May 1, 2009 AMSC 663/664 1

Image Reconstruction from Non-Uniformly Sampled

Spectral Data

Alfredo Nava-Tudela

AMSC 664, Spring 2009

Final Presentation

Advisor: John J. Benedetto

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Signals and their spectral decomposition

• A signal can be decomposed in harmonics that reveal the frequency or spectral content contained in that signal

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Signals and their spectral decomposition

• Often times, we have spectral information and we need to convert back to spatial information, for example Magnetic Resonance Imaging

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Problem Statement

• We are particularly interested in the

reconstruction of images given spectral

information• More specifically, we are interested in

image reconstruction given non- uniformly sampled spectral data

• Given a two dimensional spectral data set, reconstruct an image in the spatial domain that matches as closely as possible that data set in the spectral domain

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The Algorithm

• Stage one:

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The Algorithm

• Stage two:

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The Algorithm

• Stage three:

ImageReconstructed

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CG Experiments - Using the DFTxSinc input data

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CG Experiments - Using the DFTxSinc input data

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CG Experiments - Using the DFTxSinc input data

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CG Experiments - Using the DFTxSinc input data

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CG Experiments - Using the DFTxSinc input data

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CG Experiments - Time of one iteration vs image size

If N = 16 = 2^4, then ln(16)/ln(2) = 4.Time is given in seconds.

Time of one iteration (Tol = 1e-1)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

3 4 5 6 7 8

ln(N)/ln(2)

log(time)

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CG Experiments - Runtime vs Precision

Scaling

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 2 3 4 5 6

Precision := -log(tol)

log(time)

N = 16

N = 32

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CG Experiments - Number of iterations vs Precision

Number of Iterations (N=16)

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7

Precison := -log(tol)

log(# of iterations)

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CG Experiments - Convergence and time results

Results for N=16

0

100

200

300

400

500

600

700

800

900

1000

1 2 3 4 5 6

Precision := -log(tol)

seconds, units

time

norm

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CG Experiments - Convergence and time results

Computation time for N=16

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6

Precision := -log(tol)

log(time)

The time is given in seconds

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CG Experiments - Convergence and time results

Results for N=32

0

2000

4000

6000

8000

10000

12000

1 2 3 4 5 6

Precision := -log(tol)

seconds, units

time

norm

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CG Experiments - Convergence and time results

The time is given in seconds

Computation time for N=32

1

1.5

2

2.5

3

3.5

4

4.5

1 2 3 4 5 6

Precision := -log(tol)

log(time)

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CG Experiments - Memory usage

• One iteration of the CG method issues 2 calls to the function A_times() and 2 calls to the function A_star_times().

• Both functions, by implementation, use the same amount of memory.

• The CG method also has bookkeeping variables that require memory.

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CG Experiments - Memory usage

A call to either A_times() or A_star_times()uses the following memory:

Name Size Class Attributes

KL N^2x2 double M 1x1 double N_square 1x1 double S Mx2 double a Mx1 double complex f N^2x1 double m 1x1 double n 1x1 double sum 1x1 double complex

Which gives a sub-total for each call of:

3xN^2 + 4xM + 6 words

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CG Experiments - Memory usage

A call of the CG code, without the previous taken intoaccount, uses the following memory:

Name Size Class Attributes KL N^2x2 double S Mx2 double alpha 1x1 double complex beta 1x1 double d N^2x1 double complex delta_0 1x1 double delta_new 1x1 double delta_old 1x1 double f_hat Mx1 double complex iteration 1x1 double q N^2x1 double complex r N^2x1 double complex tol 1x1 double x N^2x1 double y N^2x1 double complexWhich gives a sub-total of:

11xN^2 + 4xM + 8 words

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CG Experiments - Memory usage

Combined, we obtain the following grand total of:

14xN^2 + 8xM + 14 words

needed to run our code.

The direct method that saves the matrices A and itsadjoint A* would need O(N^2 x M) words of memory.

Clearly the CG method is the way to go memory wise!

Memory Usage - Words vs Image Size

110

1001000

100001000001E+061E+071E+081E+091E+101E+111E+121E+13

0 128 256 384 512 640 768 896 1024

N

Words

Direct Method

CG Method

We assume M = N^2, best case scenario

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CG Experiments - Convergence N=16

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CG Experiments - Convergence N=32

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References

• Richard F. Bass and Karlheinz Groechenig “Random Sampling of Multivariate Trigonometric Polynomials”

• Zhou Wang, Alan C. Bovik, Hamid R. Sheikh, and Eero P. Simoncelli “Image Quality Assessment: From Error Measurements to Structural Similarity”, IEEE Transactions on Image Processing, Vol. 13, No. 1, January 2004

• Conjugate Gradient Method: http://en.wikipedia.org/wiki/Conjugate_gradient_method

• Jonathan Richard Shewchuk, “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain”. August 4, 1994.

• Adi Ben-Israel and Thomas N. E. Greville. Generalized Inverses. Springer-Verlag, 2003.

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References

• John J. Benedetto and Paulo J. S. G. Ferreira. Moderm Sampling Theory: Mathematics and Applications. Birkhauser, 2001.

• J. W. Cooley and J. W. Tukey. An algorithm for the machine computation of complex Fourier series. Math. Comp., 19:297-301, 1965.

• E. H. Moore. On reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society, 26:85-100, 1920.

• Diane P. O’Leary. Scientific computing with case studies. Book in preparation for publication, 2008.

• Roger Penrose. On best approximate solution to linear matrix equations. Proceedings of the Cambridge Philosophical Society, 52:17-19, 1956.