The name is absent

24

lens (60 mm). Thus we obtain a pixel size of ∆x = 1.406 mm in both dimensions, at
the chosen wavelength.

In order to estimate the resolution of the reconstructed image in Figure 3.2(a),
a 5 × 15 region containing the left “leg” of the “R” is selected. After averaging
the 5 selected rows, the resulting trace can be approximated by the convolution of
a rectangular function with a Gaussian function with unknown variance (σ²). The
width of the left “leg” of the “R”, i.e., the width of the rectangular function, is 8mm
(measured by a ruler). One can then estimate σ² of the Gaussian function to fit the
average trace. The estimate for σ² is around lmm², equivalent to a Full-Width Half-
Maximum (FWHM) of 2.354mm or 1.68 pixels, which is defined as the resolution of
our reconstructed image. This 2D Fourier inversion technique requires measurements
at all 4096 pixel locations, and is therefore slow.

3.2.2 CS reconstruction with partial dataset

Image reconstruction using CS can achieve good image reconstruction quality from
only a small randomly chosen subset of these 4096 pixels, thus speeding up the imaging
process. Consider the object mask a Iength-A^r signal x of dimension indexed as æ(n),
n ∈ {1,2, ,N}. In this case, x is a 2D image with pixels ordered in a N× 1 vector,
where N = 4096. View the Fourier measurements as projections, y(m) = {x,<ff_n),
of the signal x onto a set of Fourier basis functions {≠_m}, m ∈ {1,2,, M} where
<f>m denotes the transpose of φ_m and (∙,∙) denotes the inner product. Direct 2D
Fourier inversion requires the full dataset, M = N = 4096 measurements for image
reconstruction. However, CS uses only a much smaller number of measurements
than the number of pixels in the image, i.e., M < N. In matrix notation, the
CS system measures у = Фж, where у is an M × 1 column vector of measurements

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