Gradients with Gaussian smoothing

These functions carry out gradient estimation using Gaussian smoothing and symmetric differencing. They can be used to support, for example, the Canny edge detector, and may form the initial stage of many image and data processing operations.
The gradient functions accept different kinds of data:
gradients_x: a vector

gradients_xy: a 2-D array, typically an image. The two components of the gradient are returned.

gradients_xyt: two 2-D arrays, each typically an image. Spatial and temporal gradients are returned.

gradients_n: an N-D array. The gradient along each axis is returned.

The supporting smoothing functions carry out Gaussian smoothing, taking as inputs:

gsmooth: a vector

gsmooth2: a 2-D array

gsmoothn: an N-D array

The functions offer anisotropic smoothing if required. (That is, a different smoothing constant for each axis.)

Particular care is taken of how elements close to the array boundaries are treated. By default, the output arrays are smaller than the input arrays so that only valid values need be computed. However, an option allows the outputs to be the same size as the inputs; values close to the boundaries are then computed by extrapolation of the image using reflection or tiling.

The convenience function exindex simplifies the code for extrapolation.