A study of the coherence parameter of the progressive compressive imager based on radon transform

A study of the coherence parameter of the progressive compressive

imager based on Radon transform

Vladimir Farbera, Elman Eduard

a, Yair Rivenson

b, and Adrian Stern

a

aDept. of Electro-Optical Engineering, Ben-Gurion University of the Negev, Israel;

bElectrical Engineering Department, Ben-Gurion University of the Negev, Israel

ABSTRACT

We have recently introduced a progressive compressing sensing method based on an appropriate sampling

scheme of optical Radon projections. By choosing the sampling steps to be of the size of the golden angle

new information is optimally acquired with each angular sampling step thus permitting gradual improvement

of the reconstructed image. A comparison between the progressive sampling scheme with the conventional

one based on uniform sampling is given in terms of their coherence parameter.

Keywords: Compressive sensing, coherence parameter, Progressive Compressive Imaging, Radon

Projections, Ordered Sets Optimization

1. INTRODUCTION

Compressive Sensing, a.k.a. Compressed Sensing, (CS)1,2

is a relatively new and emerging field in the area of digital

signal sensing and processing. The main idea behind CS is direct acquisition of a compressed version of the signal, in a

way that it can be virtually completely reconstructed by employing appropriate reconstruction techniques. Unlike the

conventional sampling approach according to which in the acquisition step as much as possible samples are taken, and

then the non-informative data is discarded, CS seeks to collect only the relevant information already in the acquisition

step. A natural application of CS is for imaging because the large amount of data involved in typical imaging process.

Several Compressive Imaging (CI) techniques were developed for various purposes, such as reducing hardware

requirements3-6

, shortening image scanning times3,7

, increasing the resolution8-11

or improving other imaging

performances12

. A review of CI techniques may be found in Ref.13 .

Principles of CI system design deviate from the principles used for conventional imaging11

. Conventional imaging seeks

to perform isomorphic mapping; that is, to create images that are exact replica of the object. Ideally, each object point is

mapped to a single pixels sensor so that, besides simple geometrical transformation (e.g., inversion), the captured image

is a sharp copy of the object. In contrast, CS acquisition guidelines prescribe some way of mixing the information so

that multiple image points are projected in a single pixel sensor11

. The preferred type of projection is random projections,

such that all object points are randomly spread on the image sensors. It can be shown that such projections are universal,

that is, they apply for all type of human–ineligible images. CI with random projections has been used with the first time-

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multiplexed CI system, the single pixel camera3, and with the first single–shot CI setup

11 . However, random optical

projections have several applicative shortcomings. It requires long acquisition time with time-multiplexed CI systems or

requires delicate optical design and calibration with spatial multiplexing CI systems. From computational point of view,

it involves large systems matrices which set severe challenges in terms of memory and processing time.

The above mentioned implementing shortcomings can be significantly reduced by using some kind of structured pseudo-

random projections scheme14,15

. Such kinds of projections were proposed in Ref. 7 for imaging where image lines, rather

entire image, are projected on each pixels sensor. The overall sampling pattern, although is not random, exhibits a non-

regular pattern relative to the image rectangular grid. Image lines can be projected on the sensors with help of

anamorphic optical design, using cylindrical lenses, for instance. Figure 1 illustrates the way a cylindrical lens projects

lines from the object plane on a pixel in a line array of sensors.

The concept of Radon transform based CI is shown in Error! Reference source not found.2. The Radon transform

consists of Radon projections which are produced by a cylindrical lens L and collected by a line-array sensor S during

the rotation around the optical axis of lens L17 . In the setup of Fig. 2 the scanning process is performed by concomitant

rotation of the cylindrical lens together with the sensor. By employing an l1 minimization type reconstruction1,2

, much

fewer samples are taken than conventionally. An alternative way of scanning is by keeping the cylindrical lens and the

sensor stationary and rotating an aerial intermediate image generated before the cylindrical lens by means of proper relay

optics was implemented in our research group16

. Using this optical system we were able to demonstrate for the first time,

to the best for our knowledge, reconstructing of compressive Mega-Pixel size images.

Optical Radon transformer (Fig. 1) was used also recently for motion compressive motion detection17

. There, since the

task is simpler than the task of imaging, no scanning is used and all elements are static.

Based on the CI architecture in Ref. 7 we have recently introduced in Ref. 16 a progressive compressive imaging optical

implementation. With this progressive sampling scheme, quasi-independent new projections may be added to those

collected in a previous scan, and the reconstruction image can be gradually refined. Therefore, the angular sampling

density can be increased in a continuous way, canceling the need to resample the object when insufficient measurements

are originally taken. This gives a high flexibility in choosing the initial number of samples, which translates into

acquisition time saving. This and other advantages of the progressive sampling scheme are described with more details in

Sec. 2.

In this work we investigate the reconstruction performance of the progressive sensing scheme proposed in Ref.16 and

compare it to that achieved with the original system in Ref. 7. When applying the compressive sensing framework, a

common measure used to characterize the sensing system performance is the mutual coherence18

. The common

definition of the mutual coherence is:

Figure 1. Object lines are mapped on a sensor by a cylindrical lens.

First point

Object

line

Object plane

First line

Sensor line array

Cylindrical

lens



( ) |⟨ ⟩|

‖ ‖ ‖ ‖ (1)

where Ω is a matrix relating the measurements to the sparse coefficients and are its i's and j's columns. The

definition in (1) is commonly used to evaluate the compressibility of a system. The lowest is , the higher is the

compressibility. Here, we are interested to test behavior of coherence parameter only of the discrete Radon system

matrix , that is, (in the general case , where is a sparsifying operator (e.g., wavelet), so our choice is

equivalent to the definition of sparsifying operator to be the identity matrix, I ). We assume that by applying the

sparsifying operator, the object of interest, which is composed of N total number of pixels, can be sparsely represented

by S<<N coefficients. The total number of compressive measurements is M, such that S<M<N.

2. THE PROGRESSIVE COMPRESSIVE IMAGER

The reconstructed image quality and the total number of required samples for the system in Fig. 2 depend on the number

of Radon projections taken. With the implementation in Ref.7 the number of projections, , is set prior to the

acquisition in accordance to the sparsity of the scene to be imaged. Once the number of projections is set, Radon

projections with an angular step of are taken in the range [ ]. With the progressive sampling procedure is

not set a priori. To achieve progressive acquisition , the angular step needs to be: a) an irrational number of 2π so that no

new projection will overlap a previous one, and, b) such that any new angular sample is as de-correlated as possible from

previous ones.

In Ref.16 we have shown that sampling step, which best fulfils these two requirements is a golden angle sampling step

, which is equal to 0.7639 . Golden angle is illustrated in Fig. 3 and it represent the smaller of two angles created by

sectioning the circumference of a circle according to the golden section; that is we get two arcs such that ratio of the

angle comprised by smaller arc to that of larger arc is equal to the ratio of the angle of the longer arc to the angle of the

full circumference:

(2)

Figure 2: Concept of the optical Radon projections acquisition. The line-array sensor S collects the

ray sum of the object by means of cylindrical lens L1 [7].

Imaging system

Relay optics

L1

f(x,y)

S l

l x

x' y

y'

l x

x' y

y'

Image plane Object plane

g(x,y)

x

x' y

y'



https://www.researchgate.net/publication/5870151_Compressed_imaging_system_with_linear_sensors?el=1_x_8&enrichId=rgreq-e9018b7b-856d-40b8-a911-de774f2e8367&enrichSource=Y292ZXJQYWdlOzI2NTIxNTkyMjtBUzoxMzY5MTcwMTQyOTA0MzJAMTQwOTY1NDk2ODEzMg==

Figure 3. Golden angle

Figure 4 demonstrates reconstructions obtained with the progressive sampling imager described in Ref.16. The

reconstructions were carried out with two-step iterative shrinkage/thresholding (TwIST) algorithm20

. The progressive

image quality improvement in Fig. 4 is evident.

The progressive sampling scheme has several benefits:

a. It removes the need of prior knowledge of the number of samples required. The CS theory indicates the minimal

number of samples for the good reconstruction with a given sparsity of the object. However, in practice, the

sparsity of the object is almost always unknown; therefore one has to guess the number of samples needed. If

this guess is wrong, the entire sampling process should be repeated with higher density, according to a new

guess of the signal sparsity, which in some cases might be wrong as well. For instance, with Radon CI in Ref. 7,

in order to refine the reconstructed image one would need to repeat the entire acquisition process with smaller

angular sampling step (larger ). Obviously, this process would be very time consuming. On the other hand,

with the progressive scheme proposed in Ref.16 one only needs to continue collecting projections from any

point it stopped. This makes it possible to save the otherwise lost time by removing the need of resampling.

b. The nature of the progressive sampling with the golden angle step makes the acquisition process immune to

sudden stops, which otherwise are intolerable with a fixed uniform sampling scheme. In Ref.16 it is

demonstrated that with fixed uniform sampling, a sudden stop yields to a set of Radon projection only within a

limited angular range, which obviously leads to poor reconstruction due to missing angular information. On the

other hand, even after a sudden stop occurring, projections taken with the golden angle step cover almost

uniformly all the angular range.

c. It remedies the computational burden typical for large images. The progressive addition data can be viewed not

only as addition of quasi-independent projections, but also as addition of quasi-independent sets of projections.

This allows employing an Ordered Sets Optimization (OSO) reconstruction algorithm. OSO was first

introduced for this purpose in Ref.16, enabling, what is to best of our knowledge, first optical megapixel size

image CI demonstration.



(a)

(e)

(b)

(f)

(C)

(g)

(d)

(h) Figure 4. Progressive reconstruction with 1.56% (a, e), 3.13% (b, f), 4.3 (c, g) and 5.5% (d, h) of the nominal number of

projections. Images size is 1280x1280 pixels.

In Ref.16we compared the performance of the two angular sampling schemes for the Radon based CI: a) constant

angular step prescribed by a prior knowledge of the number of projections, b) golden angular step with the same number

of projections. There the comparison is done by evaluating the reconstruction mean square error for real and simulated

data. In the next section we describe an object independent comparison by comparing the coherence parameter (1) of the

two sensing mechanisms.

3. THE COHERENCE PARAMETER FOR THE VARIOUS RADON SAMPLING SCHEMES

The optical Radon projection acquisition scheme is shown in Fig. 1. Each pixel sensor integrates the intensity over a

perpendicular line in the object plane. Its adjacent pixel integrates intensity over adjacent image line. Mathematically,

the sensing process is described by the matrix . The sensing matrix can be written as [

]

,

where is number of radial projections and are submatrices describing the Radon projection for a given angle k .

The sensing matrix is sparse21

as demonstrated in Fig.5 , showing the Radon matrix appropriate for acquisition of 5

Radon projections from an image of size 32x32. The sparsity of can be explained by fact that every row describes

contribution of all image pixels into a given sensor, which is equal to sum of elements on straight line image pixels.

Now, since this straight line have a thickness which is equal to effective width of sensor, then each beam collected by

the sensors covers between 2N and 3N (depending on the projection angle) from elements that are not zero for image

of size ..



Figure 5. Grayscale image representing the system matrix for 5 Radon rojections for image of size

.

Here, we wish to calculate the mutual coherence, µ, of the Radon sensing matrix . Technically, this can be performed

by taking the absolute value of maximum off-diagonal element of Gram-matrix , where matrix is

normalization operator defined in following way

‖ ‖

(3)

Mathematical software, such as MATLAB are limiting the size of the matrices for which that can be calculated.

This restriction may be relaxed if we take advantage of the sparisity of the system matrix. ( see Fig. 5).

We have conducted a numerical investigation of the coherence parameter. In the numerical experiment we created a

sequence of matrices for different number of Radon projections. For example, in the uniform sampling scheme when 4

projections are taken, the sensing matrix is [

] , and for 5 projections (see Fig. 5), the

sensing matrix is [

]

.

Computation of the coherence parameter for large images is difficult. The computational load is briefly described in the

following. Let our image have pixels. Then, its Radon matrix will have columns and rows. So,

Radon matrix will have elements. We add factor √ to adjust to the fact that normally the field of view is

rectangular while the Radon transform captured with the sensor S follow a polar geometry (circle support). To compute

the coherence parameter we must perform approximately O( ) multiplication operations. For example, let

and , then number of operations . A single processor performs operations per second,

thus this computation will take seconds, that is, days. But, since Radon matrices have sparse

representation, in this example, each matrix contains at most only 5% not zero values, so we have final time that is equal

to hours for computation the coherence parameter of this Radon matrix.



i

1.05

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

-uniform---gold

10 15 20 25 30 35 40 45 50 55 60projections

Figure 6. Coherence parameter for Radon matrices for uniform and golden angle projections for object size of

256x256 pixels.

Figure 6 shows the coherence parameter of the Radon matrices as a function of the number of projections , for the two

angular sampling schemes, where the solid line represent numerical results for uniform angle projections and dashed line

represent results for golden angle projections. It can be seen that the behavior of both curves is similar in terms of

convergence rate. This implies that the uniform and golden angle sampling scheme perform similarly. As expected, as

more projections are added, the mutual coherence drops till it reaches an asymptotic value. However, we may observe

that the asymptote value is quite large. The coherence parameters for both sampling schemes converge to a value of

approximately 0.7. This fails to explain the good reconstruction results obtained experimentally with these Radon

matrices (see for example Fig.4). The high coherence value can be attributed to two reasons. The first is related to the

finite size of the sensor pixels, therefore each pixel collects the image intensity over a strip rather a line, thus inducing

correlation between projections. The second reason is due the fact that Radon matrix is strictly non-negative. It is known

conventional mutual coherence (1) fails to give a good description of sensing operators with only non-negative entries22

.

We have tried to reduce the effect of positivity by normalizing the Radon matrix, by subtracting the mean of each

column from its entries. Such an operation is motivated in Ref.22, where the preconditioning matrix :

(4)

is applied on the sensing matrix . In equation (4) expressed a matrix of size M that consist of ones.



1

0.95

Ú 0.8

0.75

0.7

0.655 10 15 20 25 30 35 40 45 50 55 60

Projections

Figure 7. Coherence parameter for Radon matrices for uniform angle sampling scheme, with (dashed line) and

without normalization (solid line) for an image of size 64x64.

Figure 7, like a figure 6, shows the coherence parameter of the Radon matrices as a function of the number of projections

, for the uniform angular sampling scheme, but this time dashed line represents results for Radon matrices multiplied

with precondition matrix P, and the solid line represents results only for Radon matrices. We see though, that, the effect

of subtracting the mean reduces only a little the coherence parameter, unlike the case when the system is described by a

random matrix demonstrated in Ref. 22. This may be attributed to the fact that, unlike the random positive matrices in

Ref.22, the Radon matrix exhibits very sparse columns. Therefore, on the one hand we have coherence reduction due to

bipolar values of the preconditioned Radon matrix , but on the other all zero elements in the original Radon matrix

(which are typically more than 95% of the total matrix entries) get a specific negative value (~1/M) and since being the

majority of the matrix elements they introduce correlation between the columns.

4. CONCLUSIONS

We have characterized the performance of the Radon transform based compressive imaging system, with uniform

sampling and progressive sampling, where the latter implements the golden ratio-based angular sampling step. The

performance was analyzed using mutual coherence parameter, which is commonly used in compressive sensing

literature. We have shown that the mutual coherence of both schemes share the same behavior, i.e., both converge to

about the same asymptotic value, at the same rate. Further addition of projections beyond the asymptotic value, will not

improve the object’s reconstruction quality. One of the conclusions from this results is that golden angle scheme can

replace the fixed uniform sampling without compromising the reconstruction quality, meanwhile gaining the benefits of

the progressive scheme. This conclusions obtained here via the analysis of the mathematical system operator validates

previous similar conclusion obtained empirically. We also found that the mutual coherence obtained in our simulations

yield a quite large value, failing to describe the good imaging results obtained with our imager.

This research was partially supported by Israel Science Foundation (grant No.1039/09), and by Harbour foundation.



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A study of the coherence parameter of the progressive compressive imager based on radon transform

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