Sparsity-optimized Harmonic Wavelets for Compressed Sensing MRI - - PowerPoint PPT Presentation

Sparsity-optimized Harmonic Wavelets for Compressed Sensing MRI

Ruediger Willenberg (ECE) JEB1433 Project Presentation April 22, 2010

References

n M.Lustig, D.Donoho, J.M.Pauly: „Sparse MRI: The

Application of Compressed Sensing for Rapid MR Imaging“, 2009

n D.E.Newland: „Harmonic Wavelet Analysis“, 1993 n D.E.Newland: „Harmonic and Musical Wavelets“, 1994 n B. Liu, „Adaptive Harmonic Wavelet Transform with

applications in vibration analysis“, 2002

n R.R.Coifman, M.V. Wickerhauser, „Entropy-based

Algorithms for Best Basis Selection“, 1992

Motivation

n Strong technical, medical and economic

incentives for sampling minimal data

n Compressed sensing provides the

mathematical tools for that

n A sparse transform domain must exist to be

able to sample minimal data

Sparse transforms

Basic Idea

n Can we find optimized transforms for „families“

• f similar images?

Basic Idea

n Can we find optimized transforms for „families“

• f similar images?

n No easy recipe to build an orthonormal basis

with optimal sparsity

n We need a transform that can be easily

reconfigured

Wavelet Transforms

n Wavelets:

Locally concentrated

• scillating functions,

scalable and translatable

Wavelet Transforms

n Wavelet transforms

break down signals

• r images in

frequency and spatial information

Harmonic Wavelet Transform

n Introduced by D.E.Newland in 1993 for signal

analysis

n Approach: Cleanly separated wavelet spectra

HWT: Complex Wavelet Components

HWT: Wavelet scaling & translation

Frequency scaling (j) and translation (k): define vj,k(x) = w(2jx-k)

HWT: Wavelet scaling & translation

HWT: Coefficients & Expansion

n Coefficients: n Expansion formula:

HWT: Base conditions

n Orthogonality: <em,en> = 0 n Orthonormality: <em,em> = 1 n Parseval Identity:

energy of coefficients = energy of function

HWT: Orthogonality

HWT: Orthonormality

HWT: Parseval

HWT: Discretization

n Basis functions must be periodic on the

unit interval:

HWT: Discrete Transform

Back to the Idea: Optimized Bases

n Freely chosen harmonic wavelets instead

• f a fixed series:

Is that still an orthonormal basis?

n Orthogonality, orthonormality and Parseval

were proven

n Newland proposed it in 1994 and called this

„General harmonic wavelets“

How to find the optimal base?

n Take data sample to compress n Try out all possible spectrum divisions n Select the most sparse one n Problems: Complexity too high?

Sparsity metric? 2-dimensional-images groups of images

Lower complexity approximation

n Binary tree search:

• Compute sparsity for each subdivision
• Compare sparsity of ‚node‘ with sum of child

sparsities

• Choose minimal set

Possible sparsity metrics

n Shannon entropy:

• Σi pi log(pi) with pi=|ci|2 / ||f||2

n L1-Norm: Sum of absolutes n L0-Norm: Number of non-zero coefficients

(Reality: non-zero coefficients -> cutoff?)

2-dimensional images

n Sequential application of

FFT, HWT for vertical and horizontal data

n Since directional

characteristics different: separate sparsity analysis, different bases

2-dimensional images, image groups

n Two approaches for line,

image combination: a) average all line FFTs, then build base spectrum b) build base spectrum for each line, average base spectra

Success metric

n l1magic: Min-l1 with quadratic constraints too

slow for whole images

n Instead robustness for image reconstruction

from sparse data:

• Transform image
• Throw away 95% smallest coefficients
• Retransform
• Compare FFT, Optimized HWT,

Original HWT, JPEG2K

Results

n Shannon Entropy, l0norm, l1norm all heavily

prefer minimal frequency bands => HWT degenerates to FFT

Results

n Through explicit tweaking, higher level

frequency bands can be forced, but no consistent behavior is observable

Original

Original HWT

General HWT

FFT

JPEG 2000

Original

Original HWT

General HWT

FFT

JPEG 2000

Conclusions

n Approach to simplistic; averaging spectral

information over a whole picture (or more) gives weak results

n Harmonic Wavelet Transform in general lacks

sparsity, especially in competition with JPEG2000 (here: LeGall 5/3 wavelet)

Conclusions

Thank you for your attention!

n Questions?