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CSCE 641 Computer Graphics: Image Sampling and Reconstruction. Jinxiang Chai. Review: 1D Fourier Transform. A function f(x) can be represented as a weighted combination of phase-shifted sine waves How to compute F(u) ?. Inverse Fourier Transform. Fourier Transform. Review: Box Function.

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## CSCE 641 Computer Graphics: Image Sampling and Reconstruction

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**CSCE 641 Computer Graphics:Image Sampling and Reconstruction**Jinxiang Chai**Review: 1D Fourier Transform**A function f(x) can be represented as a weighted combination of phase-shifted sine waves How to compute F(u)? Inverse Fourier Transform Fourier Transform**Review: Box Function**f(x) x |F(u)| u If f(x) is bounded, F(u) is unbounded**Review: Cosine** -1 1 If f(x) is even, so is F(u)**Review: Gaussian**If f(x) is gaussian, F(u) is also guassian.**Review: Properties**Linearity: Time shift: Derivative: Integration: Convolution:**Review: Properties**Linearity: Time shift: Derivative: Integration: Convolution:**Review: Properties**Linearity: Time shift: Derivative: Integration: Convolution:**Outline**2D Fourier Transform Nyquist sampling theory Antialiasing Gaussian pyramid**Extension to 2D**Fourier Transform: Inverse Fourier transform:**Building Block for 2D Transform**Building block: Frequency: Orientation: Oriented wave fields**Building Block for 2D Transform**Building block: Frequency: Orientation: Oriented wave fields A function f(x,y) can be represented as a weighted combination of phase-shifted oriented wave fields. Higher frequency**Some 2D Transforms**From Lehar**Some 2D Transforms**From Lehar**Some 2D Transforms**From Lehar**Some 2D Transforms**From Lehar**Some 2D Transforms**From Lehar**Some 2D Transforms**Why we have a DC component? From Lehar**Some 2D Transforms**Why we have a DC component? - the sum of all pixel values From Lehar**Some 2D Transforms**Why we have a DC component? - the sum of all pixel values Oriented stripe in spatial domain = an oriented line in spatial domain From Lehar**2D Fourier Transform**Why? - Any relationship between two slopes?**2D Fourier Transform**Why? - Any relationship between two slopes? Linearity**2D Fourier Transform**Why? - Any relationship between two slopes? Linearity The spectrum is bounded by two slopes.**Online Java Applet**http://www.brainflux.org/java/classes/FFT2DApplet.html**2D Fourier Transform Pairs**Gaussian Gaussian**2D Image Filtering**Fourier transform Inverse transform From Lehar**2D Image Filtering**Fourier transform Inverse transform Low-pass filter From Lehar**2D Image Filtering**Fourier transform Inverse transform Low-pass filter high-pass filter From Lehar**2D Image Filtering**Fourier transform Inverse transform Low-pass filter high-pass filter band-pass filter From Lehar**Aliasing**Why does this happen?**Aliasing**How to reduce it?**f(x)**x fs(x) x … … -2T -T 0 T 2T Sampling Analysis Sampling**f(x)**x fs(x) x … … -2T -T 0 T 2T Sampling Analysis Sampling Reconstruction**f(x)**x fs(x) x … … -2T -T 0 T 2T Sampling Analysis What sampling rate (T) is sufficient to reconstruct the continuous version of the sampled signal? Sampling Reconstruction**Sampling Theory**• How many samples are required to represent a given signal without loss of information? • What signals can be reconstructed without loss for a given sampling rate?**fs(x)**x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function ? discrete signal (samples)**fs(x)**x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function ? What happens in Frequency domain? discrete signal (samples)**fs(x)**x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function ? What happens in Frequency domain? discrete signal (samples)**Review: Dirac Delta and its Transform**f(x) x |F(u)| 1 u Fourier transform and inverse Fourier transform are qualitatively the same, so knowing one direction gives you the other**Review: Fourier Transform Properties**Linearity: Time shift: Derivative: Integration: Convolution:**Fourier Transform of Dirac Comb**T 1/T - Fourier transform of a Dirac comb is a Dirac comb as well. - Moving the spikes closer together in the spatial domain moves them farther apart in the frequency domain!**fs(x)**x … … -2T -T 0 T 2T Sampling Analysis: Spatial Domain f(x) X … … x -2T -T 0 T 2T x continuous signal comb function What happens in Frequency domain? ? discrete signal (samples)**Review: Properties**Linearity: Time shift: Derivative: Integration: Convolution:**F(u)**fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u**F(u)**fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u How does the convolution result look like?**F(u)**fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u**F(u)**fmax u -fmax Sampling Analysis: Freq. Domain … -1/T 0 1/T … u

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