Filtering (II)

Dr. Chang Shu

COMP 4900CWinter 2008

Image FilteringModifying the pixels in an image based on some

functions of a local neighbourhood of the pixels

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Linear Filtering – convolution

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Filter Output

The output is the linear combination of the neighbourhood pixels

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The coefficients come from a constant matrix A, called kernel.This process, denoted by ‘*’, is called (discrete) convolution.

Handle Border Pixels

• Set the value of all non-included pixels to zero.

• Set all non-included pixels to the value of the corresponding pixel in the input image.

Near the borders of the image, some pixels do not have enoughneighbours. Two possible solutions are:

Smoothing by Averaging

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Convolution can be understood as weighted averaging.

Gaussian Filter

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Discrete Gaussian kernel:

Gaussian Filter

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Gaussian Kernel is Separable

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Gaussian Kernel is SeparableConvolving rows and then columns with a 1-D Gaussian kernel.

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The complexity increases linearly with instead of with .m 2m

Gaussian vs. Average

Gaussian Smoothing Smoothing by Averaging

Noise Filtering

Gaussian Noise

After Gaussian Smoothing

After Averaging

Noise Filtering

Salt-and-pepper noise

After Gaussian smoothing

After averaging

Nonlinear Filtering – median filter

Replace each pixel value I(i, j) with the median of the valuesfound in a local neighbourhood of (i, j).

Median Filter

Salt-and-pepper noise After median filtering

Salt-and-Pepper Noise Removal by Median-type Noise Detectors and Edge-preserving Regularization Raymond H. Chan, Chung-Wa Ho, and Mila NikolovaIEEE Transactions on Image Processing, 14 (2005), 1479-1485.