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Filtering is a technique for modifying or enhancing an image. For example, you can filter an image to emphasize certain features or remove other features. Image processing operations implemented with filtering include smoothing, sharpening, and edge enhancement.

Filtering is a *neighborhood operation,* in
which the value of any given pixel in the output image is determined
by applying some algorithm to the values of the pixels in the neighborhood
of the corresponding input pixel. A pixel's neighborhood is some set
of pixels, defined by their locations relative to that pixel. (See Neighborhood or Block Processing: An Overview for
a general discussion of neighborhood operations.) *Linear
filtering* is filtering in which the value of an output
pixel is a linear combination of the values of the pixels in the input
pixel's neighborhood.

Linear filtering of an image is accomplished through an operation
called *convolution*. Convolution is a neighborhood
operation in which each output pixel is the weighted sum of neighboring
input pixels. The matrix of weights is called the *convolution
kernel*, also known as the *filter*.
A convolution kernel is a correlation kernel that has been rotated
180 degrees.

For example, suppose the image is

A = [17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9]

and the correlation kernel is

h = [8 1 6 3 5 7 4 9 2]

You would use the following steps to compute the output pixel at position (2,4):

Rotate the correlation kernel 180 degrees about its center element to create a convolution kernel.

Slide the center element of the convolution kernel so that it lies on top of the (2,4) element of

`A`

.Multiply each weight in the rotated convolution kernel by the pixel of

`A`

underneath.Sum the individual products from step 3.

Hence the (2,4) output pixel is

Shown in the following figure.

**Computing the (2,4) Output of Convolution**

The operation called *correlation* is closely
related to convolution. In correlation, the value of an output pixel
is also computed as a weighted sum of neighboring pixels. The difference
is that the matrix of weights, in this case called the *correlation
kernel*, is not rotated during the computation. The Image Processing Toolbox™ filter
design functions return correlation kernels.

The following figure shows how to compute the (2,4) output pixel
of the correlation of `A`

, assuming `h`

is
a correlation kernel instead of a convolution kernel, using these
steps:

Slide the center element of the correlation kernel so that lies on top of the (2,4) element of A.

Multiply each weight in the correlation kernel by the pixel of

`A`

underneath.Sum the individual products.

The (2,4) output pixel from the correlation is

**Computing the (2,4) Output of Correlation**

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