Intensity spreads over the image corners after Fourier based convolution
When I convolve two images the resulting intensity is spread over the corners of the image.
This phenomenon is caused by the following two properties of the FFT:
- Images are interpreted as periodic, i.e. the image has infinite size but repeats itself in each dimension with periods that are equal to the size of the original image in the corresponding dimension.
- The frequency origin (frequency space) is located at the zero voxel (bottom-left-plane) of the 3D image. The positive frequencies are located in the first octant of the image. Depending on the type of transform you selected, `complex' or `real', the negative frequencies are present in the other octants.
This means that you will find spatial frequency 0,0,0 at voxel (0,0,0) of the transform, and not at the center. For visualization purposes it is often desirable to have the zero frequency centered. Use To/from optic rep. from the Restoration menu in the Operations window to move the zero frequency to the center.
`Real' Fourier transforms contain only the positive frequencies in the u-direction (with u, v, w , spatial frequencies corresponding with the x, y, z axis). However, they do contain negative frequencies in the other dimensions. For visualization purposes real-FFTs are therefore less suited than complex-FFTs.
As a consequence of defining the frequency origin (frequency space) at the (0,0,0) voxel, convolutions with funtions which are not centered around (0,0,0) will cause a shift in the resulting image. For example, when you convolve an image with a sphere located at the center ( xc , yc , zc ) of the second image, the result will be shifted over a vector ( xc , yc , zc ). Because images are interpreted as periodic all octants in the resulting image will appear `swapped'.
You can prevent this from happening by using the following methods:
- Center the image with which you are going to convolve around (0,0,0). When this image, for instance a sphere generated with Generate sphere , is centered use To/from optic rep. to shift it to the origin. In other cases use the following method:
- (Alternative centering method) Determine the Center of Mass (CM) of the second image with Image statistics. Then use Shift image to move the CM to the origin. Since Shift image also interprets the image as periodic, this will produce the desired result. An advantage of this method is that you can shift the CM over a non-integer distance.
- When you have convolved with a function located at the center of the image, you can undo the shifting effect by applying To/from optic rep. to the convolution result.
Keywords: convolving FFT
Categories: Faq Deconvolution, Huygens Faq, Imported Faqs
Platforms: Irix Linux
Related products: Hu Pro
