8  Neighbourhood operations and image filtering

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8.1 Convolution and correlation

8.1.1 Discrete convolution in two dimensions

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8.1.2 Separable and rank-one kernels

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8.1.3 Boundary handling: zero, reflect, replicate, wrap

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8.1.4 Computational cost and efficient implementations

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8.2 Smoothing filters

8.2.1 Box (mean) filter and integral images

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8.2.2 Gaussian filter and σ selection

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8.2.3 Median and rank-order filters

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8.2.4 Bilateral and edge-preserving filters

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8.2.5 Non-local means

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8.3 Derivative and edge filters

8.3.1 First-order operators: Sobel, Prewitt, Scharr

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8.3.2 Second-order operators: Laplacian and Laplacian of Gaussian

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8.3.3 Gradient magnitude and orientation

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8.3.4 Sub-pixel edge localisation

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8.4 The frequency domain

8.4.1 The 2D discrete Fourier transform

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8.4.2 Magnitude, phase and their perceptual roles

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8.4.3 The convolution theorem

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8.4.4 Low-pass, high-pass and band-pass filtering

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8.4.5 Frequency-domain inspection of periodic structure

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8.4.6 The sampling theorem and aliasing

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8.5 Scale space and image pyramids

8.5.1 Scale-space theory in outline

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8.5.2 The Gaussian pyramid

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8.5.3 The Laplacian pyramid

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8.5.4 Multi-resolution processing patterns

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8.6 Choosing a filtering strategy

8.6.1 Matching a filter to the noise model and the feature

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8.6.2 Preserving edges while suppressing noise

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8.6.3 Spatial versus frequency-domain: when each wins

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8.6.4 Ordering operations in a preprocessing pipeline

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