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Menu:   Processing > Filters… > Advance->Fourier  
Script:   fourier  
Use this function to convert a grayscale image from the spatial domain into the frequency domain and vice versa, by applying the forward and inverse Fourier transform.

The discrete 2D Fourier transform is defined by the following equation:

where N is the horizontal size of the image or selection in pixels, M is the vertical size. The resulting image F is a complex image each pixel of which represents the amplitude and phase of a certain frequency. Different distances of pixels from the origin (the center of the image) represent different spectral frequencies, different directions represent different orientations in the original image, and the amplitude value at each locations shows how much of that frequency and orientations is present in the original image. The farther from the center of the image the pixel is, the higher spatial frequency it represents.

The inverse Fourier transform converts a frequency image back to the spatial domain:

Typically, you perform the direct Fourier transform first, perform some image editing and processing operations on the frequency image and then convert it back to the original grayscale type by applying the inverse transform. For instance, if the original image contains a periodic noise (see Generate Sine
), you can remove the unwanted frequency it by locating two peaks in the spectral image and erasing them with the Brush Tool

Computing the discrete Fourier transform require a great deal of processing power, but if
N and M are powers of 2, a fast algorithm called FFT available for computations.

If you choose this command, the following options will become available:

Displays the input image frame number. If you want to apply the Fourier transform to another image, type or select the corresponding value.  

Displays the number of the frame in which the output image will be created. Depending on the Preferences. ImageWarp will set it either to the first available value or to the Input frame number. Type or select another value if you want the output image to be created in a different frame.  
Select this check box to watch the Preview of the output image.  

Lets you select the direction of the transform.  
Select Direct to apply the direct Fourier transform and convert the image from the spatial domain into the frequency domain.  
Select Inverse to apply the inverse Fourier transform and convert the image from the frequency domain into the spatial domain.  
Select this option to apply the Fast Fourier transform algorithm. The FFT algorithm is applied automatically if the horizontal and vertical sizes of the Input image or selection are powers of 2. If this is not the case, selecting the FFT option will cause the image to be expanded horizontally and vertically to the closest width and height which are powers of 2. This is done by surrounding the image with the black margins prior to the application of the function.  
When the inverse transform is applied to the frequency image generated with the enforced FFT algorithm, the resulting image will retain the size of the original grayscale image.