To Do

Advanced Computer Graphics §. Sign up for Piazza

CSE 190 [Spring 2015], Lecture 3 §. Assignment 1, Due Apr 24. §. Anyone need help finding partners? Ravi Ramamoorthi §. Any issues with skeleton code? §. Please START EARLY http://www.cs.ucsd.edu/~ravir

Outline Sampling and Reconstruction §. An image is a 2D array of samples §. Basic ideas of sampling, reconstruction, aliasing §. Discrete samples from real-world continuous signal §. Signal processing and Fourier analysis §. Implementation of digital filters (second part of homework): next lecture

§. Section 14.10 of FvDFH 2nd edition (should read) §. Readings: Chapter 13 (color) and 14.10

Some slides courtesy Tom Funkhouser

Sampling and Reconstruction (Spatial) Aliasing

1 (Spatial) Aliasing Sampling and Aliasing

§. Jaggies probably biggest aliasing problem §. Artifacts due to undersampling or poor reconstruction §. Formally, high frequencies masquerading as low §. E.g. high frequency line as low freq jaggies

Image Processing pipeline Outline

§. Basic ideas of sampling, reconstruction, aliasing §. Signal processing and Fourier analysis §. Implementation of digital filters (second part of homework): next lecture

§. Section 14.10 of FvDFH 2nd edition (should read) §. Readings: Chapter 13 (color) and 14.10

Some slides courtesy Tom Funkhouser

Motivation Ideas

§. Formal analysis of sampling and reconstruction §. Signal (function of time generally, here of space) §. Important theory (signal-processing) for graphics §. Continuous: defined at all points; discrete: on a grid §. Also relevant in rendering, modeling, animation §. High frequency: rapid variation; Low Freq: slow variation §. Images are converting continuous to discrete. Do this sampling as best as possible. §. Signal processing theory tells us how best to do this §. Based on concept of frequency domain Fourier analysis

2 Sampling Theory Fourier Transform Analysis in the frequency (not spatial) domain §. §. Sum of sine waves, with possibly different offsets (phase) Tool for converting from spatial to frequency domain §. Each wave different frequency, amplitude §. Or vice versa §. One of most important mathematical ideas §. Computational algorithm: Fast Fourier Transform §. One of 10 great algorithms scientific computing §. Makes Fourier processing possible (images etc.) §. Not discussed here, but look up if interested

Fourier Transform Fourier Transform

§. Simple case, function sum of sines, cosines §. Simple case, function sum of sines, cosines +∞ +∞ f (x) = ∑ F(u)e2πiux f (x) = ∑ F(u)e2πiux u=−∞ u=−∞ 1 1 F(u) = f (x)e−2πiuxdx F(u) = f (x)e−2πiuxdx ∫0 ∫0 §. Continuous infinite case §. Discrete case ∞ x=N−1 −2πiux ⎡ ⎤ Forward Transform: F(u) = f (x)e dx F(u) = ∑ f (x)⎣cos(2πux / N) − i sin(2πux / N)⎦, 0 ≤ u ≤ N −1 ∫−∞ x=0 +∞ u=N−1 2πiux 1 ⎡ ⎤ Inverse Transform: f (x) = F(u)e du f (x) = ∑ F(u)⎣cos(2πux / N) + i sin(2πux / N)⎦, 0 ≤ x ≤ N −1 ∫−∞ N u=0

Fourier Transform: Examples 1 Fourier Transform Examples 2 ∞ −2πiux Forward Transform: F(u) = f (x)e dx Single sine curve ∫−∞ +∞ (+constant DC term) 2πiux Inverse Transform: f (x) = F(u)e du ∫−∞ §. Common examples

f (x) F(u)

−2πiux0 δ(x − x0 ) e

+∞ 1 δ(u) 2πiux f (x) = F(u)e 2 2 2 ∑ −ax π −π u /a u=−∞ e e 1 a F(u) = f (x)e−2πiuxdx ∫0

3 Fourier Transform Properties Sampling Theorem, Bandlimiting ∞ −2πiux Forward Transform: F(u) = f (x)e dx §. A signal can be reconstructed from its samples, ∫−∞ if the original signal has no frequencies above +∞ 2πiux Inverse Transform: f (x) = F(u)e du half the sampling frequency – Shannon ∫−∞ §. Common properties §. The minimum sampling rate for a bandlimited §. Linearity: F(af (x) + bg(x)) = aF(f (x)) + bF(g(x)) function is called the Nyquist rate

∞ §. Derivatives: [integrate by parts] F(f '(x)) = f '(x)e−2πiuxdx ∫−∞ = 2πiuF(u) §. ∞ 2D Fourier Transform ∞ −2πiux −2πivy Forward Transform: F(u,v) = f (x,y)e e dxdy ∫ ∫−∞ −∞ ∞ +∞ 2πiux 2πivy §. Convolution (next)I nverse Transform: f (x,y) = F(u,v)e e dudv ∫ ∫−∞ −∞

Sampling Theorem, Bandlimiting Antialiasing

§. A signal can be reconstructed from its samples, if §. Sample at higher rate the original signal has no frequencies above half §. Not always possible the sampling frequency – Shannon §. Real world: lines have infinitely high frequencies, can’t sample at high enough resolution §. The minimum sampling rate for a bandlimited §. Prefilter to bandlimit signal function is called the Nyquist rate §. Low-pass filtering (blurring) §. A signal is bandlimited if the highest frequency is §. Trade blurriness for aliasing bounded. This frequency is called the bandwidth §. In general, when we transform, we want to filter to bandlimit before sampling, to avoid aliasing

Ideal bandlimiting filter Outline

§. Formal derivation is homework exercise §. Basic ideas of sampling, reconstruction, aliasing §. Signal processing and Fourier analysis §. Convolution §. Implementation of digital filters (second part of homework): next lecture

§. Section 14.10 of FvDFH 2nd edition (should read) §. Readings: Chapter 13 (color) and 14.10

Some slides courtesy Tom Funkhouser

4 Convolution 1 Convolution 2

Convolution 3 Convolution 4

Convolution 5 Convolution in Frequency Domain ∞ −2πiux Forward Transform: F(u) = f (x)e dx ∫−∞ +∞ 2πiux Inverse Transform: f (x) = F(u)e du ∫−∞ §. Convolution (f is signal ; g is filter [or vice versa]) +∞ +∞ h(y) = ∫ f (x)g(y − x)dx = ∫ g(x)f (y − x)dx −∞ −∞ h = f * g or f ⊗ g §. Fourier analysis (frequency domain multiplication) H(u) = F(u)G(u)

5 Practical Image Processing Outline §. Discrete convolution (in spatial domain) with filters for various digital signal processing operations §. Basic ideas of sampling, reconstruction, aliasing §. Easy to analyze, understand effects in frequency domain §. Signal processing and Fourier analysis §. E.g. blurring or bandlimiting by convolving with low pass filter §. Implementation of digital filters (second part of homework): next lecture

§. Section 14.10 of FvDFH 2nd edition (should read) §. Readings: Chapter 13 (color) and 14.10

Some slides courtesy Tom Funkhouser

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