# Fourier Transform ## Basics The Fourier transform consists in representing an image by its frequential components. You can then process the frequential components image (also called the Fourier image) to perform specific filtering. ![image](assets/images/easyimage/fourier_transform.webp) Use the [`EFourierTransformer`](../../../doxygen_cpp/classeVision_1_1EFourierTransformer.html) [.NET](../../../doxygen_cs/classeVision_1_1EFourierTransformer.html) [Python](../../../doxygen_python/classeVision_1_1EFourierTransformer.html) class to perform the transformation between a spatial and a Fourier image. The Fourier image contains complex numbers and is represented by a floating point image ([`EFourierImage`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html)). * Use the method [`EFourierTransformer::DirectTransform`](../../../doxygen_cpp/classeVision_1_1EFourierTransformer.html#ab4e067777aa90e7b3cb0bb86b136113c) [.NET](../../../doxygen_cs/classeVision_1_1EFourierTransformer.html#a01865db59f93ed9ea8ad599fc87d6105) [Python](../../../doxygen_python/classeVision_1_1EFourierTransformer.html#aa0e9f3c3ca34d69f2a8c698ce022620d) to convert a spatial image to an [`EFourierImage`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html). * Use the method [`EFourierTransformer::InverseTransform`](../../../doxygen_cpp/classeVision_1_1EFourierTransformer.html#a3f1e83bdc1eba1dfbc7fde487432844d) [.NET](../../../doxygen_cs/classeVision_1_1EFourierTransformer.html#ac7f14ccf7998295b23424eb41a9224f5) [Python](../../../doxygen_python/classeVision_1_1EFourierTransformer.html#aaaac9928520db749a2ee274dc68382cc) to convert images the other way around. #### HINT An [`EFourierImage`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html) may contain high values due to the nature of the applied transformation and this may be difficult to display on a screen. For an easier visualization, use [`EFourierImage::ComputeMagnitude`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html#a7f23a695c126620b0fb050717539a699) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html#a7a9f67e0361235346ebb60cfc5cb8f64) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html#a7f3e78abd6cf525741f390326948b2a2) and [`EFourierImage::ComputePhase`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html#afaa73ba8ab88712ab3b87cc806da6f5b) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html#a7be152aff7859482ed6b8fa22e83a696) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html#ada5295fce6fef2d3a0c5c3d30587d370) to generate an image that can be drawn or saved. ## Frequential Format As the Fourier images contain redundant data, there are different possible representations: ***Packed*** and ***Complex Extended*** formats. The format is set and used when calling the methods of [`EFourierTransformer`](../../../doxygen_cpp/classeVision_1_1EFourierTransformer.html) [.NET](../../../doxygen_cs/classeVision_1_1EFourierTransformer.html) [Python](../../../doxygen_python/classeVision_1_1EFourierTransformer.html). The format of the [`EFourierImage`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html) can be get and set using [`EFourierImage::GetFrequentialDomainFormat`](../../../doxygen_cpp/classeVision_1_1EFourierImage.html#a30b83f9368285274d28a1f0c514b4c5f) [.NET](../../../doxygen_cs/classeVision_1_1EFourierImage.html#a42c8817477d6af0a4d80a1ea57ca50dc) [Python](../../../doxygen_python/classeVision_1_1EFourierImage.html#aeaf674cacac7e540b60d1c52d778f5aa), but setting a different format does not trigger a conversion. ### The Complex Extended Format The ***Complex Extended*** is the simplest of these two formats. * The only subtlety is that the complex numbers are stored in a floating point array, so the complex extended image is twice larger than the spatial image. * Each odd column contains the imaginary part of a complex number. * The following table is an illustration of an image in ***Complex Extended*** format, where the complex pixel at row i and column j is: $$Re (i,j) + i × Im (i,j)$$ #### A Fourier Image in the Complex Extended Format | | | | | |---------------|---------------|----|-----------------| | $$Re(0,0)$$ | $$Im(0,0)$$ | … | $$Im(0,w-1)$$ | | $$Re(1,0)$$ | $$Im(1,0)$$ | … | $$Im(1,w-1)$$ | | … | … | … | … | | $$Re(h-1,0)$$ | $$Im(h-1,0)$$ | … | $$Im(h-1,w-1)$$ | ### The Packed Format The ***Packed*** format is also called ***Complex Conjugate Symmetrical***. * It takes benefit of the conjugate symmetric properties of the Fourier transform of a real image to reduce the Fourier image size and to be the same as the spatial image size. * The following table is an illustration of an image in ***Packed*** format, where the complex pixel at row i and column j is: $$Re (i,j) + i \times Im (i,j)$$ There is a difference between images of odd and even height: #### A Fourier Image in Packed Format, Odd Height | $$
Re(0,0)
$$ | $$
Re(0,1)
$$ | $$
Im(0,1)
$$ | … | $$
Re(0,(w-1)/2)
$$ | $$
Im(0,(w-1)/2)
$$ | $$
Re(0,w/2)
$$ | |-------------------------|-------------------------|-------------------------|-----|-------------------------------|-------------------------------|---------------------------| | $$
Re(1,0)
$$ | $$
Re(1,1)
$$ | $$
Im(1,1)
$$ | … | $$
Re(1,(w-1)/2)
$$ | $$
Im(1,(w-1)/2)
$$ | $$
Re(1,w/2)
$$ | | $$
Im(1,0)
$$ | $$
Re(2,1)
$$ | $$
Im(2,1)
$$ | … | $$
Re(2,(w-1)/2)
$$ | $$
Im(2,(w-1)/2)
$$ | $$
Im(1,w/2)
$$ | | … | … | … | … | … | … | … | | $$
Re(h/2,0)
$$ | $$
Re(h-2,1)
$$ | $$
Im(h-2,1)
$$ | … | $$
Re(h-2,(w-1)/2)
$$ | $$
Im(h-2,(w-1)/2)
$$ | $$
Re(h/2,w/2)
$$ | | $$
Im(h/2,0)
$$ | $$
Re(h-1,1)
$$ | $$
Im(h-1,1)
$$ | … | $$
Re(h-1,(w-1)/2)
$$ | $$
Im(h-1,(w-1)/2)
$$ | $$
Im(h/2,w/2)
$$ | #### A Fourier Image in Packed Format, Even Height | $$
Re(0,0)
$$ | $$
Re(0,1)
$$ | $$
Im(0,1)
$$ | … | $$
Re(0,(w-1)/2)
$$ | $$
Im(0,(w-1)/2)
$$ | $$
Re(0,w/2)
$$ | |---------------------------|-------------------------|-------------------------|-----|-------------------------------|-------------------------------|-----------------------------| | $$
Re(1,0)
$$ | $$
Re(1,1)
$$ | $$
Im(1,1)
$$ | … | $$
Re(1,(w-1)/2)
$$ | $$
Im(1,(w-1)/2)
$$ | $$
Re(1,w/2)
$$ | | $$
Im(1,0)
$$ | $$
Re(2,1)
$$ | $$
Im(2,1)
$$ | … | $$
Re(2,(w-1)/2)
$$ | $$
Im(2,(w-1)/2)
$$ | $$
Im(1,w/2)
$$ | | … | … | … | … | … | … | … | | $$
Re(h/2-1,0)
$$ | $$
Re(h-3,1)
$$ | $$
Im(h-3,1)
$$ | … | $$
Re(h-3,(w-1)/2)
$$ | $$
Im(h-3,(w-1)/2)
$$ | $$
Re(h/2-1,w/2)
$$ | | $$
Im(h/2-1,0)
$$ | $$
Re(h-2,1)
$$ | $$
Im(h-2,1)
$$ | … | $$
Re(h-2,(w-1)/2)
$$ | $$
Im(h-2,(w-1)/2)
$$ | $$
Im(h/2-1,w/2)
$$ | | $$
Re(h/2,0)
$$ | $$
Re(h-1,1)
$$ | $$
Im(h-1,1)
$$ | … | $$
Re(h-1,(w-1)/2)
$$ | $$
Im(h-1,(w-1)/2)
$$ | $$
Re(h/2,w/2)
$$ | * The other coefficients are obtained by using the following properties: * For $$i = 1,…,h-1$$ and $$j = 1,…,w-1$$: * $$ Re (i,j) = Re (h-i,w-j) $$ * $$ Im (i,j) = -Im (h-i,w-j) $$ * For $$j = 1,…,w-1$$: * $$ Re (0,j) = Re (0,w-j) $$ * $$ Im (0,j) = -Im (0,w-j) $$ * For $$i = 1,…,h-1$$: * $$ Re (i,0) = Re (h-i,0) $$ * $$ Im (i,0) = -Im (h-i,0) $$ ## See Also Easy to understand, ready to copy code snippets. Frequency-selective spatial filtering for texture analysis using the Gabor wavelet.