The Fourier transform of a sequence is in general complex-valued, and the unique representation of a sequence in the Fourier transform domain requires both the phase and magnitude of the Fourier transform. It is often desirable to synthesize or reconstruct a signal from only partial Fourier domain information.
The Fourier transform of h(n1, n2) is said to converge to in the mean square sense when
Consider a 2-D sequence x(n1,n2) with Fourier transform so that
It has been observed that a straightforward signal synthesis from the Fourier transform phase alone often captures most of the intelligibility of the original signal x(n1,n2). A straightforward synthesis from the Fourier transform magnitude alone, however, does not generally capture the original signal’s intelligibility. To illustrate this, we synthesize the phase-only signal xP(n1,n2) and the magnitude-only signal xm(n1,n2) by
where F-1[.] represents the inverse Fourier transform operation. In phase-only signal synthesis, the correct phase is combined with an arbitrary constant magnitude. In the magnitude-only signal synthesis, the correct magnitude is combined with an arbitrary constant phase. In this synthesis, often preserves the intelligibility of while does not. An example of this is shown in Figure 1.31. Figure 1.31(a) shows an original image , and Figures 1.31(b) and (c) show and , respectively.
An experiment which more dramatically illustrates the observation that phase-only signal synthesis captures more of the signal intelligibility than magnitude-only synthesis can be performed as follows. Consider two images and . From these two images. we synthesize two other images and by
In this experiment, captures the intelligibility of , while captures the intelligibility of . An example is shown in Figure 1.32.
Figures 1.32(a) and (b) show the two images and and Figures 1.32(c) and (d) show the two images and .
The high intelligibility of phase-only synthesis raises the possibility of exactly reconstructing a signal from its Fourier transform phase . This is known as the magnitude-retrieval problem. In fact, it has been shown that a sequence is uniquely specified within a scale factor if x(n1,n2) is real and has finite extent, and if its Fourier transform cannot be factored as a product of lower-order polynomials in and . Typical images are real and have finite regions of support. In addition, the fundamental theorem of algebra does not apply to 2-D polynomials, and their Fourier transforms cannot generally be factored as products of lower-order polynomials in and . Typical images, then, are uniquely specified within a scale factor by the Fourier transform phase alone.
Two approaches to reconstructing a sequence from its Fourier transform phase alone have been considered. The first approach leads to a closed-form solution and the second to an iterative procedure. In the first approach, is expressed as
where Rx is the region of support of . Rewriting (1.47), we have
Equation (1.48) is a linear equation for the unknown values in for each frequency . If there are N2 unknown values in , we can obtain a set of N2 linear equations for by sampling at N2 points. If the frequencies are sampled at distinctly different points, noting that is an odd function and is periodic with a period of 2p x 2p. The solution to the set of N linear equations can be shown to be k, where k is an arbitrary real scaling factor. An example of signal reconstruction from phase using (1.48) is shown in Figure 1.33. Figure 1.33(a) shows an image of 16 x 16 pixels, and Figure 1.33(b) shows the reconstruction. The scaling factor of the reconstructed sequence in the figure is chosen such that the reconstruction will match the original sequence.
The reconstruction algorithm discussed above is reasonable for a small size image, but is not practical for an image of typical size. For example, reconstructing an image of 512 x 512 pixels using (1.48) requires the solution of approximately a quarter of a million linear equations. An alternate approach is to recognize that the solution to the phase-only reconstruction problem must satisfy constraints in both the spatial and frequency domains. Specifically, the solution
· must be real,
· must be zero outside the known region of support, and
· must have a nonfactorable Fourier transform.
In addition, the phase of the Fourier transform of the solution must be the same as the given. A useful approach to solving such a problem is an iterative procedure, in which we impose the spatial and frequency domain constraints separately in each domain. An iterative procedure for the phase-only reconstruction is shown in Figure 1.34. In the procedure, we begin with an initial estimate of the signal. This can be any real sequence with the same region of support as x(n1,n2). We next compute its Fourier transform. We then replace the Fourier transform with the given . The Fourier transform magnitude is not affected. We then compute the inverse Fourier transform of the modified Fourier transform. Due to the modification in the Fourier transform domain, the sequence is no longer zero outside the known region of support of x(n1,n2). We now impose the spatial domain constraint by setting the sequence to zero outside the known region of support. The resulting sequence is a new estimate of the solution. This completes one iteration in the iterative procedure. When the initial estimate of the sequence chosen is real, the constraint that the solution is real will automatically be satisfied. The above algorithm can be shown to converge to the desired solution. An example of signal reconstruction from phase using the iterative procedure in Figure 1 .34 is shown in Figure 1.35. Figure 1.35(a) shows an original imageof 128 x 128 pixels. Figures 1.35(b), (c), and (d) show the results of the iterative procedure after one iteration [phase-only synthesis of (1.39)], 10 iterations, and 50 iterations. The initial estimate used is d(n1,n2).
Although the magnitude-only synthesis of (1.44) does not capture the intelligibility of typical signals, almost all typical images are also uniquely specified by the Fourier transform magnitude. Specifically, if x(n1,n2) is real, has finile extent, and has a nonfactorable Fourier transform, then x(n1,n2) is uniquely specified by its Fourier transform magnitude within a sign factor, translation, and rotation by 180° degrees. This raises the possibility of exactly reconstructing x(n1,n2) from within a sign factor, translation and rotation by 180° degrees. This is known in the literature as the phase-retrieval problem, and has many more potential applications than the phase-only reconstruction problem. Unfortunately, none of the algorithms developed to date are as straightforward or well-behaved as the algorithms developed for the phase-only reconstruction problem. It is possible to derive a closed-form algorithm or a set of linear equations that can be used in solving for x(n1,n2) from , but their derivation is quite involved. In addition, the closed-form solution is not practical for an image of reasonable size due to the large number of linear equations that must be solved. It is also possible to derive an iterative procedure similar to that in Figure 1.34, which was developed for the phase-only reconstruction. The only modification required is to replace the Fourier transform magnitude with the given rather than to replace the Fourier transform phase with the given when the frequency domain constraints are imposed. The algorithm has been observed to converge to the desired solution when the initial estimate used is quite accurate or the signal x(n1,n2) has a special characteristic such as a triangular region of support. The magnitude-only reconstruction problem specifies x(n1,n2) within a sign factor, translation, and rotation by 180°, and, therefore, more than one solution is possible. Imposing an initial estimate sufficiently close to a possible solution or imposing additional constraints such as a triangular region of support appear to prevent the iterative procedure from wandering around from one possible solution to another, in general, however, the algorithm does not converge to the desired solution. Figure 1.36 shows an example of signal reconstruction from the magnitude using a closed-form algorithm. Figures 1.36(a) and (b) show the original and the reconstruction respectively. Developing a practical procedure that can be used to reconstruct x(n1,n2) from remains a problem for further research.