3D Filter (Frequency domain)

Overview

This program will apply a Butterworth, Gabor, or weighted combination of Gaussian filters to each xyz volume in the input. These filters are applied in the frequency domain; the advantages or disadvanteges of this technique as opposed to the filtering in the spatial domain that is done with 3D Filter are

• For a filter which has a large spatial extent, the frequency domain techniques will be generally faster. For an input with N elements and a filter with M elements where M is less than or equal to N, the spatial domain techniques generally perform on the order of N*M calculations and the frequency domain techniques perform N*log(N) calculations.
• With the frequency domain techniques, care needs to be taken in choosing the size of the input region: certain sizes (those that have small prime factors) are handled much more efficiently. By default, this is done automatically for you by adding additional elements so that the input can be handled efficiently and then stripping these additional elements off before writing the result. To change this behavior, modify the edge handling and output size inputs.
• Frequency domain techniques treat the right edge (after padding) as contiguous with the left edge; the top and bottom are handled similarly. This can be problematic when the values along the edges are very different. Preprocessing the data to remove background trends which cause the edges to be different or padding the data with additional elements so that the number of elements added is greater than the approximate spatial extent of the filter are two possible approaches to mitigate this problem.

Also of practical note is that the types of filters offered by this application and by the applications that filter in the spatial domain are quite different, and the availability of the filter you want will frequently limit your choice to one or the other.

The controls in the user interface include a standard set of controls for selecting the input file, output file, and region of interest and for starting and interrupting processing. The other controls are organized as follows:

• The type of filter is selected by the option menu just below the control for the output mode. To display and change the parameters for the current filter type, press the button labeled Set parameters which is next to the option menu.
• Some of the filters are radially symmetric. To generate a radial profile of the filter function which can be displayed with 2D Plot, press the button labeled Create filter profile to select the output file and write the result.
• Use the button labeled "Set preprocessing" to display and change the parameters which control the processing of the data before it is padded and filtered. By default, no preprocessing is done.
• Use the button labeled "Set edge handling" to control the number and values of the elements that are appended to the input region before filtering. By default, the input data is padded with zeroes so that it is a size which can be efficiently processed.
• Use the button labeled "Set output size" to open a dialog with the controls for trimming the output region. By default the output region is the same size as the input region and any padding that was added is stripped off after filtering.

Topics

Overview | Region processing | Filter types | Preprocessing | Edge handling | Output size

Related Priism Topics

Priism | 2D Filter (spatial domain) | Convolution | 2D Filter (frequency domain) 3D Filter (spatial domain)

Filter types

Three different types of filters are currently supported: Butterworth, Gabor and weighted combination of Gaussians. Descriptions of the forms of these filters and the input parameters specific to each are given below.

Butterworth

A Butterworth filter has the following characteristics:

• The response to an input consisting solely of a single spike (impulse function) is infinite in extent
• The frequency response over the set of frequencies that are only slighly attenuated (the passband) is maximally flat as is the frequency response over the set of frequencies that is highly attenuated (the stopband).

Two variants of Butterworth filters are provided, a smoothing filter which passes low frequencies and attenuates the high frequency components of the input and a sharpening filter which attenuates the low frequency components. The frequency which marks the transition between the slightly attenuated and highly attenuated frequencies is the cutoff frequency: the frequency response at the cutoff frequency is 1 / sqrt(2). In the input controls this frequency is specified as a fraction of the Nyquist frequency. The sharpness of the transition between the slightly and highly attenuated frequencies increases as the order of the filter increases.

Gabor

A Gabor filter is a good choice for selecting the components of an image that contribute to a limited range of frequencies. The range selected has two components: one centered at a (kx, ky, kz) of

  cf * (cos(a) * sin(b), sin(a) * sin(b), cos(b))

where kx is the x frequency, ky is the y frequency, kz is the z frequency, cf is the central frequency magnitude specified in the dialog, a is the central frequency theta, b is the central frequency phi. The other is at the same distance from zero frequency but radially opposite from the first. Both components are limited by an elliptical Gaussian envelope. One axis of the envelope is oriented along the radial direction from the zero frequency to the central frequency. Another axis (the "theta direction") is parallel to the plane kz = 0 and perpendicular to the radial axis. The width of the envelope along the axes is given in term of the sigma for the Gaussian.

Gaussian

The frequency response of this filter is radially symmetric about the zero frequency and is the weighted sum of up to four Gaussian envelopes. The number of Gaussians summed to give the filter is controlled by the field labeled Number of filters to combine. For each Gaussian there is central frequency, a width specified in terms of the Gaussian's sigma, and the amplitude at the central frequency. One common choice for the amplitudes of the Gaussians ensures that the frequency response at zero frequency is one. This generates a filter that does not change the mean of the data.

Topics

Overview | Region processing | Filter types | Preprocessing | Edge handling | Output size

Preprocessing

The input region can be processed to remove an offset or trend before the data is padded and filtered. To display the controls for this processing, press the button labeled Set preprocessing in the main dialog. There are five types of processing that can be done; select one by pressing the appropriate toggle in the dialog. The processing options are:

None

The input is not modified before filtering and padding.

Subtract offset

The specified value is subtracted from all the input values.

Subtract mean

For each 3D (xyz) input region, the mean of the region is subtracted from all values in the region.

Remove linear trend

An average slope for each 3D (xyz) input region is estimated by averaging values from the first and last third of each dimension; the linear trend corresponding to that average slope and the mean of the region is subtracted from the input data. The algorithm used is the average slope method extended to three dimensions (Digital data analysis procedures, Bendat and Piersol, 1st Edition, 1971, page 288).

Remove trends up to order

A polynomial of the given order is fit to each 3D (xyz) input region. That polynomial evaluated at each input point is then subtracted from the input before padding and filtering.

Topics

Overview | Region processing | Filter types | Preprocessing | Edge handling | Output size

Edge handling

The dimensions of the input region can be extended by adding elements at the end of each dimension. This is typically done for two reasons:

• Adding the extra elements gives a size that is efficiently handled by Fourier transforms.
• Adding the extra elements reduces the blending of opposite edges of the data which might be troublesome if the values at the opposite edges are very different.

To display the controls for the amount of padding and the values of the elements added, press the button labeled Set edge handling in the main dialog. For each dimension the Width field sets the number of elements that will be added. When the toggle lableled Use default width is on, the Width field will automatically be set so the padded size can be efficiently handled. There are three ways in which the values of the added elements are set. These are:

Pad with value

Use the given value for all the elements added to a dimension. The imaginary component of the value is only used if the input data is complex.

Pad with mean

Use the mean of the 3D (xyz) input region being processed when padding the dimension.

Pad with linear ramp

For each line in x, (or column in y or z-line in z) the padding elements added to the line (or column or z-line) are a linear combination of the first and last element of that input line (or column).

Topics

Overview | Region processing | Filter types | Preprocessing | Edge handling | Output size

Output size

The size of the region written to the output file may be different than the input size. The size is influenced by the controls to pad the input data before filtering, and the controls, described here, to trim the filtered result before saving it. To display the trimming controls, press the button labeled Set output size in the main dialog. This will open a dialog in which there is a set of controls for each dimension filtered. First the input size and the padded size of the region are listed. Below that are the toggles which control how the size in that dimension is trimmed. There are two mutually exclusive options for trimming:

Same as input size

Trim the dimension (if necessary) so the output has the same size in that dimension as the input region.

Trim padded size by

Removes the specified non-negative number of elements from the dimension. The removed elements are from the end of the dimension so elements added by padding are removed first and then, if necessary, elements that correspond to positions in the input region.

Topics

Overview | Region processing | Filter types | Preprocessing | Edge handling | Output size