...sense).

A random process is second-order stationary if the expected value of any quadratic function of the process random variables is invariant under shifting. This guarantees that the autocorrelation is well-defined.

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... .

This is because each iteration requires operations, and at least O(N) iterations are required for information to traverse the length of the image.

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... .

Blurring of course relies on pixels which lie outside a local window. However, this does not pose a problem: the dependence on other pixels can be integrated out. If the blurring depends on a set of pixels which is a superset of , it is simple to express in terms of and . Naturally, information from pixels outside the 3x3 window useful in restoring the central pixel is lost.

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...distance

Measured using expected squared difference

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...equations.

Note that although it is possible to create polynomials bases which are orthogonal under uniform measure (e.g. Legendre polynomials), creating such bases which are orthogonal under an unknown density function is impossible.

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...samples.

A typical image contains about 0.25 million pixels. It does not take many training images to get millions of training samples.

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...price.

There is a slight decrease in filtering time as the number of coefficients increases, due to cache effects.

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...term.

Approximating the number of grid points using the volume of the hypersphere works well for a small number of dimensions. However, at a certain critical dimension, hyperspheres stop growing in volume and start shrinking. This is because is due to the term in the denominator. However, the number of grid points does not shrink, but behaves asymptotically as , where and L is the extent of the grid in each dimension. This is still much better than the full hypercube grid, in which the number of grid points behaves asymptotically as .

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...coefficients.

Without the sparse representation and symmetry reductions, this filter would require 7355827511386641 (about 7x169#52) coefficients.

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...corners.

The keen reader will note that the division of the square into two triangles (Figure 2.11) can be done in two ways; rather than drawing the diagonal line from to , it could be drawn from to . This type of interpolation introduces an anisotropy: the basis functions have a definite orientation to them. By sacrificing isotropy, substantial computational gains are made.

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...satisfy.

These properties are

• Associativity: (ab)c = a(bc)
• Identity: There is an identity element e such that ae = ea = a for all a.
• Inverses: For every element a, there is an inverse of a such that .
• Closure: For every a and b in the group, ab is also in the group.

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...filters,

Under the mild assumption that 274#79 for almost all applications

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...filters

For noise with symmetric distributions only

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...updates.

A rank-1 update of a matrix has the form .

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...costs.

Note: the system of equations is typically large enough that storing as a dense matrix is impossible. Some filters described consisted of 16000 grid points, which would require about 2 Gb of RAM were a dense matrix representation used. A sparse matrix representation of the same matrix fits comfortably into 64 Mb of RAM.

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...unpredictable.

This is what happens with polynomial filters and outliers: polynomial approximations can have wild oscillations outside the region of training samples, and run off to plus or minus infinity.

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...signal.

This is because a 3x3 window contains 9 observations; the residual variance is therefore 800/9=88.89.

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