3D Fourier Transform

Overview

This program will perform a 3D Fourier transform to each xyz volume in the input with the option to process the data beforehand to remove trends and smooth discontinuities at the edges.

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 direction of the transform and form of the output is controlled by the three toggles just below the standard output mode and region selection controls. When the input data is real, the direction of the transform is always forward, but the output can be fully complex and have both positive and negative x frequencies present in the output or just have non-negative frequencies present. In the latter case there is no loss of information, since the dropped data at (kx,ky,kz) is just the complex conjugate of the data at (-kx,-ky,-kz). When the input data is complex there are more options:
- When the Fully complex toggle is off, a reverse transform is performed to get a purely real result. There is however an ambiguity about whether the result should have an even or odd number of pixels in x. If the input data after padding which was supplied to the forward transform had an even number of pixels in x, turn on the Even X toggle; otherwise, turn it off.
- When the Fully complex is on, the transform direction can be either forward or reverse depending on how the Forward toggle is set.
Use the Zero frequency centered toggle button to control the convention for the location of the zero frequency. If the toggle button is off, the zero frequency is at (0,0,0) (if a reverse transform, this is relative to the selected input region); otherwise the zero frequency is at (0, (ny - 1) / 2, (nz - 1) / 2) for real to complex or complex to real transforms and at ((nx - 1) / 2, (ny - 1) / 2, (nz - 1) / 2) for complex to complex transforms. For a forward transform, nx, ny, and nz are the input dimensions after padding; for a reverse transform, nx, ny, and nz are the input dimensions.
Use the Shift field to specify a translation that is applied in conjunction with the transformation. For a forward transform, a shift of (x, y, z) will cause pixel (x, y, z) to be shifted to the origin, (0,0,0). For a reverse transform, the same shift will place what would appear at the origin, (0,0,0), at pixel (x, y, z).
Use the button labelled "Set preprocessing" to display and change the parameters which control the processing of the data before it is padded and transformed. Preprocessing can only be done before forward transforms. By default, no preprocessing is done.
Use the button labelled "Set window/taper" to select a function used to smooth out discontinuities at the edges of the data after preprocessing and before padding and transforming the data. This option is only available if a forward transform is done. By default, no smoothing is done.
Use the button labelled "Set edge handling" to control the number and values of the elements that are appended to the input region before the transform. By default, for a forward transform the input data is padded with zeroes so that it is a size which can be efficiently processed and on the reverse transform the data is not padded.
Use the button labelled "Set output size" to open a dialog with the controls for trimming the output region. Trimming can only be done when a reverse transform is applied. By default, no trimming is done.

Because of the nature of the Fourier transform, the handling of the data header is different than most other Priism applications:

The pixel format for the output is always complex when a forward transform is used, and may be complex or real (floating-point) when the reverse transform is used. The Mode control is always ignored.
When performing a forward transform, the sizes in the header are changed, but the coordinate system information (origin, pixel spacings, tilt angles) is not converted to frequency units so that when the reverse transform is performed, the original coordinates are still available. The starting indices and the number of samples in the header are set so they refer to the origin and size of the data region before padding.
When performing a reverse transform, the sizes and number of samples in the header are updated to reflect the output size. The coordinate system information and starting indices are usually unchanged (since the forward transform did not change them) except if the size of the region has changed due to padding before the reverse transform. Then the pixel sizes are scaled by the ratio of the unpadded size to the padded size and the starting indices are reset to zero since padding before the reverse transform is equivalent to resampling the data on a finer grid.

The Fourier transforms calculated have these properties:

If the Zero frequency centered toggle button is off, the zero frequency component is at (0,0) (the lower left corner) after a forward transform, and the component at positive Nyquist in all three directions is at (nx / 2, ny / 2, nz / 2) where nx, ny, and nz are the input dimensions after padding. If the Zero frequency centered toggle button is on and the transform is real to complex, the zero frequency component is at (0, (ny - 1) / 2, (nz - 1) / 2), and the component at positive Nyquist in all three directions is at (nx / 2, ny, nz). If the Zero frequency centered toggle button is on and the transform is complex to complex in the forward direction, the zero frequency component is at ((nx - 1) / 2, (ny - 1) / 2, (nz - 1) / 2), and the component at positive Nyquist in all three directions is at (nx, ny, nz).
When the forward transform is applied, the result is divided by the number of pixels in the region. If the data was tapered it is also further scaled by a factor such that, if the input had on average a zero mean, the tapering operation would, on average, not change the variance of the data.
The sign used in the complex exponential is negative for the forward transform and positive for the reverse transform.

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Shift

The Shift field specifies a translation, measured in pixels, that is applied in conjunction with the transform. The translation is circular in the sense that something which is pushed off one side wraps around to the opposite side.

When a forward transform is performed, a shift of (x, y, z) moves pixel (x, y, z) to the origin, (0,0,0). For a reverse transform, the same shift will place what would appear at the origin, (0,0,0), at pixel (x, y, z).

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Preprocessing

The input region can be processed to remove an offset or trend before the data is tapered, padded, and transformed. To display the controls for this processing, press the button labelled 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 padding and transforming the data.
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 transforming the data.

If you are intending to taper the data and the data does not already have a zero mean, you should consider using the mean subtraction, linear trend removal, or polynomial fit options.

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Window/taper

Applying a windowing function, also called tapering, multiplies the data by a function which goes to zero (or near zero) at the edges of the data. This is done before the forward transform to remove discontinuities at the edges which can complicate the interpretation of frequency spectra; the disadvantage of applying a window is that it essentially throws away parts of the input and it broadens features in the spectra.

The windowing functions used are separable, i.e, the windowing function, W(x,y,z), can be written as the product of WX(x), WY(y), and WZ(z). WX, WY, and WZ are calculated from the same formula but with different inputs for the number of pixels, n, and the fraction, f, of the dimension that is attenuated by the windowing function. The three values in the Coverage field are the values of f divided by two for the x, y, and z directions, respectively. Commonly used values for the coverage are 0.5 (a "full taper") and 0.1 (a 10% taper). The power, x, is used by half-sine windowing function.

The functions used to calculate the windowing functions are shown below. i is the pixel index and runs from 0 to n minus 1, and e is given by

      { floor(0.5 + f / 2.0 * n)          n even
  e = {
      { floor(0.5 + f / 2.0 * (n - 1))    n odd

Rectangular

         { 0     i < e
  w(i) = { 1     e <= i <= n - e
         { 0     i > n - e

Because of the sharp cutoffs of the rectangular window, it is not particularly useful; it's provided for completeness.

Triangular

         { i / e           i < e
  w(i) = { 1               e <= i <= n - e
         { (n - i) / e     i > n - e

Half-sine^power

         { (sin(pi * i / (2 * e)))^x           i < e
  w(i) = { 1                                   e <= i <= n - e
         { (sin(pi * (n - i) / (2 * e)))^x     i > n - e

Hamming

         { 0.54 - 0.46 * cos(pi * i / e)           i < e
  w(i) = { 1                                       e <= i <= n - e
         { 0.5f - 0.46 * cos(pi * (n - i) / e)     i > n - e

Blackman

         { cos(pi * i / e)           i < e
  a(i) = { -1                        e <= i <= n - e
         { cos(pi * (n - i) / e)     i > n - e

  w(i) = 0.34 - a(i) * (0.5 - a(i) * 0.16)

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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 when filtering or cross-correlations are being performed.

To display the controls for the amount of padding and the values of the elements added, press the button labelled 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 labelled 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 for y or z-line for 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 or z-line).

If a forward transform is being done and the data was tapered, than padding with zeroes is the best choice. Padding with zeroes is also appropriate if a reverse transform is being done and you want the grid sampled on a finer grid using Fourier interpolation.

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Output size

When a reverse transform is performed, the size of the region written to the output file may be smaller than that of the straight Fourier transform of the input after padding. Press the button labelled Set output size to open a dialog with controls for trimming the data. For each dimension, there is a display of the size of the transformed data and what the application believes was the size of the data before padding and the forward transform was applied (this information is read from the number of samples in the header; it's possible that processing after the forward transform have altered this field so it may not reflect the original size). The amount of trimming for each dimension is listed in an editable field.

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3D Fourier Transform

Overview

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Related Priism Topics

Shift

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Preprocessing

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Window/taper

Rectangular

Triangular

Half-sine^power

Hamming

Blackman

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Edge handling

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Output size

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