The name is absent

26

sparse in another reconstruction basis Φ (such as wavelets), the reconstruction al-
gorithm substitutes x = Φ(9 in the constraint in equation (3.1) and solve for the
minimum ∕l-norm of θ instead. Figure 3.2(b) demonstrates the CS reconstruction
result using only 500 measurements out of the total 4096 measurements through the
SPGLl algorithm described by van den Berg et al [57]. Using CS allows a reduction
of the number of measurements required for image formation by more than a factor
of eight.

We desire to further improve the reconstruction result by removing the back-
ground profile of the phase, which is not due to the object but is inherent in the
spherical wavefront curvature of the Gaussian beam illumination of the object (see
Figure 3.2(d)). As a result, the phase in the Fourier plane is distorted by the su-
perposition of a spherically varying background. We first remove the object mask in
the setup and obtain a 64 × 64 image of the background phase of the beam through
2D Fourier inversion. These phase values at each pixel form the diagonal entries
of a matrix P. Similar to the modification suggested by Lustig et al [65], we insert
this diagonal matrix P after Φ in equation (2.3) and then solve the phase-corrected
optimization problem for image reconstruction. This phase correction procedure not
only removes the spherically varying phase profile in the reconstruction (compare
Figure 3.2(f) to (d-e)), but also improves the quality of the reconstruction (compare
Figure 3.2(c) to (b)). Using the same procedure as for Figure 3.2(a) to estimate the
reconstructed image resolution, we obtain a resolution of 4.19 pixels and 3.35 pixels
for Figure 3.2(b) and (c) respectively.

Figure 3.3 plots the mean-square error (MSE) between the magnitudes of the
reconstructed image and the reference image in Figure 3.2(a), normalized by the
energy of the reference image, against the number of measurements used in CS. Since

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