# 5.5 Single-pixel camera  (Page 2/2)

 Page 2 / 2

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 $\left\{{\phi }_{j}\right\}$ are a sequence of delta functions $\delta \left[n-j\right]$ 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 $\stackrel{^}{x}$ 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 ${\ell }_{1}$ minimization   [link] .

## Discrete formulation

Since the DMD array is programmable, we can employ arbitrary test functions ${\phi }_{j}$ . However, even when we restrict the ${\phi }_{j}$ to be $\left\{0,1\right\}$ -valued, storing these patterns for large values of $N$ is impractical. Furthermore, as noted above, even pseudorandom $\Phi$ can be computationally problematic during recovery. Thus, rather than purely random $\Phi$ , we can also consider $\Phi$ 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

${W}_{j}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}{W}_{j-1}& {W}_{j-1}\\ {W}_{j-1}& -{W}_{j-1}\end{array}\right].$

This construction produces an orthonormal matrix with entries of $±1/\sqrt{N}$ that admits a fast implementation requiring $O\left(NlogN\right)$ computations to apply. As an example, note that

${W}_{1}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$

and

${W}_{2}=\frac{1}{2}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 1& -1\\ 1& 1& -1& -1\\ 1& -1& -1& 1\end{array}\right].$

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

$\Phi =\left(\frac{1}{2},\sqrt{N},{I}_{\Gamma },{W}_{B},+,\frac{1}{2}\right)D.$

Note that $\frac{1}{2}\sqrt{N}{I}_{\Gamma }{W}_{B}+\frac{1}{2}$ merely rescales and shifts ${I}_{\Gamma }{W}_{B}$ to have $\left\{0,1\right\}$ -valued entries, and recall that each row of $\Phi$ 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 $\Phi$ combined with a wavelet basis $\Psi$ will yield a product $\Phi \Psi$ satisfying the restricted isometry property , but this approach seems to work well in practice.

#### Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
why its coecients must have a power-law rate of decay with q > 1/p. ?