``Grayscale_Dilation'' - gray-scale dilation

VFX name:

iGrayDilate

IMPROMPTU Equivalent:

Grayscale_Dilation

purpose:

Perform a gray-scale dilation [27] on an input volume, using a specified structuring element with adjustable dimensions. Grayscale dilation is defined as

$$v_{dil}(x,y,z) = \max_{i,j,k \, \in \, B} \; [v_{orig}(x-i, y-j, z-k) + b(i,j,k)]$$

where v_dil is the dilated voxel intensity, v_orig is the original voxel intensity, b is the grayscale structuring element, and B is the domain of b, or the set of all (i,j,k) values where b is defined.

Three structuring element shapes are currently available with this function:

1.

sphere - a structuring element b(x,y,z) of the form

$$b(x,y,z) = D \sqrt{1 - \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} }$$

where D is the maximum intensity value for the structuring element, and a, b, and c are the x, y, and z filter dimensions, respectively.

2.

paraboloid - structuring element b(x,y,z) of the form

$$b(x,y,z) = D \left ( 1 - \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} \right )$$

where the parameters are the same as those defined above.

3.

cone - structuring element b(x,y,z) shaped as the elliptical cone given by

$$b(x,y,z) = D \left ( 1 - \sqrt{ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} } \right )$$

where the parameters are the same as those defined above.

The domain B for these structuring elements is defined as all (x,y,z) within the size of the structuring element where the equations have a real solution.

input:

An 8-bit gray-scale volume.

output:

An 8-bit gray-scale volume.

parameters:

Default parameter menu for this function is

x_filter_dimension = 3
y_filter_dimension = 3
z_filter_dimension = 3
shape = sphere
max_gray_level = 10
iterations = 1

The ``filter_dimension'' parameters specify the size of the structuring element in voxels. The x, y, and z values specified here correspond to the a, b, and c values used to compute the structuring element (see above).
The parameter ``shape'' specifies the shape of the structuring element to be used when performing the dilation. The available structuring elements are a sphere, paraboloid, and cone.
The parameter ``max_gray_level'' corresponds to the D value (see above) used to set the highest intensity value within the structuring element.

comments:

1.

This function can also be used to perform dilations using other structuring elements, although this is not currently a menu option within the ROI system. Other routines can be written to call the grayscale dilation function directly if different structuring elements are desired.

2.

This function is very slow, taking about 3 minutes for a 100x100x100 image using a 7x7 structuring element.