Fannjiang - Revision history

imported>Projects221: /* Introduction */

2013-03-21T07:07:02Z

Introduction

← Older revision		Revision as of 07:07, 21 March 2013
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	==Introduction==		==Introduction==

	Compressed sensing, which proposes the recovery of sparse signals from highly sub-Nyquist sampling, is a recent advancement in signal processing that has always intrigued me. Just as image compression schemes like JPEG have helped lighten expensive data storage and transmission, compressed sensing could reform the expensive data acquisition that is an element of many modern imaging systems. (Romberg 2008 gives a great introduction.) For my project, I did a simple implementation of compressed sensing for an imaging system used in astronomy, known as radio interferometry. Early in the literature of compressed sensing, radio interferometry was noted in Wiaux et al. 2009 and others as an ideal candidate for application, as it conforms to the two principles compressed sensing works by. The first is signal sparsity, where most of the signal's coefficients are zero or near-zero in some transform space; the second is incoherence of the measurement matrix, or a property known as the restricted isometry property, where (roughly speaking) the underdetermined measurement matrix has columns that are "nearly" orthogonal. The measurement matrix in radio interferometry is in essence the "partial Fourier ensemble" described in Donoho 2006, or "the collection of ~~n×m~~ matrices made by sampling n rows out of the m × m Fourier matrix". The partial Fourier ensemble was one of the first measurement matrices proven to work well for compressed sensing, as shown in Candès et al. 2006, which makes radio interferometry a natural candidate.		Compressed sensing, which proposes the recovery of sparse signals from highly sub-Nyquist sampling, is a recent advancement in signal processing that has always intrigued me. Just as image compression schemes like JPEG have helped lighten expensive data storage and transmission, compressed sensing could reform the expensive data acquisition that is an element of many modern imaging systems. (Romberg 2008 gives a great introduction.) For my project, I did a simple implementation of compressed sensing for an imaging system used in astronomy, known as radio interferometry. Early in the literature of compressed sensing, radio interferometry was noted in Wiaux et al. 2009 and others as an ideal candidate for application, as it conforms to the two principles compressed sensing works by. The first is signal sparsity, where most of the signal's coefficients are zero or near-zero in some transform space; the second is incoherence of the measurement matrix, or a property known as the restricted isometry property, where (roughly speaking) the underdetermined measurement matrix has columns that are "nearly" orthogonal. The measurement matrix in radio interferometry is in essence the "partial Fourier ensemble" described in Donoho 2006, or "the collection of n × m matrices made by sampling n rows out of the m × m Fourier matrix". The partial Fourier ensemble was one of the first measurement matrices proven to work well for compressed sensing, as shown in Candès et al. 2006, which makes radio interferometry a natural candidate.

	"Sampling n rows out of the m x m Fourier matrix" leaves open the question of ''which'' n rows, however. In radio interferometry, the set of rows is determined by the placement of the radio antennas in the antenna array. Random sampling has proven to be critical to much of compressed sensing theory, which clashes with the highly patterned arrays used in radio interferometry such as the prominent Very Large Array (VLA). Wenger et al. 2010 noted this clash, and ran a few numerical simulations to see whether a randomized array or the traditional patterned array would work better for compressed sensing. The authors report that the VLA's patterned array actually outperforms a uniform random array, which I thought was an interesting and somewhat counterintuitive result. So for my project, I ran similar numerical simulations, with some deviations from the paper:		"Sampling n rows out of the m x m Fourier matrix" leaves open the question of ''which'' n rows, however. In radio interferometry, the set of rows is determined by the placement of the radio antennas in the antenna array. Random sampling has proven to be critical to much of compressed sensing theory, which clashes with the highly patterned arrays used in radio interferometry such as the prominent Very Large Array (VLA). Wenger et al. 2010 noted this clash, and ran a few numerical simulations to see whether a randomized array or the traditional patterned array would work better for compressed sensing. The authors report that the VLA's patterned array actually outperforms a uniform random array, which I thought was an interesting and somewhat counterintuitive result. So for my project, I ran similar numerical simulations, with some deviations from the paper:

imported>Projects221: /* Appendix */

2013-03-21T06:59:23Z

Appendix

← Older revision		Revision as of 06:59, 21 March 2013
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	==Appendix==		==Appendix==

	MATLAB code: [File: MATLABCode.zip]		MATLAB code: [[File: MATLABCode.zip]]

	Link to a PostScript-format download of the ''VLA Green Book'', which publishes the polar coordinates of their antennas in Chapter 1: [https://science.nrao.edu/facilities/vla/obsolete/green-book]		Link to a PostScript-format download of the ''VLA Green Book'', which publishes the polar coordinates of their antennas in Chapter 1: [https://science.nrao.edu/facilities/vla/obsolete/green-book]

imported>Projects221: /* Appendix */

2013-03-21T06:56:57Z

Appendix

← Older revision		Revision as of 06:56, 21 March 2013
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	==Appendix==		==Appendix==

			MATLAB code: [File: MATLABCode.zip]

	Link to a PostScript-format download of the ''VLA Green Book'', which publishes the polar coordinates of their antennas in Chapter 1: [https://science.nrao.edu/facilities/vla/obsolete/green-book]		Link to a PostScript-format download of the ''VLA Green Book'', which publishes the polar coordinates of their antennas in Chapter 1: [https://science.nrao.edu/facilities/vla/obsolete/green-book]

imported>Projects221: /* Protocol */

2013-03-21T06:50:24Z

Protocol

← Older revision		Revision as of 06:50, 21 March 2013
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	===Protocol===		===Protocol===

	* Calculate two versions of the measurement matrix <math>\Theta</math> using the formula given by the van Cittert-Zernike theorem, one with the VLA and one with a random Gaussian array. I roughly eyeballed the sparsity of the images, and took 1/3 the number of measurements traditionally needed for each image. For instance, for a 200 x 200 = 40,000-pixel image, I used a measurement matrix with 40,000/3 <math>\approx</math> 16,000 rows, instead of the full 40,000 rows.		* Calculate two versions of the measurement matrix <math>\Theta</math> using the formula given by the van Cittert-Zernike theorem, one with the VLA and one with a random Gaussian array. I roughly eyeballed the sparsity of the images, and took 1/3 the number of measurements traditionally needed for each image. For instance, for a 200 x 200 = 40,000-pixel image, I used a measurement matrix with 40,000 / 3 <math>\approx</math> 16,000 rows, instead of the full 40,000 rows.
	* Calculate the DCT matrix for the sparsifying matrix <math>\Psi</math>.		* Calculate the DCT matrix for the sparsifying matrix <math>\Psi</math>.
	* Calculate <math>\Phi = \Theta \times \Psi</math>.		* Calculate <math>\Phi = \Theta \times \Psi</math>.

imported>Projects221: /* Compressed Sensing in Astronomical Imaging */

2013-03-21T06:49:12Z

Compressed Sensing in Astronomical Imaging

← Older revision		Revision as of 06:49, 21 March 2013
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	* I incorporated the discrete cosine transform (DCT) covered in class to represent images sparsely, whereas Wenger et al. just took advantage of the fact that radio images tend to be sparse in the pixel domain.		* I incorporated the discrete cosine transform (DCT) covered in class to represent images sparsely, whereas Wenger et al. just took advantage of the fact that radio images tend to be sparse in the pixel domain.
	* I used orthogonal matching pursuit (OMP), a greedy compressed sensing algorithm, whereas the original paper used SparseRI, a compressed sensing recovery scheme the authors designed specifically for radio interferometry.		* I used orthogonal matching pursuit (OMP), a greedy compressed sensing algorithm, whereas the original paper used SparseRI, a compressed sensing recovery scheme the authors designed specifically for radio interferometry.
	* I compared the VLA to a Gaussian array, as many of the massive up-and-coming interferometers like the Atacama Large Millimetre/submillimetre Array (ALMA) (which celebrated its inauguration this month!) and Square Kilometre Array (SKA) seem to be centrally condensed as well as randomized, whereas the paper compared the VLA to a uniform random array.		* I compared the VLA to a random Gaussian array, as many of the massive up-and-coming interferometers like the Atacama Large Millimetre/submillimetre Array (ALMA) (which celebrated its inauguration this month!) and Square Kilometre Array (SKA) seem to be centrally condensed as well as randomized, whereas the paper compared the VLA to a uniform random array.

	To simulate signals from astronomical radio sources, I just took images the VLA has published on their online image gallery [http://images.nrao.edu/] and vectorized them. If the radio source is thought of of as an array of point sources, then the brightness value of each pixel in the image represents the intensity of each point source.		To simulate signals from astronomical radio sources, I just took images the VLA has published on their online image gallery [http://images.nrao.edu/] and vectorized them. If the radio source is thought of of as an array of point sources, then the brightness value of each pixel in the image represents the intensity of each point source.
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	</center>		</center>

	I could not find published coordinates of the ALMA antennas, or the planned coordinates of the SKA antennas. Given their centrally condensed and randomized nature, I modeled them with a Gaussian distribution of 27 antennas (since the VLA has 27 antennas). Boone 2002 notes that today's radio astronomy community believes a Gaussian sampling of the Fourier plane to be ideal, so this model should be suitable:		I could not find published coordinates of the ALMA antennas, or the planned coordinates of the SKA antennas. Given their centrally condensed and randomized nature, I modeled them with a random Gaussian distribution of 27 antennas (since the VLA has 27 antennas). Boone 2002 notes that today's radio astronomy community believes a Gaussian sampling of the Fourier plane to be ideal, so this model should be suitable:

	<center>		<center>
	<gallery widths=400px heights=300px>		<gallery widths=400px heights=300px>
	File:GaussianAntennas.png \| Gaussian distribution of antennas, which I used to model today's prototypical array (e.g. the ALMA and the SKA) in my simulations.		File:GaussianAntennas.png \| Random Gaussian distribution of antennas, which I used to model today's prototypical array (e.g. the ALMA and the SKA) in my simulations.
	File:GaussianBaselines.png \| Baselines of a Gaussian array on the Fourier plane. Calculated by finding the displacement vectors between all pairs of antennas in the array.		File:GaussianBaselines.png \| Baselines of a random Gaussian array on the Fourier plane. Calculated by finding the displacement vectors between all pairs of antennas in the array.
	</gallery>		</gallery>
	</center>		</center>
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	===Protocol===		===Protocol===

	* Calculate two versions of the measurement matrix <math>\Theta</math> using the formula given by the van Cittert-Zernike theorem, one with the VLA and one with a Gaussian array. I roughly eyeballed the sparsity of the images, and took 1/3 the number of measurements traditionally needed for each image. For instance, for a 200 x 200 = 40,000-pixel image, I used a measurement matrix with 40,000/3 <math>\approx</math> 16,000 rows, instead of the full 40,000 rows.		* Calculate two versions of the measurement matrix <math>\Theta</math> using the formula given by the van Cittert-Zernike theorem, one with the VLA and one with a random Gaussian array. I roughly eyeballed the sparsity of the images, and took 1/3 the number of measurements traditionally needed for each image. For instance, for a 200 x 200 = 40,000-pixel image, I used a measurement matrix with 40,000/3 <math>\approx</math> 16,000 rows, instead of the full 40,000 rows.
	* Calculate the DCT matrix for the sparsifying matrix <math>\Psi</math>.		* Calculate the DCT matrix for the sparsifying matrix <math>\Psi</math>.
	* Calculate <math>\Phi = \Theta \times \Psi</math>.		* Calculate <math>\Phi = \Theta \times \Psi</math>.
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	<center>		<center>
	<gallery widths=~~350px~~ heights=~~350px~~>		<gallery widths=320px heights=320px>
	File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.		File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.
	File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array. RE = 3.78%.		File:CrabNebGaussian.png \| OMP reconstruction using the random Gaussian array. RE = 3.78%.
	File:CrabNebDCT.png \| OMP reconstruction using the VLA configuration. RE = 7.52%.		File:CrabNebDCT.png \| OMP reconstruction using the VLA configuration. RE = 7.52%.
	</gallery>		</gallery>
	</center>		</center>

	The Gaussian array clearly gives a more accurate reconstruction, while there are distinct blocking artifacts in the VLA reconstruction.		The random Gaussian array clearly gives a more accurate reconstruction, while there are distinct blocking artifacts in the VLA reconstruction.

	===Image of the Cassiopeia A Supernova Remnants===		===Image of the Cassiopeia A Supernova Remnants===

	<center>		<center>
	<gallery widths=~~350px~~ heights=~~350px~~>		<gallery widths=320px heights=320px>
	File:CassAOrig.png \| 200-by-200 pixel radio image of the Cassiopeia A supernova remnants.		File:CassAOrig.png \| 200-by-200 pixel radio image of the Cassiopeia A supernova remnants.
	File:CassAGaussian.png \| OMP reconstruction using the Gaussian array. RE = 1.19%.		File:CassAGaussian.png \| OMP reconstruction using the random Gaussian array. RE = 1.19%.
	File:CassAVLA.png \| OMP reconstruction using the VLA configuration. RE = 0.33%.		File:CassAVLA.png \| OMP reconstruction using the VLA configuration. RE = 0.33%.
	</gallery>		</gallery>
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	<center>		<center>
	<gallery widths=~~350px~~ heights=~~350px~~>		<gallery widths=320px heights=320px>
	File:M87Orig.png \| 120-by-120 pixel radio image of the galaxy M87.		File:M87Orig.png \| 120-by-120 pixel radio image of the galaxy M87.
	File:M87Gaussian.png \| OMP reconstruction using the Gaussian array. RE = 0.20%.		File:M87Gaussian.png \| OMP reconstruction using the random Gaussian array. RE = 0.20%.
	File:M87VLA.png \| OMP reconstruction using the VLA configuration. RE = 0.90%.		File:M87VLA.png \| OMP reconstruction using the VLA configuration. RE = 0.90%.
	</gallery>		</gallery>
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	<center>		<center>
	<gallery widths=~~350px~~ heights=~~350px~~>		<gallery widths=320px heights=320px>
	File:3C58Orig.png \| 160-by-160 pixel radio image of the galaxy 3C58.		File:3C58Orig.png \| 160-by-160 pixel radio image of the galaxy 3C58.
	File:3C58Gaussian.png \| OMP reconstruction using the Gaussian array. RE = 1.37%.		File:3C58Gaussian.png \| OMP reconstruction using the random Gaussian array. RE = 1.37%.
	File:3C58VLA.png \| OMP reconstruction using the VLA configuration. RE = 0.28%.		File:3C58VLA.png \| OMP reconstruction using the VLA configuration. RE = 0.28%.
	</gallery>		</gallery>
	</center>		</center>

	The patterned VLA configuration consistently causes blocking artifacts in the reconstructions. For these images, the dominant DCT coefficients in each block are typically the low-frequency coefficients. Therefore, when too few DCT coefficients are recovered by OMP, the missing coefficients (typically the high-frequency ones) create breaks in features that should span the blocks continuously. It appears that the patterned VLA configuration aggravates this phenomenon, while the Gaussian array gives much more accurate reconstructions without blocking artifacts.		The patterned VLA configuration consistently causes blocking artifacts in the reconstructions. For these images, the dominant DCT coefficients in each block are typically the low-frequency coefficients. Therefore, when too few DCT coefficients are recovered by OMP, the missing coefficients (typically the high-frequency ones) create breaks in features that should span the blocks continuously. It appears that the patterned VLA configuration aggravates this phenomenon, while the random Gaussian array gives much more accurate reconstructions without blocking artifacts.

	===DCT versus the Pixel Basis===		===DCT versus the Pixel Basis===
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	<center>		<center>
	<gallery widths=~~350px~~ heights=~~350px~~>		<gallery widths=320px heights=320px>
	File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.		File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.
	File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array and the DCT as the sparsifying basis.		File:CrabNebGaussian.png \| OMP reconstruction using the random Gaussian array and the DCT as the sparsifying basis.
	File:CrabNebPixel.jpg \| OMP reconstruction using the Gaussian array and the pixel basis.		File:CrabNebPixel.jpg \| OMP reconstruction using the random Gaussian array and the pixel basis.
	</gallery>		</gallery>
	</center>		</center>
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	==Conclusions==		==Conclusions==

	My imaging simulations confirmed that CS can be successfully applied to radio interferometry, as the literature ~~as shown~~. For each image, I took only 1/3 the number of measurements traditionally needed—and it may be possible to take less, as I was simply eyeballing the sparsity of the images—and yet OMP was able to reconstruct images with low relative error.		My imaging simulations confirmed that CS can be successfully applied to radio interferometry, as the literature has suggested. For each image, I took only 1/3 the number of measurements traditionally needed—and it may be possible to take less, as I was simply eyeballing the sparsity of the images—and yet OMP was able to reconstruct images with surprisingly low relative error.

			In all my simulations, the random Gaussian array provides far more accurate reconstructions than the patterned VLA configuration. Though this result aligns with the principle of random sampling predominant in CS theory, it contradicts the results of Wenger et al. 2010, which reported that the patterned VLA configuration gave more accurate CS reconstructions. There are several possible explanations for this deviation: the first is that the paper compared the VLA to a uniform random distribution of antennas, not a centrally condensed distribution such as a Gaussian. Though both a uniform random array and random Gaussian array have the randomized nature that the VLA lacks, the difference between uniform and centrally condensed coverage of the Fourier plane may have been critical. Secondly, Wenger et al. used a CS recovery scheme they developed themselves, called SparseRI. The paper did not describe the nature of the recovery scheme, though it does appear to be specialized for radio interferometry, rather than a general, widely used CS framework such as OMP. Thirdly, the paper did not use a sparsifying basis like the DCT and ran the reconstructions in the pixel basis. These differences in our methods may explain why Wenger et al. preferred the patterned VLA configuration for CS,while my simulations preferred a random Gaussian array.

	===Future Work===		===Future Work===

			The deviations between our simulations provide many ideas for future work. Finding an optimal sparsifying basis for radio images is particularly important, and multiresolution schemes such as wavelet bases can be considered. Radio interferometry in general has not been practiced outside of the pixel basis, and these simulations highlight the important of doing so before CS can be applied. The CS recovery algorithm is also a key variable: I used the OMP largely because it is much more computationally efficient than <math>\ell_1</math>-minimization approaches like basis pursuit and LASSO, but these may respond to antenna arrays differently. As we look to deal with new floods of data from massive arrays, including the ALMA and the SKA, optimizing these factors for CS in radio interferometry will become more and more important.

	==References==		==References==

imported>Projects221: /* Compressed Sensing in Astronomical Imaging */

2013-03-21T06:29:52Z

Compressed Sensing in Astronomical Imaging

← Older revision		Revision as of 06:29, 21 March 2013
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	<center>		<center>
	<gallery widths=~~300px~~ heights=~~300px~~>		<gallery widths=350px heights=350px>
	File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.		File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.
	File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array. RE = 3.78%.		File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array. RE = 3.78%.
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	<center>		<center>
	<gallery widths=~~300px~~ heights=~~300px~~>		<gallery widths=350px heights=350px>
	File:CassAOrig.png \| 200-by-200 pixel radio image of the Cassiopeia A supernova remnants.		File:CassAOrig.png \| 200-by-200 pixel radio image of the Cassiopeia A supernova remnants.
	File:CassAGaussian.png \| OMP reconstruction using the Gaussian array. RE = 1.19%.		File:CassAGaussian.png \| OMP reconstruction using the Gaussian array. RE = 1.19%.
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	<center>		<center>
	<gallery widths=~~300px~~ heights=~~300px~~>		<gallery widths=350px heights=350px>
	File:M87Orig.png \| 120-by-120 pixel radio image of the galaxy M87.		File:M87Orig.png \| 120-by-120 pixel radio image of the galaxy M87.
	File:M87Gaussian.png \| OMP reconstruction using the Gaussian array. RE = 0.20%.		File:M87Gaussian.png \| OMP reconstruction using the Gaussian array. RE = 0.20%.
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	<center>		<center>
	<gallery widths=~~300px~~ heights=~~300px~~>		<gallery widths=350px heights=350px>
	File:3C58Orig.png \| 160-by-160 pixel radio image of the galaxy 3C58.		File:3C58Orig.png \| 160-by-160 pixel radio image of the galaxy 3C58.
	File:3C58Gaussian.png \| OMP reconstruction using the Gaussian array. RE = 1.37%.		File:3C58Gaussian.png \| OMP reconstruction using the Gaussian array. RE = 1.37%.
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	<center>		<center>
	<gallery widths=~~300px~~ heights=~~300px~~>		<gallery widths=350px heights=350px>
	File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.		File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.
	File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array and the DCT as the sparsifying basis.		File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array and the DCT as the sparsifying basis.

imported>Projects221: /* DCT versus the Pixel Basis */

2013-03-21T06:28:32Z

DCT versus the Pixel Basis

← Older revision		Revision as of 06:28, 21 March 2013
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	</center>		</center>

	Given the same set of measurements, the pixel basis gives a disastrous reconstruction, barely recovering enough pixels to confirm the nebula's existence. Natural images ~~such as~~ these are typically far sparser~~, or~~ far more compressible, in the DCT basis than in the natural pixel basis, which is critical to successful CS.		Given the same set of measurements, the pixel basis gives a disastrous reconstruction, barely recovering enough pixels to confirm the nebula's existence. Natural images like these are typically far sparser and far more compressible in the DCT basis than in the natural pixel basis, which is critical to successful CS.

	==Conclusions==		==Conclusions==

imported>Projects221: /* DCT versus the Pixel Basis */

2013-03-21T06:27:58Z

DCT versus the Pixel Basis

← Older revision		Revision as of 06:27, 21 March 2013
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	</gallery>		</gallery>
	</center>		</center>

			Given the same set of measurements, the pixel basis gives a disastrous reconstruction, barely recovering enough pixels to confirm the nebula's existence. Natural images such as these are typically far sparser, or far more compressible, in the DCT basis than in the natural pixel basis, which is critical to successful CS.

	==Conclusions==		==Conclusions==

imported>Projects221: /* DCT versus the Pixel Basis */

2013-03-21T06:25:26Z

DCT versus the Pixel Basis

← Older revision		Revision as of 06:25, 21 March 2013
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	File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.		File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.
	File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array and the DCT as the sparsifying basis.		File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array and the DCT as the sparsifying basis.
	File:CrabNebPixel.~~jp2~~ \| OMP reconstruction using the Gaussian array and the pixel basis.		File:CrabNebPixel.jpg \| OMP reconstruction using the Gaussian array and the pixel basis.
	</gallery>		</gallery>
	</center>		</center>

imported>Projects221: /* Compressed Sensing in Astronomical Imaging */

2013-03-21T06:23:56Z

Compressed Sensing in Astronomical Imaging

← Older revision		Revision as of 06:23, 21 March 2013
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	===Protocol===		===Protocol===

	* Calculate two versions of the measurement matrix <math>\Theta</math> using the formula given by the van Cittert-Zernike theorem, one with the VLA ~~baselines~~ and one with ~~the~~ Gaussian array ~~baselines~~.		* Calculate two versions of the measurement matrix <math>\Theta</math> using the formula given by the van Cittert-Zernike theorem, one with the VLA and one with a Gaussian array. I roughly eyeballed the sparsity of the images, and took 1/3 the number of measurements traditionally needed for each image. For instance, for a 200 x 200 = 40,000-pixel image, I used a measurement matrix with 40,000/3 <math>\approx</math> 16,000 rows, instead of the full 40,000 rows.
	* Calculate the DCT matrix for the sparsifying matrix <math>\Psi</math>.		* Calculate the DCT matrix for the sparsifying matrix <math>\Psi</math>.
	* Calculate <math>\Phi = \Theta \times \Psi</math>.		* Calculate <math>\Phi = \Theta \times \Psi</math>.
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	The patterned VLA configuration consistently causes blocking artifacts in the reconstructions. For these images, the dominant DCT coefficients in each block are typically the low-frequency coefficients. Therefore, when too few DCT coefficients are recovered by OMP, the missing coefficients (typically the high-frequency ones) create breaks in features that should span the blocks continuously. It appears that the patterned VLA configuration aggravates this phenomenon, while the Gaussian array gives much more accurate reconstructions without blocking artifacts.		The patterned VLA configuration consistently causes blocking artifacts in the reconstructions. For these images, the dominant DCT coefficients in each block are typically the low-frequency coefficients. Therefore, when too few DCT coefficients are recovered by OMP, the missing coefficients (typically the high-frequency ones) create breaks in features that should span the blocks continuously. It appears that the patterned VLA configuration aggravates this phenomenon, while the Gaussian array gives much more accurate reconstructions without blocking artifacts.

			===DCT versus the Pixel Basis===

			As CS revolves around the sparsity or compressibility of signals, one cannot take full advantage of CS without finding an optimal sparse representation of the signal. To highlight the advantage of using the DCT as a sparsifying basis instead of the pixel basis, as Wenger et al. 2010 did, I ran OMP on the simulated measurements of the Crab Nebula using the pixel basis. I simply set the sparsity matrix <math>\Psi</math> to the identity matrix, instead of the DCT matrix.

			<center>
			<gallery widths=300px heights=300px>
			File:CrabNebOrig.png \| 120-by-120 pixel radio image of the Crab Nebula.
			File:CrabNebGaussian.png \| OMP reconstruction using the Gaussian array and the DCT as the sparsifying basis.
			File:CrabNebPixel.jp2 \| OMP reconstruction using the Gaussian array and the pixel basis.
			</gallery>
			</center>

	==Conclusions==		==Conclusions==

			My imaging simulations confirmed that CS can be successfully applied to radio interferometry, as the literature as shown. For each image, I took only 1/3 the number of measurements traditionally needed—and it may be possible to take less, as I was simply eyeballing the sparsity of the images—and yet OMP was able to reconstruct images with low relative error.

	===Future Work===		===Future Work===