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Aerial view of the single-pixel camera in the lab.

The single-pixel design reduces the required size, complexity, and cost of the photon detector array down to a single unit, which enables the use of exotic detectors that would be impossible in a conventional digital camera. Example detectors include a photomultiplier tube or an avalanche photodiode for low-light (photon-limited) imaging, a sandwich of several photodiodes sensitive to different light wavelengths for multimodal sensing, a spectrometer for hyperspectral imaging, and so on.

In addition to sensing flexibility, the practical advantages of the single-pixel design include the facts that the quantum efficiency of a photodiode is higher than that of the pixel sensors in a typical CCD or CMOS array and that the fill factor of a DMD can reach 90% whereas that of a CCD/CMOS array is only about 50%. An important advantage to highlight is that each CS measurement receives about N / 2 times more photons than an average pixel sensor, which significantly reduces image distortion from dark noise and read-out noise.

The single-pixel design falls into the class of multiplex cameras. The baseline standard for multiplexing is classical raster scanning, where the test functions { φ j } are a sequence of delta functions δ [ n - j ] that turn on each mirror in turn. There are substantial advantages to operating in a CS rather than raster scan mode, including fewer total measurements ( M for CS rather than N for raster scan) and significantly reduced dark noise. See  [link] for a more detailed discussion of these issues.

[link] (a) and (b) illustrates a target object (a black-and-white printout of an “R”) x and reconstructed image x ^ taken by the single-pixel camera prototype in [link] using N = 256 × 256 and M = N / 50 [link] . [link] (c) illustrates an N = 256 × 256 color single-pixel photograph of a printout of the Mandrill test image taken under low-light conditions using RGB color filters and a photomultiplier tube with M = N / 10 . In both cases, the images were reconstructed using total variation minimization, which is closely related to wavelet coefficient 1 minimization   [link] .

Sample image reconstructions from single-pixel camera. (a) 256 × 256 conventional image of a black-and-white “R”. (b) Image reconstructed from M = 1300 single-pixel camera measurements ( 50 × sub-Nyquist). (c) 256 × 256 pixel color reconstruction of a printout of the Mandrill test image imaged in a low-light setting using a single photomultiplier tube sensor, RGB color filters, and M = 6500 random measurements.

Discrete formulation

Since the DMD array is programmable, we can employ arbitrary test functions φ j . However, even when we restrict the φ j to be { 0 , 1 } -valued, storing these patterns for large values of N is impractical. Furthermore, as noted above, even pseudorandom Φ can be computationally problematic during recovery. Thus, rather than purely random Φ , we can also consider Φ that admit a fast transform-based implementation by taking random submatrices of a Walsh, Hadamard, or noiselet transform  [link] , [link] . We will describe the Walsh transform for the purpose of illustration.

We will suppose that N is a power of 2 and let W log 2 N denote the N × N Walsh transform matrix. We begin by setting W 0 = 1 , and we now define W j recursively as

W j = 1 2 W j - 1 W j - 1 W j - 1 - W j - 1 .

This construction produces an orthonormal matrix with entries of ± 1 / N that admits a fast implementation requiring O ( N log N ) computations to apply. As an example, note that

W 1 = 1 2 1 1 1 - 1

and

W 2 = 1 2 1 1 1 1 1 - 1 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 .

We can exploit these constructions as follows. Suppose that N = 2 B and generate W B . Let I Γ denote a M × N submatrix of the identity I obtained by picking a random set of M rows, so that I Γ W B is the submatrix of W B consisting of the rows of W B indexed by Γ . Furthermore, let D denote a random N × N permutation matrix. We can generate Φ as

Φ = 1 2 N I Γ W B + 1 2 D .

Note that 1 2 N I Γ W B + 1 2 merely rescales and shifts I Γ W B to have { 0 , 1 } -valued entries, and recall that each row of Φ will be reshaped into a 2-D matrix of numbers that is then displayed on the DMD array. Furthermore, D can be thought of as either permuting the pixels or permuting the columns of W B . This step adds some additional randomness since some of the rows of the Walsh matrix are highly correlated with coarse scale wavelet basis functions — but permuting the pixels eliminates this structure. Note that at this point we do not have any strict guarantees that such Φ combined with a wavelet basis Ψ will yield a product Φ Ψ satisfying the restricted isometry property , but this approach seems to work well in practice.

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Source:  OpenStax, Introduction to compressive sensing. OpenStax CNX. Mar 12, 2015 Download for free at http://legacy.cnx.org/content/col11355/1.4
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