[ITK-users] Metadata Information in FFT images can be misleading.

[Matt McCormick]

Hi Pablo,

Thanks for discussing this issue and the excellent overview of the
topic. The meaning of metadata in the frequency domain could
definitely a source of confusion and bugs, and we would benefit from a
clarity and consensus on the subject.

For practical reasons of pipeline use and existing algorithm
application, re-use of itk::Image as much about seems beneficial.

What do you think about adding documentation to
itk::ImageBase::GetSpacing() [1]? We could state that the Spacing is
always in physical units.  If the image is in the frequency domain,
then Spacing is equal to 1 / Frequency Sampling and the Frequency Bin
Resolution equals 1 / (Frequency Sampling * N), where N is the Size of
the LargestPossibleImageRegion (but not in the half-Hermitian
storage).

Then, the frequency shrinkage filter would increase its output Spacing
accordingly so metadata is updated throughout an analysis pipeline.

Thanks,
Matt

[1] https://itk.org/Doxygen/html/classitk_1_1ImageBase.html#aaadef7c0a9627cf22b0fbf2de6de913c

On Tue, Mar 14, 2017 at 4:54 AM, Pablo Hernández
<pablo.hernandez.cerdan at outlook.com> wrote:
> Hey there,
>
> I am contributing with the IsotropicWavelet external module, and I am facing
> some doubts about the meaning of image metadata in frequency-domain images.
>
> Current behavior performing a FFTforward is to copy the input image
> metadata: spacing,origin, and direction to the output, even though the
> output is in the frequency domain or dual space. I guess it is better copy
> it than to lose it, but doesn't mean that the metadata is meaningful in the
> frequency domain. Spacing information can led the user to think that between
> each pixel holding a frequency value, that spacing is like a frequency
> resolution, but it is not.
>
> Some refresh of the lingo to help:
> dual space: f ~ 1/T  (where T can represent time or space)
> Units of frequency are: Hertz = 1 / [ T ], or Radians (Rad = 2pi Hz)
> Spacing (or Sampling in the lingo of digital signaling processing) in a
> spatial domain image is associated with the resolution of the image, and
> represents the pixel width in physical units.
> The Origin is an array holding some relative units to world coordinates that
> are meaningful for the experimenter.
>
> What does spacing mean in a frequency domain image? The frequency resolution
> after an FFT is related to the size of the FFT (the size of the image) and
> the sampling rate of the original image (more here).
>
> Freq_bin_resolution = Freq_sampling / N. Where N is the size of the image,
> and Freq_sampling = 1 / Spacing. All these variables are arrays, with size
> equal to the dimension of the image.
>
> And the origin, in the case of the output of an FFT, depends on the layout
> of that particular FFT algorithm. VNL and FFTW share the same layout, where,
> for example, the zero frequency bin is stored the first index {{0}} , (also
> note that the physical units of that first index is always 0.0 Hz,
> regardless of origin, or spacing of the original image).
>
> If we set the Freq_sampling to {{1.0}} (corresponding to a Spacing of
> {{1.0}}, then the freq resolution will be {{1/N}} (depending on the size!),
> and the range of frequencies will always be: (-0.5, 0.5] Hz, or (-pi, pi]
> rads (independent of the size).
>
> So right now, in the wavelet module that works in the frequency domain, and
> does some shrinkage in this domain, I have chosen to ignore all this
> metadata, and let the user recover it if he/she needs it, but it might be
> worth to think about this.
>
> For example, in a shrinkage by a factor of 2, the width of a pixel
> representing a frequency, doubles. F_bin_resolution =1/(N/2) = 2/N, and the
> range of allowed frequencies keeps the same: (-Freq_sampling/2.0,
> Freq_sampling/2.0] or (-0.5, 0.5] or (-pi,pi] if Freq_sampling =1.0
>
> This is all good, but hard to grasp when the metadata can be misleading. If
> you have any thoughts on how to improve this or be clearer to the user,
> happy to hear, meanwhile I think I will keep ignoring any metadata
> associated with the spatial domain when dealing with frequency domain
> images.
>
> Cheers,
> Pablo
>
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