ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011 Digital Image Processing Wavelets and Multi Resolution Processing 1 “If you painted a picture with a sky, clouds, trees, and flowers, you would use a different size brush depending on the size of the features. Wavelets are like those brushes.” Ingrid Daubechies
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20111
Digital Image Processing
Wavelets and Multi Resolution Processing
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“If you painted a picture with a sky,clouds, trees, and flowers, you would usea different size brush depending on thesize of the features. Wavelets are like thosebrushes.”
Ingrid Daubechies
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Wavelets and Multi Resolution Processing
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• Image: A non‐stationary Phenomenon
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• Image Pyramid
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• Gaussian (up) and Laplacian (down) Pyramid
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• Subband Coding (1D)
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• Subband Coding (2D)
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• W, W‐1: Forward and Inverse wavelet transform• D (.,λ): Thresholding operator (λ being the threshold)
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• Motivation for Thresholding:– Small coefficients: Dominated by noise.– Large coefficients: Dominated by signal.– Then replacing small coefficients with zero!
• Some Assumption:– Wavelet de‐correlating property generate a sparse signal.– Noise spreads out equally along all coefficients.– The noise level is NOT too high.
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• Hard and Soft Thresholding:– Hard:
– Soft:
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• Threshold Selection:– The most important question.– Very Low threshold: Noisy‐Like result– Very High Threshold: Too smooth result.– Several methods proposed:
• VisuShrink• SureShrink• …
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• VisuShrink (Universal Thresholding):
– N: Sample (Signal/Image) size (# of pixels in image)– : Noise variance
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• SureShrink (Adaptive Thresholding):– Sub‐band adaptive thresholding (each detail sub‐band)– Based on Stein’s Unbiased Estimator for Risk (SURE), a method for estimating the loss in an unbiased fashion.
– :Wavelet coefficients in the jth sub‐band– For the soft threshold estimator:
– We have:
– Optimal threshold:
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• NormalShrink:– For maximum J scale and for scale k:
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