Section 4.1 Vectors in ℝ n. ℝ n Vectors Vector addition Scalar multiplication.
Vector Spaces Space of vectors, closed under addition and scalar multiplication.
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Transcript of Vector Spaces Space of vectors, closed under addition and scalar multiplication.
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Vector Spaces
• Space of vectors, closed under addition and scalar multiplication
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Image Averaging as Vector addition
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Scaler product, dot product, norm
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Norm of Images
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Orthogonal Images, Distance,Basis
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Roberts Basis: 2x2 Orthogonal
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Cauchy Schwartz InequalityU+V≤U+V
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Schwartz Inequality
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Quotient: Angle Between two images
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Fourier AnalysisFourier Analysis
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Fourier Transform Pair
• Given image I(x,y), its fourier transform is
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Image Enhancement in theFrequency Domain
Image Enhancement in theFrequency Domain
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Complex Arithmetic
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Fourier Traansform of an Image is a complex matrix
Let F =[F(u,v)]
F = ΦMM I(x,y) ΦNN I(x,y)= Φ*MM F Φ*MM
Where
ΦJJ (k,l)= [ΦJJ (k,l) ] and
ΦJJ (k,l) = (1/J) exp(2Πjkl/J) for k,l= 0,…,J-1
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Fourier Transform
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Properties
• Convolution Given the FT pair of an image
f(x,y) F(u,v) and mask pair h(x,y) H(u,v)
• f(x,y)* h(x,y) F(u,v). H(u,v) and
• f(x,y) h(x,y) F(u,v)* H(u,v)
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Properties of Fourier TransformProperties of Fourier Transform
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Properties of Fourier TransformProperties of Fourier Transform
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Properties of Fourier TransformProperties of Fourier Transform
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Properties of Fourier TransformProperties of Fourier Transform
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Image Enhancement in theFrequency Domain
Image Enhancement in theFrequency Domain
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Design of H(u,v)
İdeal Low Pass filter
H(u,v) = 1 if |u,v |< r
0 o.w.
Ideal High pass filter
H(u,v) = 1 if |u,v |> r
0 o.w
Ideal Band pass filter
H(u,v) = 1 if r1<|u,v |< r2
0 o.w
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İmage EnhancementSpatial SmoothingLow Pass Filtering
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Ideal Low pass filterIdeal Low pass filter
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Ideal Low Pass FilterIdeal Low Pass Filter
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Output of the Ideal Low Pass FilterOutput of the Ideal Low Pass Filter
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Gaussian Low Pass FilyerGaussian Low Pass Filyer
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Gaussian Low Pass FilterGaussian Low Pass Filter
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Gaussian Low Pass FilterGaussian Low Pass Filter
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Gaussian Low PassFilterGaussian Low PassFilter
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High Pass Filter: Ideal and GaussianHigh Pass Filter: Ideal and Gaussian
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Ideal High PassIdeal High Pass
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Fourier Transform-High Pas Filtering
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Frequency Spectrum of Damaged CircuitFrequency Spectrum of Damaged Circuit
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Gaussian Low Pass and High PassGaussian Low Pass and High Pass
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Output of Gaussian High Pass Output of Gaussian High Pass
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Gaussian Filters: Space and Frequency DomainGaussian Filters: Space and Frequency Domain
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Spatial Laplacian Masks and its Fourier Transform
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Laplacian FilterLaplacian Filter
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Laplacian FilteringLaplacian Filtering