FeatureJ: Derivatives

General Description

This plugin enables you to compute multi-dimensional, Gaussian-scaled derivatives of images, which form the basis of differential-geometric descriptions of image features as can be found in the literature on human and computer vision [1,2].

Description of Dialog Components

x-, y-, z-order of differentiation. The differentiation order can be specified for the x-, y-, and z-dimension. Currently the largest supported order of differentiation is five. An order of zero in any dimension implies that only smoothing is applied in that dimension. The differentiation operation is carried out for every time frame and channel in a 5D image.

Smoothing scale. The smoothing scale is equal to the standard deviation of the Gaussian (derivative) kernel and must be larger than or equal to zero. If you have indicated in the Options dialog that you would like to apply (physically) isotropic Gaussian image smoothing, then in each dimension the scale is divided by the sampling interval in that dimension (that is, the pixel width/height/depth as specified in ImageJ > Image > Properties).

Algorithmic Details

The derivative images are obtained by dimension-separated convolution either with the derivative of the multi-dimensional Gaussian kernel, or with the Gaussian kernel itself followed by finite difference operators, depending on what is indicated in the Options dialog regarding separation of smoothing and differentiation. Since the kernel is truncated, the former method results in an image which is all zero if the derivative smoothing scale becomes too small, whereas the latter method still gives the expected output. In either case, the algorithm uses mirror-boundary conditions to obtain values outside the image, which implies that if the order of differentiation is larger than zero for a given dimension and the image size equals one in that dimension, the result is always zero.

References