Introduction

Image formation is typically described as a linear process. Such systems are described by the Fredholm equation of the first kind,

where, for 2-dimensional systems, the object coordinates are and the image coordinates are .

Such a general description of a linear system is termed shift-variant, because the shape of , the impulse response (or point spread function, PSF), might change as the object position, , is shifted. The Hubble Space Telescope (HST) is an example of a shift-variant imaging system. The primary contribution to image degradation in the HST is spherical aberration. This aberration varies only with the exit pupil geometry, and not with field position - it is intrinsically space-invariant. The exit pupil geometry itself, however, changes with field position. This is due to the different longitudinal positions (for the unfolded system) of the stop (primary mirror), the central obscuration, and the primary and central obscuration of the Cassegrain relays in the WF/PC, all of which undergo effective lateral shifts for different field positions (causing vignetting). Hence, the exit pupil geometry changes with field position, so the system response changes with field position (i.e., is shift-variant). The field extent (thus the vignetting in the exit pupil) for the WF/PC is large enough that the shift-variance of the imaging system is substantial.

If the shape of does not change as the object position is changed, then is said to be shift-invariant (alias space-invariant, isoplanatic, or stationary), and the problem is substantially simplified. Of course, the position of in the image plane changes as the object position changes. The formulation of image formation after this simplification is

Shift-invariant systems are much more easily inverted (given , find ) than shift-variant systems. All shift-invariant systems have complex exponentials as eigenfunctions (Gaskill 1978), so Fourier transforms reduce the above convolution into a simple scalar multiplication. Inversion, within the pass-band of the system, then consists simply of a scalar division (in the complex exponential basis), possibly combined with a regularization term to attack the problem of noise amplification. Since very efficient algorithms are available to take these transforms for discrete systems (FFTs), the solution time of is quick in comparison to space-variant inversion. Shift-invariance also has the advantage of being well and widely understood; there are many restoration and processing tools in place which depend upon this simplification (Wiener filtering (Jain 1989), statistical estimation (Jain 1989), and super-resolution (Gerchberg 1989 and Papoulis 1975), to name a few).

Space-variant inversion, excluding null functions, can be achieved from a singular value decomposition (SVD) of (Andrews and Hunt 1977). This method involves finding two sets of eigenvectors, and transforming into one of the eigenvector bases and into the other. Although theoretically this is the same process as discussed above, in practice it is intractable for large systems due to the need to explicitly solve for two (typically large) sets of eigenvalues (whereas one set, complex exponentials, were needed for shift-invariant systems, and were known a priori).

The Landweber algorithm (Landweber 1951) provides an iterative solution to this space-variant problem without having to resort to finding the eigenvectors and eigenvalues needed for SVD. However, even if the impulse response does not change, the algorithm must be fully repeated for each image. This algorithm is an inefficient way to restore a large number of images generated by the same imaging system.

A formulation of space-variant imaging is desired for which the image and object can both be expressed in a common basis (complex exponentials are preferred, since efficient mechanisms for their manipulation are already in place). This amounts to reformulating the space-variant problem into a space-invariant problem. Sawchuck (1972, 1973, 1974), Robbins (1970), and Robbins and Huang (1972) found that for certain problems (coma-like aberrations and motion blur), the image plane could be remapped so that the space-variant problem could be exactly transformed into a space-invariant form. But only impulse responses which could be decomposed into

where the coordinate transformations are

could be transformed. No general method for finding this decomposition was given; this is the goal of this paper. The advantage of obtaining this formulation of imaging is clear: the hard work is done up front - once an impulse response is given and the coordinate transforms are found, the restoration can be done using the space-invariant tools already mentioned. In general, it is impossible to decompose every possible shift-variant impulse response into equivalent shift-invariant responses. If this exact representation is unattainable, then an approximate shift-invariant response is sought.

From Eq. 3, note that the object and image planes can be warped by different transformations ( and , respectively). Different transformations are desired only if there is a warping caused by the imaging system (e.g., the Seidel distortion aberration, non-perpendicular imaging system geometries, magnification, etc.). Under many imaging processes (e.g., blurring, for which the centroid of the impulse response is at the intended image location), the sample positions in the restoration plane must be the same as the sample positions in the image plane. For most restoration procedures, the sample positions in the image plane define the sample positions in the object plane. A geometric warping caused by the imaging system may be corrected separately. For non-warped imagery, which is assumed in all that follows, the object and image planes will undergo the same transformation: . The dimensionality of the problem is then reduced by one-fourth. , , and still need to be found.