image =	backwardDiscreteFourierTransform3dImg (inDFTImg3d1,inDFTImg3d2)
image =	backwardDiscreteFourierTransform3dImg (inDFTImg3d1,inDFTImg3d2,inOptDFTConfig)

Detailed Description

Backward Discrete Fourier Transform for an input 3d image.

This algorithm allows to compute the real part of Backward Discrete Fourier Transformation of an input complex 3d image reprensented by its two components $ InDFTImg3d1 $ and $ InDFTImg3d2 $ .

In three dimensions, the real part of Backward Discrete Fourier Transformation (BDFT) of input images $ InDFTImg3d1 $ and $ InDFTImg3d2 $ with size $\left \{ S_x, S_y, S_z \right \}$ is defined as the following real image:

$bdft(x, y, z) = Re \left\{ \sum_{j=0}^{S_z}{\sum_{k=0}^{S_y}{\sum_{l=0}^{S_x}{InDFTImg3d(l, k, j)\exp^{+2\pi i \left (\frac{xl}{S_x}+\frac{yk}{S_y}+\frac{zj}{S_z} \right )}}}} \right\}$

This transformation allows to retrieve forward domain and can be usefull for example in case of filtering of Fourier Frequencies. The following image is an illustration of Backward Discrete Fourier Transformation results:

As in case of Forward Discrete Fourier Transform (see Forward Discrete Fourier Transform 3d), $InDFTConfig$ parameter allows to setup:

input images coordinates domain : if the input images are expressed using polar coordinates we proceed to a polar to cartesian conversion (see Polar to cartesian transformation)
input quadrants policy : if the input images quadrants are centered on lowest frequencies we proceed to a swap operation.
scale policy: during processing, Backward Discrete Fourier Transformation can be scaled by a constant value :
$bdft(x, y, z) = \lambda Re \left (\sum_{j=0}^{S_z}{\sum_{k=0}^{S_y}{\sum_{l=0}^{S_x}{InDFTImg3d(l, k, j)\exp^{+2\pi i \left (\frac{xl}{S_x}+\frac{yk}{S_y}+\frac{zj}{S_z} \right )}}}}\right )$
- eDFTScalePolicy::eDFTSP_Default for default behavior with $\lambda=\frac{1}{S_x}$ along x axis, $\lambda=\frac{1}{S_y}$ along y axis and $\lambda=\frac{1}{S_z}$ along z axis.
- eDFTScalePolicy::eDFTSP_Unitary for unitary BDFT with $\lambda=\frac{1}{\sqrt{S_x}}$ along x axis $\lambda=\frac{1}{\sqrt{S_y}}$ along y axis and $\lambda=\frac{1}{\sqrt{S_z}}$ along z axis.
  See also
  https://en.wikipedia.org/wiki/Discrete_Fourier_transform#The_unitary_DFT
- eDFTScalePolicy::eDFTSP_Custom for a user custom scale factor defined for each axis.

See also: https://en.wikipedia.org/wiki/Discrete_Fourier_transform

Example of Python code :

Example imports

import PyIPSDK

import PyIPSDK.IPSDKIPLIntensityTransform as itrans

Code Example

    # opening of input images
    inImg1 = PyIPSDK.loadTiffImageFile(inputImgPath1)
    inImg2 = PyIPSDK.loadTiffImageFile(inputImgPath2)
    
    # Backward Discrete Fourier Transform of input images
    outImg = itrans.backwardDiscreteFourierTransform3dImg(inImg1, inImg2)

Example of C++ code :

Example informations

Header file

#include <IPSDKIPL/IPSDKIPLIntensityTransform/Processor/BackwardDiscreteFourierTransform3dImg/BackwardDiscreteFourierTransform3dImg.h>

Code Example

    // opening of input images
    ImagePtr pInImg1 = loadTiffImageFile(inputImgPath1);
    ImagePtr pInImg2 = loadTiffImageFile(inputImgPath2);
    // Backward Discrete Fourier Transform of input images
    ImagePtr pOutImg = backwardDiscreteFourierTransform3dImg(pInImg1, pInImg2);