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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 4260–4271
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Progressive compressive imaging from Radon projections

Sergei Evladov, Ofer Levi, and Adrian Stern  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 4260-4271 (2012)
http://dx.doi.org/10.1364/OE.20.004260


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Abstract

In this work we propose a unique sampling scheme of Radon Projections and a non-linear reconstruction algorithm based on compressive sensing (CS) theory to implement a progressive compressive sampling imaging system. The progressive sampling scheme offers online control of the tradeoff between the compression and the quality of reconstruction. It avoids the need of a priori knowledge of the object sparsity that is usually required for CS design. In addition, the progressive data acquisition enables straightforward application of ordered-subsets algorithms which overcome computational constraints associated with the reconstruction of very large images. We present, to the best of our knowledge for the first time, a compressive imaging implementation of megapixel size images with a compression ratio of 20:1.

© 2012 OSA

1 Introduction

Compressive Sensing (CS) is a new and emerging field in the signal processing world; its main idea is direct acquisition of a signal in a compressed form [1

1. A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007). [CrossRef]

7

7. A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010). [CrossRef]

]. Compressive Imaging (CI) [1

1. A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007). [CrossRef]

, 4

4. Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” Computational Optical Sensing and Imaging, COSI OSA Technical Digest (CD), paper CTuA4 (2009).

7

7. A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010). [CrossRef]

] is an optical implementation of CS where new optical designs replace conventional imaging systems with the aim of saving on sensor cost, time of acquisition or both.

Conventional digital compressing techniques use the redundancy feature of natural images. The compressed image is represented by small set of coefficients, compared to the number of pixels used for the acquisition. The theory of compressed sensing states that an image can be reconstructed with an overwhelming probability from fewer measurements than the conventional number of pixels provided it is compressible in some known basis (e.g. Fourier or wavelet) and an appropriate sensing mechanism is applied.

Optical implementation of random Φ involves several severe challenges [4

4. Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” Computational Optical Sensing and Imaging, COSI OSA Technical Digest (CD), paper CTuA4 (2009).

, 5

5. R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011). [CrossRef]

]. In [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] several compressive imaging (CI) architectures were proposed that compromise randomness for the benefit of optical implementation feasibility, yet exhibiting good performance. Those architectures are based on direct or indirect angular sub-sampling of the Fourier plane. One of the implementations proposed in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] uses optical Radon projections of the object plane onto a line array sensor (vector sensor). Radon projections were used to demonstrate the CS concept in the seminal work of Candes et al. [8

8. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006). [CrossRef]

], while heuristics of their incoherence can be found in [9

9. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008). [CrossRef]

]. The principle of CI based on Radon projections [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] is briefly summarized in subsection 2.1. In this work we present an extension of the concept introduced in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] by considering angular sampling that permits progressive acquisition of the image information. Whereas in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] a uniform angular sampling is proposed, here we propose an alternative sampling scheme in which it is possible to progressively add information. This scheme is based on the irrational golden angle sampling step [10

10. H. Niederreiter, Uniform Distribution of Sequences (Dover Publications, 2006).

, 11

11. M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

].

Progressive sampling is beneficial when the sparsity of the signal is unknown in advance and therefore the number of samples required to be captured is not known prior to capturing. If the number of samples required is not known a-priori one would adopt a scheme that permits successive image improvement with respect to the numbers of samples. The sampling process can be stopped by the user once desired image quality has been achieved. Another benefit of a progressive sampling scheme is immunity to sudden stops during the sampling process; in such a case the partial samples suffice to provide a visual informative image. This point is demonstrated in sub-section 2.2. Another important advantage of the progressive sampling scheme proposed here is that it fits well with ordered-subset reconstruction algorithms, which as we show in Sec. 3 makes CI and reconstruction of large images feasible.

The structure of the paper is as follows. In Sec. 2 we review briefly the compressive sensing technique based on optical Radon projections presented in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

], and then explain the new angular sampling scheme for progressive compressive imaging. Section 3 presents the proposed ordered-sets (OS) estimation algorithm, while Sec. 4 shows the results of the computer simulations and real experiments done with the progressive sampling technique. Section 5 concludes and summarizes this work.

2 Progressive compressive sensing with fixed step angular sampling

2.1 Compressive sensing with optical Radon projections

The CI technique presented in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] relies on capturing Radon projections of the image plane. Unlike Radon transform commonly used in medical tomography, the Radon transform used here is in the object plane, that is, perpendicular to the optical axis. If f(x,y)is the object plane, the Radon transform is given by
f(s,θ)=g(s,θ)=xyf(x,y)δ(sxcos(θ)ysin(θ))dydx,
(1)
where s is the radial coordinate and θis the projection angle. The optical Radon projector is based on a cylindrical lens as shown in Fig. 1
Fig. 1 The principle of optical Radon Projections. The detector S collects the ray sum of the object by means of the cylindrical lens.
. The Radon projection for a given angle g(s) is captured with a linear sensor which may be a one dimensional array of pixels. The linear sensor S is aligned with the imaging axis y’ of the cylindrical lens L and is located in the image plane of the system. In order to obtain the Radon projections at various angles, the lens L and the sensor S rotate concomitantly around the optical axis z.

The original image can be reconstructed from Radon projections which have been captured with the angular sampling technique described above. By employing nonlinear sparsity-promoting algorithms, signals sampled with a compression ratio of approximately 1:10 were reconstructed in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

, 7

7. A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010). [CrossRef]

].

In general, the reconstruction process involves minimization of the cost function J(a)in the form of [5

5. R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011). [CrossRef]

]:
J(a)=12gA(Ψa)2+λΦ(a),
(2)
where ais a sparse vector representing the original image coefficients in lexicographic order in a basis Ψ(e.g. wavelet). A is the operator performing forward projection and in our case is a Radon transform, gis the measured projections vector, and Φis a regularization penalty function, which could be a TV (Total Variation) norm or l1norm, weighted by parameterλ.

Technically, the forward operator Amay be implemented explicitly by an appropriate transition matrix or implicitly, for instance as a function handle using Matlab’s Radon transform. Using A in the form of a matrix has the advantage of accessibility to the precise adjoint AT, which is necessary for iterative reconstruction algorithms. However, its disadvantage is its huge size when large images are considered. For example, if the matrix operator is used with megapixel size images, the number of coefficients in a typical A matrix is of the order O(1012) . Such matrices require memory of the order of terabytes and even in compact format available due the matrix sparsity, it requires gigabytes of memory. Yet, it is clear that in order to handle such a matrix large computational power is required. Fortunately, in Sec. 3 we show how this problem of dimensionality can be remedied significantly owing to an OS reconstruction algorithm available with the progressive sampling scheme proposed here.

2.2 Angular sampling for progressive compressive imaging

The natural and most straightforward angular sampling scheme is one that uses constant angular steps and that samples the object uniformly. Such a scheme requires a-priori knowledge of the number of projections Q that will be sufficient for the reconstruction.

If one samples the object uniformly, with fixed angular step, then these steps can be written in radians asdθ=2πα, withαbeing a normalized step. Since the circle is being sampled repeatedly, starting at the originθ=0, the position of the qth sample can be written as

θq=2παq,q=0,1...,.
(3)

Figure 2(a) (color online) demonstrates the two sampling schemes. The red cross marks relate to the uniform and fixed angular step withθ=2π/20 radians; i.e., according to (3) withα=1/20. The circle is being sampled uniformly by definition. The blue circles relate to the second sampling scheme, where the angular step is an irrational number known also as the “golden angle”θga137.45[deg], an angle subtended by the small arc (b) in Fig. 3
Fig. 3 Arcs, a, and, b, are in a golden ratio relation. Golden angle θgais subtended by arc b.
that obeys

φ=ab=a+ba.
(4)

The ratio  between the two arcs is the golden ratio [12

12. M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Broadway Books, 2003).

] which can be found as the solution of the quadratic equation:φ2φ1=0.

The golden ratio is well known to humanity since ancient times and was thought to be the divine sign, since it appears in nature in snail’s shells, the leaves of the sunflower and more [12

12. M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Broadway Books, 2003).

]. For this reason, ancient artists, architects and philosophers used it extensively in their works. Here we choose to use the sampling step to be the golden angle θgabecause as shown in [11

11. M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

] it has the smallest discrepancy value. The discrepancy criterion [10

10. H. Niederreiter, Uniform Distribution of Sequences (Dover Publications, 2006).

] D(Θ/2π)is used to measure how close the sequence Θ=[θ0,θ1,...,θQ1]is to a uniform sampling and is found to be the closest for the golden angle sampling.

Figure 2 demonstrates that the discrepancy between the two sampling schemes is negligible. It can be seen that sampling with golden angle steps does not yield local aggregation of points and in fact it is almost as uniform as with angular steps of θ=2π/20 .

Figure 2(b) shows angular samples obtained in a scenario in which the scanning process is suddenly stopped in the middle, so only 8 angular samples are captured. It can be seen that despite the small number of samples the angular sampling in steps of θga spans the circle fairly uniformly, whereas sampling with a step of (Q+ΔQ)α=1/20[deg]naturally fails to cover the circle.

2.3 Advantages of angular sampling with golden angle step for progressive compressive imaging

Performing angular sampling with golden angle steps makes it possible to add information progressively. Since the samples do not overlap, the information addition is very simple compared to the regular uniform fixed step sampling. Progressive information addition may be useful when gradual quality improvement is acceptable. This may save time of the overall process, since the detail of interest may be identified at an early stage of acquisition, and the process may then be stopped completely or continued until the result is satisfactory. Consider for example the case in which, because of wrong assumption about object’s sparsity, Q angular samples are taken, leading to an unsatisfactory reconstructed image. In order to refine the image quality, with the sampling scheme using angular sampling steps of θgaone just needs to continue sampling till the reconstruction is adequate. In contrast, with the uniform sampling scheme one needs to guess the number of additional samples required,ΔQ and perform a new scan taking additional new (Q+ΔQ)samples with an angular step ofθ'=2π/(Q+ΔQ).

Progressive sampling is also useful also for transmission purposes; the receiver does not have to wait until all the samples are received and only then reconstruct the image, but can generate gradually improving images from partial data arriving continuously.

The second advantage of the progressive sampling scheme is robustness to sudden system scanning failure as demonstrated in Fig. 2(b). The case considered here is a sudden sampling process cessation, when only 40% of the required projections have been taken. It can be seen that while the red cross signs, denoting sampling step of θ=2π/20 have completed sampling only half of the circle, the golden angle steps denoted by the blue signs have covered the whole circle. The meaning of this is that in spite of the malfunction of the system, the user still gets a reasonable representation, albeit of a degraded quality. We note that for illustrative purposes, in this example we did not account for the symmetry of the Radon transform and we demonstrate the point on a full circle. Obviously, the uniformity of golden samples holds also on part of the circle. Another important advantage of the non-overlapping property of the samples is the applicability of OS reconstruction algorithms, as explained in the following section.

3 Reconstruction with ordered subsets

3.1 The Ordered Subsets reconstruction concept

Reconstruction algorithms working on ordered subsets are probably best known in the realm of medical tomography (e.g., [13

13. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994). [CrossRef] [PubMed]

]). Ordered subsets or block-iterative methods have been applied together with a number of iterative algorithms, with Ordered Subsets Estimation Maximization (OS-EM) being the most widely used. The essential difference between the OS-EM algorithm and conventional algorithms (e.g., Maximum Likelihood Estimation Maximization, ML-EM) is the use of only a subset of projections for updating the estimate rather than using the entire set of measured projections. For OS-EM, the use of only part of the data during the updating process is termed a sub-iteration and the term single iteration usually refers to the use of all the data at once. With OS-EM the reconstruction proceeds by utilizing subsets of the projections, chosen in a specific order that attempts to maximize the new information being added in sub-iterations. The iteration process proceeds by using different projections in each subsequent subset until all projections are used. OS-EM was found to be much faster than the whole set of ML-EM with similar reconstruction quality results. It is important to note that the subset partition should be such that subsets are balanced, meaning that projections are as uniform distributed as possible within the subset [14

14. H. Zaidi, Quantitative Analysis in Nuclear Medicine Imaging (Springer, 2006).

].

3.2 Ordered sets obtained by golden angle sampling

Here we follow the main idea used within the statistical algorithm OS-EM. However, instead of performing Estimation Maximization (EM) on the ordered sets [13

13. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994). [CrossRef] [PubMed]

] we do an l1 type minimization (2), as prescribed by the CS theory. The captured projections are divided into subsets and iteration proceeds by using different projections in each subsequent subset until all projections are used [14

14. H. Zaidi, Quantitative Analysis in Nuclear Medicine Imaging (Springer, 2006).

].

The proposed OS algorithm works as follows. The algorithm stages are divided into inner sub-iterations and outer iterations. Each sub-iteration, q, acts on the sub-set Gq. One full iteration is defined when all the sub-sets have been optimized once. The first outer iteration sets the initial guess as zero and passes it to the first inner iteration, which feeds the set of measurements G1 into the solver, which performs a CS optimization. The output of the first inner iteration is passed to the second inner iteration as an updated guess, while the set of the measurements is changed to G2. This process continues until all the sub-sets have been optimized. After all the sets have passed the inner iteration, the minimization result is passed back to the outer iteration and is set as an initial guess for the following outer iteration. We found empirically that three to four outer iterations are sufficient, and there is no additional improvement with a larger number of outer iterations.

Besides the advantage of using OS in reducing the processing time, there is a significant advantage in terms of memory requirements. Recall that, as explained in section 2.1, the dimensions of the forward matrix A representing the set of Radon projections may be extremely large. However, when the measurements are divided into sets according to the golden angle sampling step, like in OS, the problem ofA's dimensionality mentioned in section 2.1 can be avoided, and the computations may be performed on a regular PC. Recall that the golden angle sampled subsets are built in a progressive way, while containing partial information about the whole image and not a fraction of it. Thus a much smaller A(G)which can be created and stored separately from the other parts of is Afacilitated.

4 Results

In order to demonstrate the progressive compressive imaging technique we performed simulations and optical experiments, where in both cases the image was reconstructed from a different number of projections acquired with the proposed sampling scheme. The reconstruction process is done using the TwIST solver [15

15. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007). [CrossRef] [PubMed]

] on the simulated or optically acquired measurements.

4.1 Simulated experiment

The simulation relates to the acquisition process. The purpose of the simulation is to verify the progressive improvement of the reconstruction with the increasing number of samples when a golden angle step, θga, is used. For comparison, we also simulated in parallel an experiment in which Radon projections were taken with a uniform sampling scheme; i.e., with steps ofθu=2π/Q radians. This latter auxiliary experiment refers to the case when the number of the samples that should be taken for good reconstruction quality is known in advance.

For the numerical test we used the synthetic image in inset Fig. 4(c)
Fig. 4 Reconstruction quality as function of number of Radon projections. (a) Reconstruction in the wavelet domain. (b) Reconstruction in the space domain, (c) Synthetic image used in simulation (1280 x 1280 pixels)
which is highly sparse in the “Haar” wavelet basis with sparsity K = 1.4% non-zero elements. In the simulation of the acquisition part, the Radon projections of the synthetic image are generated at an increasing number of angles by means of the forward projection matrix. The reconstruction part computes the approximations from the acquired projections, which are then compared to the original image in terms of PSNR:
PSNR=20log10(MAXIMSE),
(5)
where R is the reference image (Fig. 3(c)) and I is the reconstructed image, both of size m×n pixels and MSE denotes the mean-square-error given by

MSE=1mni=0m1j=0n1[R(i,j)I(i,j)]2.
(6)

Figure 4 shows the reconstruction PSNR, measured in [dB], versus the number of projections for the golden angle and uniform sampling scheme.

Figure 4(a) shows the results for signal reconstruction simulation in the space domain, where the vector ain Eq. (2) is the signal itself andΨ=I. The gradual improvement in the reconstruction quality with increasing number of samples is clearly seen in Fig. 4(b). The golden angle scheme is best suited for this scenario since its advantage is in reducing acquisition time by adding new samples, in contrast to the uniform angle scheme, where re-sampling is required.

Figure 4(b) shows the reconstruction performed in the wavelet domain, i.e., with Ψ in (2) being Haar wavelet transform. It can be seen that the improvement of the reconstruction quality with increasing number of samples is faster in Fig. 4(b) than in Fig. 4(a) because the image is much sparser in the wavelet domain. The improvement rate is similar for both sampling schemes (withθuandθga), which implies that a similar number of samples is needed in order to achieve a given reconstruction quality.

4.2 Optical experiment

We built an optical setup following the incoherent CI approach outlined in [6

6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

] with the ability to automatically capture projections and with controllable and precise angular sampling. The scheme of the system is shown in Fig. 5
Fig. 5 Schematic diagram of the automatic CI optical system. Radon projections are recorded with the linear detector aligned with the Y axis and located in the image plane of the system.
. Unlike the concept presented in Fig. 1, where both the lens, L, and the sensor, S, are rotated synchronously about the axis, the only rotating element in the system of Fig. 5 is a prism, which rotates an aerial image at the input of the cylindrical lens. The schematic drawing in Fig. 5 shows an object, which sends a bundle of rays into the beam splitter. Half of the rays are directed into the rotating prism, which reflects them back through the beam splitter and through the cylindrical lens onto the linear detector located in the image plane of the lens. The rotating right angle prism is mounted on the computer-controlled rotating stage with z being the axis of rotation. Each position of the prism enables a specific angle of the Radon projection.

As in the simulation part, we took several sets of projections from 1.5% to 5.5% of the nominal reconstruction number of samples. For an image of size n×nwe defined the nominal number of projections to be n, so that the entire Radon transform will have approximately n2 point samples. For the quantitative evaluation of the reconstruction we compare the reconstructed images to the virtually perfect reference image (“golden standard”) taken to be a reconstruction made from 20% of the nominal number of the projections. This choice is reasonable since according to our experiments and simulations good reconstruction quality has been achieved with 20% of the projections. For instance, in Fig. 5 it can be seen that with 20% of the nominal projections (256 projections) the asymptotically best image is reached.

As explained in section 2.3, the golden angle sampling scheme permits OS reconstructions with the explicit Radon forward operator even for large images. The advantage of using the precise forward matrix, A, instead of an implicit forward operator is demonstrated in Fig. 7
Fig. 7 (a) is the reference image, (b) is the reconstruction in the golden sampling scheme, image (c) is the reconstruction in the uniform sampling scheme, image (d) is OS reconstruction using the explicit Radon transition matrix. All images are cropped to 1024 x 1024 pixels.
. Figure 7(a) shows the real life object of size 1280 x 1280 pixels. Figures 7(b)7(d) show the reconstructions obtained from 70 projections, yielding a compression ratio of approximately 20:1. Figures 7(b) and 7(d) show the reconstructions obtained by applying the TwIST algorithm to the entire set of projections. The reconstruction is carried out with an implicit forward Radon operator implemented as a function handle referring to Matlab’s Radon transform function. The image in Fig. 7(b) is obtained from uniform samples and Fig. 7(c) from projections taken with golden angle sampling scheme. Figure 7(d) shows the reconstruction obtained from the OS reconstruction applied on the same number of projections captured following the golden angle sampling scheme (Sec. 2.2), where the number of sub-sets is 14 and each sub-set contains 5 projections. Here, the explicit Radon transition matrix, A, is appropriate for the subset. It can be seen that the visual quality of OS reconstruction in Fig. 7(d) is higher compared to the similar CS reconstruction without OS shown in Fig. 7(b) and Fig. 7(c). The OS result has less radial artifacts which are present in non-OS reconstructions. The difference between the reconstructions is attributed to the difference between the projecting operators. The transition matrix yields less numerical errors and permits a precise adjoint operator (simply the adjoint of the matrix A), and thus yields a stable and accurate reconstruction.

5 Conclusions

The simulation and laboratory results show that in spite of the pseudo-randomness of the sampling pattern, the reconstruction results display insignificant differences in quality compared to the standard uniform sampling scheme. We have demonstrated optical compression ratios of 20:1 with satisfying reconstruction quality. The progressive sampling scheme, upon which we rely, enables excellent tradeoff between the compression and the quality of reconstruction.

It is also shown that the angular sampling used for progressive CI fits well an ordered subset (OS) reconstruction approach. The OS algorithm accelerates the reconstruction and facilitates the processing of large images.

Acknowledgment

This research was supported by the Israel Science Foundation (grant No. 1039/09).

References and links

1.

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007). [CrossRef]

2.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008). [CrossRef]

3.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). [CrossRef]

4.

Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” Computational Optical Sensing and Imaging, COSI OSA Technical Digest (CD), paper CTuA4 (2009).

5.

R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011). [CrossRef]

6.

A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007). [CrossRef] [PubMed]

7.

A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010). [CrossRef]

8.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006). [CrossRef]

9.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008). [CrossRef]

10.

H. Niederreiter, Uniform Distribution of Sequences (Dover Publications, 2006).

11.

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

12.

M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Broadway Books, 2003).

13.

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994). [CrossRef] [PubMed]

14.

H. Zaidi, Quantitative Analysis in Nuclear Medicine Imaging (Springer, 2006).

15.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007). [CrossRef] [PubMed]

OCIS Codes
(110.0110) Imaging systems : Imaging systems
(110.1758) Imaging systems : Computational imaging

ToC Category:
Imaging Systems

History
Original Manuscript: December 20, 2011
Manuscript Accepted: January 27, 2012
Published: February 6, 2012

Citation
Sergei Evladov, Ofer Levi, and Adrian Stern, "Progressive compressive imaging from Radon projections," Opt. Express 20, 4260-4271 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4260


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References

  1. A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol.3(3), 315–320 (2007). [CrossRef]
  2. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag.25(2), 21–30 (2008). [CrossRef]
  3. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006). [CrossRef]
  4. Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” Computational Optical Sensing and Imaging, COSI OSA Technical Digest (CD), paper CTuA4 (2009).
  5. R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng.50(7), 072601 (2011). [CrossRef]
  6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett.32(21), 3077–3079 (2007). [CrossRef] [PubMed]
  7. A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser.206, 012019 (2010). [CrossRef]
  8. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006). [CrossRef]
  9. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag.25(2), 72–82 (2008). [CrossRef]
  10. H. Niederreiter, Uniform Distribution of Sequences (Dover Publications, 2006).
  11. M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process.18, 2091–2098 (2010).
  12. M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Broadway Books, 2003).
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