kernels: Kernel-generating functions

In mmand: Mathematical Morphology in Any Number of Dimensions

Kernel-generating functions

Description

These functions can be used to generate kernels for morphological, smoothing or resampling operations. There are two types of kernels: kernel arrays, which are used with morph, and kernel functions, which are used with resample.

Usage

isKernel(object)

isKernelArray(object)

isKernelFunction(object)

kernelArray(values)

shapeKernel(width, dim = length(width), type = c("box", "disc", "diamond"),
  binary = TRUE, normalised = FALSE)

gaussianKernel(sigma, dim = length(sigma), size = 6 * sigma,
  normalised = TRUE)

sobelKernel(dim, axis = 1)

kernelFunction(name = c("box", "triangle", "mitchell-netravali", "lanczos"),
  ...)

boxKernel()

triangleKernel()

mitchellNetravaliKernel(B = 1/3, C = 1/3)

mnKernel(B = 1/3, C = 1/3)

lanczosKernel()

Arguments

object	Any object.
values	A numeric vector or array, containing the values of the kernel array.
width	A numeric vector giving the width of the shape in each dimension, in array elements. Does not need to be integer-valued, or equal for all dimensions. Will be recycled to length dim if that parameter is also specified.
dim	An integer value giving the dimensionality of the kernel. Defaults to the length of width or sigma, where available.
type	A string giving the type of shape to produce. In one dimension, these shapes are all equivalent.
binary	If FALSE, the value of the kernel at each point represents the proportion of the array element within the shape. If TRUE, these values are binarised to be 1 if at least half of the element is within the shape, and 0 otherwise.
normalised	If TRUE, the sum of non-missing elements of the kernel will be unity. Note that this is the default for gaussianKernel, but not for shapeKernel.
sigma	A numeric vector giving the standard deviation of the underlying Gaussian distribution in each dimension, in array elements. Does not need to be equal for all dimensions. Will be recycled to length dim if that parameter is also specified.
size	A numeric vector giving the width of the kernel in each dimension, which will be rounded up to the nearest odd integer. Defaults to four times the corresponding sigma value.
axis	The axis along which the gradient operator will be applied.
name	A string giving the name of the kernel function required.
...	Parameters for the kernel function.
B, C	Mitchell-Netravali coefficients, each of which must be between 0 and 1.

Details

There are two forms of kernel used by this package. Kernel arrays, otherwise known in mathematical morphology as structuring elements, are numeric arrays with class kernelArray. They are defined on a grid of odd width, and are used by morph and related functions. Kernel functions, by contrast, are represented in R as a list containing a name and, optionally, some parameters. The real implementation is in C++. They are defined everywhere within the support of the kernel, and are used by resample and friends. The key distinction is in whether the kernel will always be centred exactly on the location of an existing value in the data (for kernel arrays) or not (for kernel functions).

The kernelArray and kernelFunction functions create objects of the corresponding classes, while isKernelArray and isKernelFunction test for them. In addition, isKernel returns TRUE if its argument is of either kernel class.

The remaining functions generate special-case kernels: shapeKernel generates arrays with nonzero elements in a box, disc or diamond shape for use with morphology functions; gaussianKernel generates Gaussian coefficients and is used by gaussianSmooth; sobelKernel generates the Sobel-Feldman gradient operator, for use by sobelFilter; boxKernel is used for “nearest neighbour” resampling, and triangleKernel for linear, bilinear, etc. The Mitchell-Netravali kernel, a.k.a. BC-spline, is based on a family of piecewise-cubic polynomial functions, with support of four times the pixel separation in each dimension. The default parameters are the ones recommended by Mitchell and Netravali as a good trade-off between various artefacts, but other well-known special cases include B=1, C=0 (the cubic B-spline) and B=0, C=0.5 (the Catmull-Rom spline). mnKernel is a shorter alias for mitchellNetravaliKernel. Finally, the Lanczos kernel is a five-lobe windowed sinc function.

Value

For isKernel, isKernelArray and isKernelFunction, a logical value. For kernelArray, shapeKernel, gaussianKernel and sobelKernel, a kernel array. For kernelFunction, boxKernel, triangleKernel, mitchellNetravaliKernel and mnKernel, a kernel function.

Author(s)

Jon Clayden <[e-mail]>

References

The Mitchell-Netravali kernel is described in the following paper.

D.P. Mitchell & A.N. Netravali (1988). Reconstruction filters in computer graphics. Computer Graphics 22(4):221-228.

See Also

morph for general application of kernel arrays to data, morphology for mathematical morphology functions, resample for resampling, and gaussianSmooth for smoothing. Also see sampleKernelFunction for kernel sampling and plotting.

Examples

shapeKernel(c(3,5), type="diamond")
gaussianKernel(c(0.3,0.3))
mnKernel()

mmand documentation built on Feb. 16, 2023, 9:22 p.m.