Image gradients

In general kernelfun can be any function which satisfies the following interface:

where kern_d is the kernel for producing the derivative with respect to the th dimension of an -dimensional array. The parameter extended[i] is true if the image is of size > 1 along dimension . The parameter kern_d may be provided as a dense or factored kernel, with factored representations recommended when the kernel is separable.

Some valid kernelfun options are described below.

At the image edge, border is used to specify the padding which will be used to extrapolate the image beyond its original bounds. As an indicative example of each option the results of the padding are illustrated on an image consisting of a row of six pixels which are specified alphabetically: . We show the effects of padding only on the left and right border, but analogous consequences hold for the top and bottom border.

The gradient of a continuous analogue image at location is defined as the vector

where is the unit vector in the -direction. The gradient points in the direction of maximum rate of change of at the coordinates . The gradient can be used to compute the derivative of a function in an arbitrary direction. In particular, the derivative of in the direction of a unit vector is given by , where denotes the dot product.

In practice, we acquire a digital image where the light intensity is known only at a discrete set of locations. This means that the required partial derivatives are undefined and need to be approximated using discrete difference formulae [1].

A straightforward way to approximate the partial derivatives is to use central-difference formulae

and

However, the central-difference formulae are very sensitive to noise. When working with noisy image data, one can obtain a better approximation of the partial derivatives by using a suitable weighted combination of the neighboring image intensities. The weighted combination can be represented as a discrete convolution operation between the image and a kernel which characterizes the requisite weights. In particular, if ( represents a kernel, then

The kernel is frequently also called a mask or convolution matrix.

The choice of weights determines the magnitude of the approximation error and whether the finite-difference scheme is isotropic. A finite-difference scheme is isotropic if the approximation error does not depend on the orientation of the coordinate system and anisotropic if the approximation error has a directional bias [2]. With a continuous analogue image the magnitude of the gradient would be invariant upon rotation of the coordinate system, but in practice one cannot obtain perfect isotropy with a finite set of discrete points. Hence a finite-difference scheme is typically considered isotropic if the leading error term in the approximation does not have preferred directions.

Most finite-difference schemes that are used in image processing are based on kernels, and as noted by [7], many can also be parametrized by a single parameter as follows:

where

A kernel is called separable if it can be expressed as the convolution of two one-dimensional filters. With a matrix representation of the kernel, separability means that the kernel matrix can be written as an outer product of two vectors. Separable kernels offer computational advantages since instead of performing a two-dimensional convolution one can perform a sequence of one-dimensional convolutions.