Announcements Fourier Transform • Send email to the TA for the mailing list • Analytic geometry gives a coordinate aster@ umiacs . umd . edu system for describing geometric objects. • For problem set 1: turn in written answers to • Fourier transform gives a coordinate problems 2 and 3. • Everything printed, plus source code emailed system for functions. to the TA. • “What kind of geometric object?”, eg ., circle, line, …. • It is possible that the first problem set is kind of hard. Start early. Basis Example ∀ ∃ • P=(x,y) means P = x(1,0)+y(0,1) c , a , a such that : • Similarly: 1 2 θ + = θ + θ cos( c ) a cos a sin θ = θ + θ f ( ) a cos( ) a sin( ) 1 2 1 1 1 2 + θ + θ + a cos( 2 ) a sin( 2 ) Matlab 2 1 2 2 • Matlab Note, I’m showing non - standard basis, these are from basis using complex functions. Orthonormal Basis 2D Example • ||(1,0)||=||(0,1)||=1 • (1,0).(0,1)=0 • Similarly we use normal basis elements eg : θ cos( ) θ = π θ θ 2 cos( ) ∫ cos 2 d θ cos( ) 0 • While, eg : θ θ θ = π 2 ∫ cos sin d 0 0 1
Remember Convolution Convolution Theorem X X X X X X 11 10 0 0 1 10 ⊗ = 10 X X − 9 10 1 0 1 f g T F * G 11 1 O X X 10 9 10 0 2 1 I X X 11 10 9 9 11 10 F X X 9 10 11 9 99 11 • F,G are transform of f,g X X X X X X 10 9 9 11 10 10 1 1 1 That is, F contains coefficients, when 1 1 1 1/9 we write f as linear combinations of 1 1 1 harmonic basis. 1/9.(10x1 + 11x1 + 10x1 + 9x1 + 10x1 + 11x1 + 10x1 + 9x1 + 10x1) = 1/9.( 90) = 10 Examples Examples θ ⊗ θ = cos cos ? • Transform of box filter is θ ⊗ θ = cos cos 2 ? sinc . • Transform of θ ⊗ = cos f ? Gaussian is Gaussian. θ + θ + θ ⊗ = (cos . 2 cos 2 . 1 cos 3 ) f ? ( Trucco and Verri ) 2
Example: Smoothing by Implications Averaging • Smoothing means removing high frequencies. This is one definition of scale. • Sinc function explains artifacts. • Need smoothing before subsampling to avoid aliasing. Smoothing with a Gaussian Sampling Boundary Detection - Edges • Boundaries of objects – Usually different materials/orientations, intensity changes. 3
We also get: Boundaries of materials Boundaries of surfaces properties Boundaries of lighting Edge is Where Change Occurs • Change is measured by derivative in 1D • Biggest change, derivative has maximum magnitude • Or 2 nd derivative is zero. Noisy Step Edge Optimal Edge Detection: Canny • Gradient is high everywhere. • Assume: • Must smooth before taking gradient. – Linear filtering – Additive iid Gaussian noise • Edge detector should have: – Good Detection. Filter responds to edge, not noise. – Good Localization: detected edge near true edge. – Single Response: one per edge. 4
So, 1D Edge Detection has Optimal Edge Detection: Canny (continued) steps: 1. Filter out noise: convolve with • Optimal Detector is approximately Gaussian Derivative of Gaussian. 2. Take a derivative: convolve with • Detection/Localization trade - off [- 1 0 1] – More smoothing improves detection 3. Find the peak. – And hurts localization. • Matlab • This is what you might guess from • We can combine 1 and 2. (detect change) + (remove noise) • Matlab 5
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