Fpssft can be viewed as a lowcomplexity, fourier projectionslice approach for signals that are sparse in the frequency domain. The central slice theorem dictated that if 1dft of projections is added at the center rotated at the corresponding theta then the 2dift of the resultant data in fourier domain is equivalent to the back projections in spacial domain. Photographs focused at different depths correspond to slices at different trajectories in the 4d space. The central slice algorithm1 used in the ct package is wrong. Sep 10, 2015 the fourier slice theorem is the basis of the filtered backprojection reconstruction method. The fourier slice theorem is extended to fanbeam geometry by zhao in 1993 and 1995. Accelerating drr generation using fourier slice theorem on. In fpssft, the dft of an 1d slice of the data is the projection of the dd dft of the data on that same line along which the timedomain slice was taken. Tomography radon transform, slice projection theorem.
Consequently, the formulae expressing important theorems e. The theorem is valid when the inhomogeneities in the object are only weakly scattering and. Pdf generalized fourier slice theorem for conebeam image. Xray computed tomography radon transform fourier slice. Efficient rendering of digitally reconstructed radiographs.
Nonuniform fast fourier transforms using minmax interpolation jeffrey a. It follows from the theorem that a reconstruction can be obtained by a 3d inverse fourier transform. Abstractwe have recently proposed a sparse fourier transform based on the fourier projectionslice theorem fpssft, which is an ef. It is easy clear that the slice projection theorem is still true for directions aligned with the coordinate axes. The fourier slice theorem is the basis of the filtered backprojection reconstruction method. Osa image registration under translation and rotation in. This video is part of the computed tomography and the astra toolbox training course, developed at the vision lab at the university of antwerp, belgium. Fourier transform theorems addition theorem shift theorem convolution theorem similarity theorem rayleighs theorem differentiation theorem. Our result exploits that a slice in frequency space is an isoparameter set in polar coordinates. Consequently, the drr image of a 3d signal or a scalar volume eld is equivalent to the inverse fourier transform of a central slice. Accelerating drr generation using fourier slice theorem on the gpu.
When a function has circular symmetry, its twodimensional fourier transform can be expressed as a hankel transform in terms of the single radial variable. Interpolate onto cartesian grid then take inverse transform. The projection of wavelets defined in these coordinates therefore yields a sequence that is closed under projection and with analytically described functions. Jun 15, 2018 the implementation reconstructs a tomographic image i. This theorem allows the generation of attenuationonly renderings of volume data in on2 log n time for a volume of size n3. Dct vs dft for compression, we work with sampled data in a finite time window. That being said, most proofs are quite straightforward and you are encouraged to try them. The simplest example of the fourier slice theorem is given for a projection at 8 0. An introduction to fourier analysis fourier series, partial di. I sampled a slice of radial spoke of 2d dft of a rectagular image. Instead of using the radon transform fourier slice theorem or algebraic reconstruction techniques art, their method uses random walks to reconstruct the volume from an array of cameras.
The fourier slice theorem is derived by taking the onedimensional fourier transform of a parallel projection and noting that it is equal to a slice of the twodimensional fourier transform of the original object. Ece637 tomographic reconstruction fourier slice theorem s. Direct fourier tomographic reconstruction imagetoimage filter. If the drr is generated as a direct projection of threedimensional ct volume in the spatial domain, it can. The radon transform and the mathematics of medical imaging. A mathematical theorem stating that a periodic function fx which is reasonably continuous may be expressed as the sum of a series of sine or cosine terms called the fourier series, each of which has specific amplitude and phase coefficients kn. Fourier slice theorem reconstruction fourier space. The fourier transform is of fundamental importance in a broad range of applications, including both ordinary and partial di erential equations, quantum mechanics, signal and image processing, control theory, and probability, to name but a few. Instead, convolution back projection is the most commonly used method to recover the image and this will be the topic of discussion in the next section. Our result exploits that a slice in frequency space is an isoparameter set in spherical coordinates. Fourier transform theorems addition theorem shift theorem. The main objective of this paper is to develop an algorithm based on fourier slice theorem for measuring the simultaneous rotation and. This theorem allows us to compute integrals over volumes by extracting slices from a frequency domain representation of the volume. The fourier slice theorem fst holds for parallel xray beams and does not hold for divergent sources.
Fourier slice theorem states that fourier transform of your projections are equal to slices of 2d fourier transform. Fourier slice theorem relates 1d fourier transform of the projection with 2d fourier transform of the original image 25. Ct reconstruction package file exchange matlab central. The horizontal line through the 2d fourier transform equals the 1d fourier transform of the vertical projection. Feb 15, 2005 hi there, i have computed the 2d fourier transform of an image and also the 1d fourier transform of the projection of the same image at 45 degrees. Computed tomography notes, part 1 challenges with projection. A slice of the 3d fourier transform of a rectangular block on the right is equivalent to the 2d fourier transform of the projection of the image of that block on. You have probably seen many of these, so not all proofs will not be presented. Fourier transform is an extremely powerful mathematical tool for the analysis of nonperiodic functions. F u, 0 f 1d rfl, 0 21 fourier slice theorem the fourier transform of a projection is a slice of the fourier. So while the fourier slice theorem illustrates a simple and beautiful relationship between the image and its projections, we cannot put it to use in practical implementation. The fourier series of fx is a way of expanding the function fx into an in nite series involving sines and cosines. Lecture notes optical engineering mit opencourseware.
For a ksparse signal, fpssft achieves sample complexity of ok and. The projection slice theorem says that the fourier transform of px is one slice through fk x, k y, along the k x axis which is parallel to the projection axis the x axis. Use of log polar transform gives a solution to this problem, but at a cost of losing the vital phase information from the image. So you have to use your obtained samples to interpolate the remaining points. First, consider the fourier transform of the object along the line in the. Fourier slice theorem the fourier slice theorem establishes a direct mapping between the projections of multidimensional spatial signals and their fourier transform 10, 11. The end result is the fourier slice photography theoremsection4. Lecture 15 fourier transforms cont d here we list some of the more important properties of fourier transforms.
Direct fourier interpolation method this method makes direct use of the central section theorem. Three different fourier transforms fourier transforms convergence of dtft dtft properties dft properties symmetries parsevals theorem convolution sampling process zeropadding phase unwrapping uncertainty principle summary matlab routines dsp and digital filters 201710159 fourier transforms. The fourier transform of the projection of fx, y has produced a slice of the 2d fourier transform f. The use of the fourierslice theorem as a tool for reprojection was noted in the late 1980s by crawford et al. Another thing we will want to do is to group together similar images and average them, so that we can see what. Projectionslice theorem for fourier series mathematics. To verify the fourier slice theorem, i will have to show that the 1d fourier transform of the projection is equal to a slice of the 2d fourier transform of the image. Therefore, the projection of suitable wavelets defined separably in these coordinates can be computed analytically, yielding a sequence of wavelets closed under. In mathematics, the projection slice theorem, central slice theorem or fourier slice theorem in two dimensions states that the results of the following two calculations are equal. As such, the summation is a synthesis of another function. The theorem states that a slice extracted from the frequency domain representation of a 3d map yields the 2d fourier transform of a projection of the 3d map in a direction perpendicular to the slice figure 1. Efficient rendering of digitally reconstructed radiographs on.
Rev multidimensional sparse fourier transform based on. It is no small matter how simple these theorems are in the dft case relative to the other three cases dtft, fourier transform, and fourier series, as defined in appendix b. The inverse fourier transform the fourier transform takes us from ft to f. Notice that it is identical to the fourier transform except for the sign in the exponent of the complex exponential. The trick will be to determine the projection direction corresponding to each of many singleparticle images. This theorem is used, for example, in the analysis of medical ct scans where a projection is an xray image of an internal organ. We look at a spike, a step function, and a rampand smoother functions too. Hi, is it true that central slice theorem holds only with fourier transform and not discrete fourier transform.
Sutton, member, ieee abstract the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finitelength signals over a set of uniformly spaced frequency locations. When you sample the projections over discrete angles the ft of projections become samples of 2d fourier transform. Projection slice theorem university of california, san diego. Direct fourier tomographic reconstruction imagetoimage. An extension of the projection slice theorem is used todirectly extract the frequencydomain image of an object as viewed from any direction. The fourier slice theorem is proposed by bracewell 1956, which leads to the fourier image reconstruction method for parallelbeam geometry. However, it is not obvious to me what a projection of fourier coefficients should be for an arbitrary direction. In mathematics, the projectionslice theorem, central slice theorem or fourier slice theorem in. The projection slice theorem suppose we have a 2d image, defined by pixel values fx,y, and its 2d fourier transform fk x,k y. Direct fourier reconstruction of a tomographic slice file. Fourier theorems in this section the main fourier theorems are stated and proved. The fourier projection slice theorem states that the inverse transform of a slice extracted from the frequency domain representation of a volume yields a projection of the volume in a direction perpendicular to the slice.
It follows that given the projection data, it should then be possible to estimate the object by simply performing the 2d. By combining the above mentioned conebeam image reconstruction theory and the above mentioned fourier slice theory of. Properties of the fourier transform properties of the fourier transform i linearity i timeshift i time scaling i conjugation i duality i parseval convolution and modulation. The intuition is that fourier transforms can be viewed as a limit of fourier series as the period grows to in nity, and the sum becomes an integral. Using the fourier slice theorem for representation of.
Central slice theorem the central slice theorem, also called projection slice or fourier slice theorem, denes the projections of multidimensional signals in terms of their fourier transforms 11, 10. With appropriate weights, one cycle or period of the summation can be made to approximate an arbitrary function in that interval or the entire function if it too is periodic. Robust sparse fourier transform based on the fourier. Informationtheoretic performance of inversion methods. Volume rendering using the fourier projectionslice theorem. Generalized fourier slice theorem for conebeam image. Each such fouriertransformed view is a planar slice of the volumetric frequency representation. Fourierslice theorem the fourier slice theorem establishes a direct mapping between the projections of multidimensional spatial signals and their fourier transform 10, 11. This video is part of the computed tomography and the astra toolbox training course, developed at the. Last time we saw how, if we know the projection directions, we can use the fourier slice theorem to make a 3d reconstruction. The classical fourier projectionslice based method either. Direct fourier reconstruction of a tomographic slice. Image recognition in the presence of both rotation and translation is a longstanding problem in correlation pattern recognition.
Similarity theorem example lets compute, gs, the fourier transform of. Mar 29, 2017 the central slice algorithm1 used in the ct package is wrong. Properties of the fourier transform properties of the fourier transform i linearity i timeshift i time scaling i conjugation i duality i parseval convolution and modulation periodic signals constantcoe cient di erential equations cu lecture 7 ele 301. Schmalz5 1 institute of biomathematics and biometry, gsf national research center for environment and health, d85764 neuherberg, germany 2 faculty of mathematics, chemnitz university of technology, d09107 chemnitz, germany. Hi there, i have computed the 2d fourier transform of an image and also the 1d fourier transform of the projection of the same image at 45 degrees. Convolutions, sampling, fourier transforms informationtheoretic view of inverse problems. Since rotating the function rotates the fourier transform, the same is true for projections at all angles.
Remembering the fact that we introduced a factor of i and including a factor of 2 that just crops up. The implementation reconstructs a tomographic image i. We present a local fourier slice equation that enables local and sparse projection of a signal. Iterative tomographic image reconstruction using nonuniform. Schmalz5 1 institute of biomathematics and biometry, gsf national research center for environment and health, d85764 neuherberg, germany 2 faculty of mathematics, chemnitz university of technology, d09107 chemnitz.
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