FeatureJ: Derivatives

General Description

This plugin enables one 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].

Dialog Description

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 10. 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 zero. See the algorithmic details below for boundary conditions in setting this parameter. If physically isotropic Gaussian image smoothing is to be applied (which can be specified in the Options dialog), then in each dimension the scale is divided by the sampling interval in that dimension (the pixel width/height/depth as specified in ImageJ > Image > Properties).

Algorithmic Details

The derivative images are obtained by dimension-separated convolution with (derivatives of) the multi-dimensional Gaussian function. This necessarily involves sampling and truncation of the continuous (derivatives of the) Gaussian function. At very small smoothing scales (say less than half the derivative order), this causes sampling artifacts, while at very large scales (where the kernel would have to be larger than twice the image size in the corresponding dimension), this causes truncation artifacts. In both cases, the actual kernel used for the discretized convolution does not faithfully represent the shape of the original continuous function (due to undersampling or truncation), and the results become meaningless. The algorithm uses mirror-boundary conditions to obtain values outside the image at (positions close to) the boundaries.

References