued function yields a read- valued Fourier Transform

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Fourier Transform For example, lots of sinusoidal waves can make up a square wave

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Fourier Transform Transform pairs are unique, one-to-one Dirac (delta) function

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Fourier Transform Properties –There are a bunch of properties that you can read about, but only one is “surprising” Convolution Property –Convolution in the spatial domain is point- by-point multiplication in the frequency domain

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Convolution Property Spatial domain –Perform a slide/multiply-accumulate operation with a kernel and the image Frequency domain –Fourier Transform kernel → spectrum –Fourier Transform image → spectrum –Multiply the two spectrums –Inverse Fourier Transform product → filtered image

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Discrete Signals So how do we apply these integrals to digital (discrete) images?

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Discrete Signals Next time

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Fourier series. The frequency domain It is sometimes preferable to work in the frequency domain rather than time –Some mathematical operations are easier.

Fourier series. The frequency domain It is sometimes preferable to work in the frequency domain rather than time –Some mathematical operations are easier.

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