CSE527-l3 - 2/3/10 Images as functions f(x,y) y x What is a...

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2/3/10 1 Images as functions x y f(x,y) What is a digital image? • Digital images: – Sample the 2D space on a regular grid – Quantize each sample • For samples being Δ apart: f [ i , j ] = Quantize{ f ( i Δ , j Δ ) } • Image: matrix of integer values 10 58 197 46 46 0 176 135 5 188 191 68 2 1 1 29 26 37 0 89 144 147 187 102 255 252 0 166 123 62 166 63 127 17 1 0
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2/3/10 2 Image Noise • Images are corrupted by “noise” mostly during acquisition • Signal Processing techniques can be used to model and remove this noise • Image Restoration: removal of noise – Additive Noise – Salt & Pepper Noise Additive Noise • In the additive noise model the signal is corrupted with random fluctuations
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2/3/10 3 Salt and Pepper Noise • Salt and pepper noise can model errors introduced by the acquisition process. (x and y are random variables in the range (0,1) and l is a constant) ( , ) ( , ) ( ) min max min I i j I i j x l i y i i x l = < + ± Linear Filters General process: – Form new image whose pixels are a weighted sum of original pixel values, using the same set of weights at each point. Properties – Output is a linear function of the input – Output is a shift-invariant function of the input (i.e. shift the input image two pixels to the left, the output is shifted two pixels to the left) Example: smoothing by averaging – form the average of pixels in a neighbourhood Example: smoothing with a Gaussian – form a weighted average of pixels in a neighbourhood Example: finding a derivative – form a weighted average of pixels in a neighbourhood
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2/3/10 4 Image Filtering • Noise reduction/image restoration • Structure extraction 7 Gaussian Noise 8
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2/3/10 5 Removing noise • Basic assumption – Noise process is independent, identically distributed – Image has a more regular underlying structure • Consider measuring the speed of wind or the value of a stock • By considering larger neighborhoods we can separate the signal from the noise 9 Moving Average in 2D 0 90
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2/3/10 6 Moving Average in 2D 0 10 90 Moving Average in 2D 20
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2/3/10 7 Moving Average in 2D 0 10 20 30 90 Moving Average in 2D 14
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2/3/10 8 Moving Average in 2D 0 90 10 20 30 40 60 50 80 Correlation Filtering Say the averaging window size is 2k+1 x 2k+1: Different weights depending on neighboring pixel’s relative position: Correlation filtering: 16
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2/3/10 9 Correlation Filtering Filtering an image – Replace each pixel by a weighted combination of its neighbors.
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CSE527-l3 - 2/3/10 Images as functions f(x,y) y x What is a...

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