1.6 Diffraction imaging
1.6.1 Phase problems
Phase problems occur in many areas of image recovery, for example, in optical astronomy, X-ray crystallography and electron microscopy. In optical astronomical imaging, the phase of the detected wavefield is distorted when propagating through regions of variable re- fractive index in the turbulent atmosphere. In X-ray crystallography, only the intensity of the diffraction pattern can be measured due to the short wavelength of X-rays. In electron microscopy, although a low resolution image can be measured, high resolution informa- tion can be obtained only in the diffraction plane and the phase of the diffracted electrons can not be measured. Various experimental and computational techniques have been de- veloped to solve phase problems. Computational techniques, which are of most relevance here, are generally referred to as “phase retrieval algorithms.”
If only the amplitude of the Fourier transform of an image is measured, an immediate question is the degree to which the data uniquely define the image. In principle, there are infinite number of possible phase functions that can be used, together with the am- plitude, to reconstruct an image. However, since an image represents a physical object, there are usually restrictions on what represents a feasible image. A particularly important constraint that applies to almost any physical object is compact support(nonzero within a finite region). This simply represents the property of any imaged object being of finite size. Such constraints limit the number of possible solutions, and ideally lead to a single unique solution. Therefore, the practical uniqueness question is: “Given the Fourier amplitude data and appropriate constraints (ora prioriinformation), is it possible to retrieve a unique phase solution?”
Some characteristics of an image are irretrievably lost when the phase is not available. This can be seen as follows: The Fourier amplitude of the images
f(x)
f(x+x1) exp(iφ1)
f∗(−x+x2) exp(iφ2)
are the same, wherex1,x2, φ1,φ2 are real constants and ∗ denotes complex conjugation.
This indicates that the information regarding the absolute position, inversion in the origin and conjugation, and a phase constant of the image are lost. However, these ambigui- ties are generally trivial in most applications and so of little significance. A more serious ambiguity occurs when the imagef(x)can be written as a convolution, i.e.,
Consider the case wheref,gandhare real. The two images
f(x) =g(x) ⊙ h(x)⇒ |F(u)|=|G(u)||H(u)|
f1(x) =g(x) ⊙ h(−x)⇒ |F1(u)|=|G(u)||H(u)|, (1.69)
therefore have the same Fourier amplitude and cannot be distinguished from the ampli- tude information alone. If an image is a convolution ofN components, then there are2N images with the same Fourier amplitude. In general however, it is unlikely that an image will have the form Eq. (1.68). The uniqueness properties of the phase problem are different in the one-dimensional images and two (and more)-dimensional cases.
The uniqueness of phase problem in one dimension was first studied by O’Neill and Walther [14, 15], which is briefly explained here. For a finite size image (f(x) with compact sup- port), the Fourier transformF(u) can be analytically continued into the complexz-plane whereu = Real{z}. The functionF(z) is called ananalyticfunction (i.e., is infinitely dif- ferentiable in the finite complex plane) and can be factorised as
F(z) =k ∞ Y n=1 1− z zn , (1.70)
wherekis a complex constant, i.e., is completely characterised by its zeroszn. The intensity
I =|F|2is then given as I(z) =F(z)F∗(z∗) =|k|2 ∞ Y n=1 1− z zn 1− z z∗ n . (1.71)
Therefore, exchanging (flipping) one of the zeros zi for zi∗ does not change the intensity
I(u). There are therefore an infinite number of images f(x) with the same Fourier am- plitude|F(u)|2and the problem is highly non-unique. Each of these images has the same
support. A similar result exists for a discrete image ofNpixels [16]. In this case the Fourier transform is anN-th order polynomial which, by the fundamental theorem of algebra, can be factorised intoN linear factors, and characterised byN zeros. Flipping each zero leads to2N possible solutions that give the same Fourier amplitude. In general then, the phase problem is highly non-unique in the one-dimensional case.
In the case for two (or more) dimensions, however, the situation is quite different, which was noted first by Bruck and Sodin [16]. In the continuous case, a two-dimensional analytic function cannot generally be factorised, and in the discrete case there is no fundamental theorem of algebra in more than one dimension, i.e., multi-dimensional polynomials are almost always irreducible (cannot be factorised) except rare cases. In general therefore, the two (or more) dimensional phase problem is unique. Uniqueness, for two (or more)
1.6 Diffraction imaging 19
dimensions, can be illustrated with the concept of zero sheets [17, 18]. They showed that the Fourier amplitude of aK-dimensional image is zero on a single continuous surface in a 2K-dimensional space, which is refereed to as a zero sheet. The zero sheet has a dimension of2K−2. The single zero sheet in the multi-dimensional case replaces the point zeros in one-dimensional case, so that only the whole sheet can be flipped.
Given that, in the multi-dimensional case (which are usually concerned), the phase prob- lem has a unique solution, the problem is to find this unique image from measurements of the Fourier amplitude. This is referred to asphase retrievalas it is equivalent to reconstruct- ing the Fourier phase [19, 20, 21].
A multi-dimensional image can be reconstructed directly from its Fourier-amplitude using the concept of zero sheets [17, 18]. However, this approach is computationally too expen- sive and too noise sensitive for practical applications. The most effective and commonly used phase retrieval algorithms are known asiterative transform algorithms (ITA). The ap- proach is to find an image that satisfies the Fourier amplitude data as well as the image- plane constraints (compact support, etc.). This is achieved by iterating between image space and Fourier space, alternately satisfying constraints in each. ITAs for phase retrieval began with the ideas of Gerchberg and Saxton [22] and later developed by Fienup [23]. The basic procedure of iterative transform algorithms is shown in Fig. 1.6. The algorithm gen- erally starts with a random imagef0′(x, y)which is Fourier transformed, and the transform modified to satisfy the Fourier constraints, i.e., the updated Fourier transformFn(u, v)at then-th iteration is given by
Fn(u, v) =|F(u, v)|exp iφFn′(u, v) , (1.72)
whereφ{}denotes the phase. For the case of a support constraint, the image is set to zero outside the support region, i.e.,
fn′+1(x, y) =fn(x, y) where the constraints are satisfied
= 0 otherwise. (1.73)
This is referred to as theerror reduction(ER)algorithmand can also be used to incorporate a positivity constraint [23]. Although this is the most obvious way of applying image space constraints, the ER algorithm often exhibitsstagnationin which progress towards the solution stalls [21]. The probability of stagnation is greatly reduced by using the hybrid- input-output algorithm [23] in which the image is updated as
fn′+1(x, y) =fn(x, y) where the constraints are satisfied
Figure 1.6: The iterative transform algorithm.
where β is a constant, usually between 0.5 and 1, called feedback parameter. The HIO algorithm is generally quite effective. Iterative transform algorithms have recently been formulated in the more general framework ofiterative projection algorithms[24, 25]. This has allowed the development of more general algorithms and a wider variety of constraints.