CS 294-6: Computational Imaging

Tentative Syllabus:

(1) Coded Aperture Imaging.

Gamma rays and high-energy X-rays cannot be refracted or reflected (except through small angles) and so image formation using methods common in optics is either impossible or very limited. Pin-hole image formation and collimation methods may be used, but have a poor trade-off between signal-to-noise ratio and resolution.

An alternative is coded aperture imaging, one view of which is that multiple pin-holes produce an image convolved with the coded aperture. Well designed apertures have flat power spectrum, (with essentially random phase).

Reconstruction in the far field case (X-ray astronomy e.g.) can be accomplished either in the Fourier or the spatial domain by inverting the transfer function of the coded aperture, often simply by correlation or convolution with the mask pattern itself. Mathematical design methodologies for coded apertures with desirable properties exist for various fill-factors, but there is no general theory.

Most mask patterns have 50% fill factor which may not be ideal for some applications, and may not yield the best signal-to-noise ratio. Also, some applications require that no two holes are adjacent on the grid for mechanical integrity of the mask.

In the near-field case (biomedical imaging e.g.) the simple-minded application of coded aperture idea lead to significant artifacts. Some methods have been developed for cancelling such artifacts by combining images taken using different apertures.

(2) Synthetic Aperture Microscopy.

A low-resolution imager can be used with many high-resolution textured illumination patterns to obtain an image set that allows one to reconstruct an image limited in resolution by the illumination rather than the imaging optics. The response at each picture cell is proportional to the integral of the point-by-point product of the surface reflectance pattern, the incident texture and the sensitivity function of the sensor cell.

One version of an instrument exploiting this approach generates the high-resolution textures by interference of a number of laser beams arriving at a surface with the same angle of elevation. In this case the depth of field is very high since the illumination field is constant perpendicular to the surface.

Issues include finding good methods for reconstruction from what are basically many unevenly spaced samples in Fourier transform domain, and finding good illumination methods to best sample the transform domain. Using regularly spaced beams leads to highly anisotropic sampling of the low and the high freuqencies. Calibration using fluorescent microbeads may be used. Extensions to three-dimensional sampling may be explored.

(3) Diaphanography or "Optical Tomography".

Tomography deals with reconstruction from line integrals of absorbing density. MRI can be viewed as dealing with reconstruction from integrals over planes. In both cases the inverse Radon transform can be used to recover the density distribution.

Visible light is scattered and absorbed in living tissue. While some of the light entering at one spot exits at another, it does not travel in a straight line between entrance and exit. After the first few scattering lengths, the direction of photons is essentially randomized and the problem can be treated by an approximation that leads to the diffusion equation in the (scalar) photon flux.

A three-dimensional resistive grid is a useful model for this system. Imagine feeding current into a point on the exterior surface of this grid while measuring the potential at a variety of other points on the surface. Repeat the process for many source positions. Combinations of numerous source positions and detector positions yield a large set of measurements from which one can attempt to recover the three-dimensional density distribution.

The inverse problem is ill-posed for thick enough tissue. Contrast between different tissue types can be enhanced by attaching pigment particles to antigen targets.

(4) Exact Cone Beam Reconstruction.

The logarithm of the ratio of X-ray source intensity to X-ray detector signal is proportional to the integral of density along the line connecting the two. Measuring such integrals along a set of parallel lines in a plane yields a projection whose one-d transform is a slice through the two-d transform of the density distribution in the plane.

Reconstruction from parallel projections thus can be accomplished using the Fourier domain. However, X-ray source output is better utilized using fan-beams, where the rays are no longer parallel. The reconstruction now requires a different approach which can be implemented directly in the spatial domain. Exact methods for fan-beams were worked out in the 70's.

Even better X-ray source output utilization can be had using an area detector. Now the rays form a cone. Presently no exact algorithm for three-dimensional reconstruction from cone-beam data has been implemented.

An approximation due to Feldkamp is widely used for small-angle cone-beams. A much better method may be possible based on a key insight by Grangeat, who discovered a trick to find the radial derivative of the Radon transform from cone beam projections, when the Radon transform itself cannot be so recovered.

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