Filter

The Filter module applies a digital filter to a uniform lattice. The lattice may be two-dimensional (images) or three-dimensional (volumes). Each filter reads the input lattice, performs a particular filtering operation on the data values in the lattice nodes, and sends the results to the output lattice. The input and output lattices are always the same size and type.

Filter module computations include data statistics such as local minimum, maximum, median, average, standard deviation; and image modification such as brightness and contrast.

Inputs

Outputs

Properties

Select the Filter module in the Network Manager
to display its properties in the Property Manager.

Use the Filter module to choose one of many
filter types to apply to the input lattice.

Input

The Input property shows the source to which the module is connected. This option cannot be changed in the Property Manager, but can be changed in the Network Manager by changing the module input.

Filter Type

The Filter type is the type of filter to apply to the input lattice. There are six categories of filters: Linear Convolution; Nonlinear - Order Statistics; Nonlinear - Moment Statistics; Nonlinear - Other; Nonlinear - Edge Detection; and Nodal. To change the filter type, click on the existing option and select the desired option from the list.

All of the filters are space-domain filters (currently, there are no frequency-domain filters). Note: This filter is applied to ALL components. Use the Subset module to apply filtering to individual components.

All linear convolution filters compute weighted averages of the neighboring input lattice nodes. The only differences between the various filters are the shape of the neighborhood and the specific weights used. Consider the computation of the output lattice value at node (a, b, c) using a linear convolution filter with a kernel size S.

where Wijk are the weights defined for the specified filter. The output lattice node value is then

When we compute the value at node (a, b, c), the weights are functions of the nodal separation distance between (a, b, c) and the neighboring nodes.

Filter Type	Description	Equations
Average (Rectangular)	A moving average with a rectangular neighborhood with dimensions of SxSxS, S being the kernel size. All of the weights in the rectangle are 1.
Average (Spherical)	A moving average with an axis-aligned spherical neighborhood and axis dimensions of S/2, S/2, S/2, S being the kernel size. All of the weights in the sphere are 1.
Distance	A distance weighted averaging filter. The resulting isoweight lines are concentric ellipsoids.	where Wijk represents the weights defined for the specified filter and
Distance (Inverse)	A distance weighted averaging filter. By definition, the central weight is 1.0. The resulting iso-weights are concentric spheres.	where
Distance ' Inf Norm'	A distance weighted averaging filter. The resulting iso-weight lines are concentric shoeboxes.	where
Gauss	A distance weighted averaging filter. The Gauss filter smooths or blurs an image by performing a convolution operation with a Gaussian filter kernel. The distance weighting function is given by a Gaussian bell-shaped curve. The resulting iso-weight lines are concentric ellipsoids.	where
Laplacian Edge Detect	Used for edge detection, this implementation uses a 3x3x3 kernel. Mathematically, there exist an infinite number of valid possibilities for the convoluted weights; those used in the filter include those illustrated on the right.
Unsharp Mask	This filter first blurs an image and then subtracts the blurred image from the original image. The difference between the blurred image and the original image (known as the "mask") is then added back to the original image. The resulting combined image appears sharper to the human eye. The sharpness parameter, c, is used to compare the original image against the blurred image.	Original image: ( W=c/(2c-1)) Blurred image: (W=(1-c)/(2c-1))

The nonlinear filters are not weighted averages of the neighboring input lattice values.

Filter Type	Description
Interquartile Range	A nonlinear filter that retains the local interquartile range (75th minus 25th percentile) in a moving shoebox-shaped neighborhood.
Maximum (dilation)	A nonlinear filter that retains the maximum in a moving shoebox-shaped neighborhood. The result is similar to a morphological dilation operation.
Median	A simple, nonlinear, edge-preserving smoothing filter that retains the median (50th percentile) in a moving shoebox-shaped neighborhood. Median filters are most effective on high-amplitude noise that has a low probability occurring. There are two ways to control the amount and scale of the removed noise: variation of kernel size; and multiple application of the filter.
Minimum (erosion)	A nonlinear function that retains the minimum in a moving shoebox-shaped neighborhood. The result is similar to a morphological dilation operation.
Quartile (lower)	A nonlinear function that retains the lower quartile (25th percentile) in a moving shoebox-shaped neighborhood.
Quartile (upper)	A nonlinear function that retains the upper quartile (75th percentile) in a moving shoebox-shaped neighborhood.
Range	A nonlinear function that retains the local data range (maximum minus the minimum) in a moving shoebox-shaped neighborhood.

Filter Type	Description	Equation
Central Moment	A nonlinear filter that retains the local data central moment in a moving shoebox-shaped neighborhood. The filter calculates the Nth centralized moment of the data in a gliding window. The second moment (power = 2) is, therefore, the local variance in the data. For some data sets, this can be used to mask out noisy regions or to detect edges.	where p is a positive integer.
Coefficient of Variation	A nonlinear filter that retains the local data coefficient of variation ( std. dev. / average) in a moving shoebox-shaped neighborhood.
Standard Deviation	A nonlinear filter that retains the local data standard deviation in a moving shoebox-shaped neighborhood. The standard deviation is the square root of the variance, which is computed in three steps, as follows: First, count the number of non-blank input lattice nodes in the neighborhood. With a neighborhood height S and width T, calculate N as shown to the right. Second, calculate the local average as shown on the right. Third, compute the local variance as shown on the right.

Filter Type	Description	Equation
Mean Removal	A simple high-pass filter that computes and then subtracts the neighborhood mean, including the voxel itself. This filter emphasizes outliers in the lattice.	where represents the mean
Median Difference	A nonlinear filter that retains the local median difference (z - median) in a moving shoebox-shaped neighborhood. The median filter is a simple edge-preserving smoothing filter. This filter emphasizes outliers in the lattice.	where represents the median
Rank	A nonlinear filter that retains the rank in the local neighborhood for the current voxel.
Threshold Averaging	A nonlinear filter that retains the local "threshold" average in a moving shoebox-shaped neighborhood.	where represents the average value
Threshold Crossing	A nonlinear filter that sets the current voxel to 1 if a value crosses the threshold somewhere within the local neighborhood. Without a crossed threshold, it sets the voxel to 0.
Zero Crossing	A nonlinear filter that sets the current voxel to 1 if there is a change within the local neighborhood. Without a change, it sets the voxel to 0. This filter is usually used in conjunction with a second derivative filter (eg, Laplacian).

Filter Type	Description
Prewitt	A nonlinear filter that applies the Prewitt edge-detector to a 3x3x3 neighborhood. This is a Prewitt "first derivative" filter. This is also a generalization of the 2D Sobel edge-detection filter, as suggested by Nikolaidis (2001, p 96) and uses a weighted central scheme to compute the directional derivatives but averages over nine neighborhood voxels to reduce noise.
Sobel Max	A nonlinear filter that applies the Sobel edge-detector to a 3x3x3 neighborhood. This is an "edge-detection" filter that keeps the maximum of the absolute value of the gradient components. This is a generalization of the 2D Sobel edge-detection filter as suggested by Nikolaidis (2001, p 96) and uses a weighted central scheme to compute the directional derivatives but averages over nine neighboring voxels to reduce noise. See also: Crane (1997, p 86), Gonzalas (1983, p 336), and Pitas (2000, p 242).
Sobel Norm	A nonlinear filter that applies the Sobel edge-detector to a 3x3x3 neighborhood. This is an "edge-detector" filter that keeps the Euclidean norm of the gradient components. This is also a generalization of the 2D Sobel edge-detection filter, as suggested by Nikolaidis (2001, p 96) and uses a weighted central scheme to compute the directional derivatives but averages over nine neighboring voxels to reduce noise. See also: Crane (1997, p 86), Gonzalas (1983, p 336), and Pitas (2000, p 242).

Filter Type	Description	Equation
Bounding	The output is filtered on a voxel-by-voxel basis. If the value is above the specified upper bound, the value is set equal to the upper bound; if the value is below the lower bound, it is set equal to the lower bound. This filter operates node by node (ie, Ni = Nj = Nk = 1). Blanks and edges have no impact on this filter (ie, blanks stay blanked).
Brightness & Contrast	Modifies the image brightness and contrast. The range increases by a factor of C and the midrange shifts by a factor of B. The output is filtered on a voxel-by-voxel basis. This filter applies an affine transformation node by node (ie, Ni = Nj = Nk = 1). Blanks and edges have no impact on this filter (ie, blanks stay blanked).	where represents brightness, represents contrast, and is the global midrange for the entire lattice.
Gamma Correction	This filter performs gamma correction on a voxel-by-voxel basis: Gamma=1 performs no adjustment; Gamma<1 reduces gamma; Gamma>1 increases gamma. Blanks and edges have no impact on this filter (ie, blanks stay blanked).	where is the maximum intensity throughout the entire lattice.

Orientation

Use the Orientation option to specify the orientation of the filter. The first three choices— XY planes, XZ planes, and YZ planes— orient the filter in the specified direction and apply it to all slices of the lattice. The 3D option applies a full three-dimensional filter. Note: when this module is connected to a two-dimensional lattice, the three-dimensional orientation is disabled and this choice defaults to the two-dimensional lattice orientation. To change the orientation, click on the existing option and select the desired option from the list.

Kernel Size

The Kernal size is the the size of the convolution kernel. The kernel is centered on the node being filtered and defines the neighborhood of surrounding nodes used to calculate the center node. If the kernel size is N, then the filtering kernel is NxNxN nodes in three dimensions. Not all filters use a convolution kernel, and for these the kernel size property is hidden. Kernel sizes must be odd to maintain symmetry. To change the Kernal size, highlight the existing value and type a new value or click the

button to increase or decrease the neighborhood size.

The neighborhood of an output lattice node is a cubic sub- array of nodes in the input lattice that is centered on the corresponding input lattice node. It has three equal dimensions, known as the kernel size. It must be noted that this concept of a neighborhood does not disallow that application of a lattice filter to a two-dimensional lattice. In the case of a two-dimensional lattice, one of the three dimensions is simply equal to 1. Since the neighborhood is centered on a node, the width, height, and depth must all be odd numbers. For example, if the kernel size is 3, the neighborhood of the output lattice node at (21, 31, 16) is the following rectangular sub-array of 27 input lattice nodes:

If kernel size of the neighborhood is represented by S, then the number of nodes in the neighborhood equals SxSxS. Furthermore, the nodes in the neighborhood of node (a,b,c) can be enumerated as

where

Components to Filter

Click the

next to the Input Components to open the components list. Check the box next to the Component to pass the checked component through the filter. Unchecked components pass through the filter unaltered (unfiltered). This option allows the alpha channel of RGBA images to be excluded since this is not usually desired.

Edge Handling

Click the

next to the Edge Handling to open the options for handling data near the edge of the lattice.

Use the Edge handling option to specify how to handle edge effects when the kernel overhangs the edge of the lattice. Many filters, e.g., convolution filters, require a neighborhood of data around the lattice node being filtered. When filtering nodes along the edge, the neighborhood is incomplete on one or more sides. To change the edge handling, click on the existing option and select the desired option from the list. These options specify how to handle this condition:

Edge Fill

When the Edge handling is set to Fill, the Edge fill option becomes available. This is the value used to apply to the edge of the lattice. To change the value, highlight the existing value and type the desired value.

Blank Handling

Click the

next to the Blank Handling to open the options for handling blanked nodes in the lattice.

Use the Blank handling option to determine how to handle blanked nodes within the lattice. To change the way blank values are handled, click on the existing option and select the desired option from the list. Options include:

Blank Fill

When the Blank handling is set to Fill, the Blank fill option becomes available. This is the value used to substitute for values in the lattice that are blanked. To change the value, highlight the existing value and type the desired value.

Filter References

Crane, R. (1990) A Simplified Approach to Image Processing. Prentice-Hall PTR, Upper Saddle River, NJ, 317 pp. ISBN 0-13-226416-1.

Gonzalas, R, & Wintz, P. (1983) Digital Image Processing. Addison-Wesley Publishing Company, Reading, MA, 431 pp. ISBN 0-201-02597-3.

Nikolaidis, N, & Pitas, I. (2001) 3-D Image Processing Algorithms. John Wiley & Sons, New York, NY, 176 pp. ISBN 0-471-37736-8.

Pitas, I. (2000) Digital Image Processing Algorithms and Applications. John Wiley & Sons, New York, NY, 419 pp. ISBN 0-471-37739-2.