performed by: ron amit supervisor: tanya chernyakova in cooperation with: prof. yonina eldar

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Performed by: Ron AmitSupervisor: Tanya Chernyakova

In cooperation with: Prof. Yonina Eldar

Sub-Nyquist Sampling in Ultrasound Imaging

Part A Final PresentationSemester: Spring 2012

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Agenda• Introduction• Project Goals• Background• Recovery Method• Image Construction• Summary• Future Goals

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Introduction

Introduction 4

Ultrasound Imaging

Introduction 5

Beamforming

Introduction 6

Problem

• Typical Nyquist rate is 20 MHz * Number of transducers * Number of image lines

• Large amount of data must be collected and processed in real time

Introduction 7

Solution

• Develop a low rate sampling scheme based on knowledge about the signal structure

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Main goal: Prove the preferability of the Xampling method for Ultrasound imaging

Part A:• Improve recovery method• Improve image construction runtime

Project Goals

Project Goals

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Background

Background 10

FRI Model

• Theoretical lower bound of sample rate:

Background 11

Unknown Phase

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x 10-6

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1h(t), Known Pulse Shape

t [sec]

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plitu

de0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10-6

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t [sec]

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• Define:

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Sampling Scheme

Receiver Elements

Low Rate Samples

RecoveryImage

Construction

Background

Block Diagram

𝒄𝑡 𝑙 ,𝑏𝑙

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Single Receiver Xample Scheme

• Unknown parameters are extracted from low rate samples.

Background

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• Combines Beamforming and sampling process.

• Samples are a group of Beamformed signal’s Fourier coefficients.

• Sampling at Sub-Nyquist rate is possible.

• Digital processing extracts the Beamformed signal parameters.

Compressed Beamforming

Background

Background 15

Using analog kernels and integrators

First Sampling Scheme :

Problem : Analog kernels are complicated for hardware implementation


Background 16

Simplified Sampling Scheme :

• Based on approximation• One simple analog filter per receiver• Linear transformation applied on samples


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Recovery Method

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Sampling Scheme

Receiver Elements

Low Rate Samples

Recovery𝑡 𝑙 ,𝑏𝑙

Image Constructio

n

Block Diagram

𝒄

19Recovery Method

Parameter Recovery• Problem : Recover and from samples

• Time delay : • Amplitude and phase:

• Complex samples: - Partial group of the Beamformed signal’s Fourier Coefficients

• The relation shown in [1]:

Compressed Sensing Formulation

Time quantization:

𝑁=⌊𝜏∆𝑠

⌋Number of times samples:

Equation Set:

, j=1,..,K

Recovery Method 20

21Recovery Method

(~𝑐1

⋮~𝑐𝐾

)=( e− 𝑖

2𝜋𝑇∆𝑠𝑘1 1

⋯ ⋯ ⋯ e− 𝑖

2𝜋𝑇∆𝑠𝑘1 𝑁

⋮ ⋯ ⋯ ⋯ ⋮

e− 𝑖 2𝜋

𝑇∆ 𝑠𝑘𝐾 1

⋯ ⋯ ⋯ e−𝑖 2𝜋

𝑇∆ 𝑠𝑘𝐾 𝑁 )(

𝑥1

⋮⋮⋮𝑥𝑁

)

Define :1≤n≤𝑁

Matrix Form:

[KxN] – Partial DFT MatrixProblem: = V , unknown

Compressed Sensing Formulation

Equation Set:

K << N

22Recovery Method

Problem: = V , unknown

OMP Algorithm

OMP Solves : , such that

Sparsity assumption:

Number of Real samples:1662 (per image line per transducer)

Standard Image

Standard Image:OMP with L=25:Number of Real samples:200 (per image line per transducer)Recovery Method : OMP-NoamRecovery Runtime : 2.4111 [Sec]

Alternative Imaging - Using PhasePSNR: 10.9991 [dB]Imaging Runtime: 0.49933 [Sec]

23Recovery Method

• The signal is reconstructed by incorporating the pulse shape

• Namely, passing trough a band-pass filter: • Conceptual Change: The signal of interest is and not .

• need to be reconstructed correctly only in the pulse pass-band bandwidth .

New Approach

24Recovery Method

• Assume includes all the Fourier coefficients in the pulse bandwidth:

o Any for which the Fourier coefficients in the pulse bandwidth are equal to will yield perfect reconstruction.

o Equivalent condition: = V exactly.

New Approach

-8 -6 -4 -2 0 2 4 6 80

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8x 10

-7 H(f), Fourier Transform of Known Pulse

f [MHz]

Am

plitu

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Proposed Solution Solve: = V

�̂�= 1N

V𝑯

~𝒄Possible Solution:

=Proof:

• Simple solution - easy to calculate

• Equivalent to building using only the sampled frequencies

Recovery Method

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Using all the 361 Fourier coefficients in the pulse bandwidth:

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8x 10


f [MHz]

Am

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Number of Real samples:722 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.30741 [Sec]


Proposed Solution - Result

Recovery Method

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Proposed Solution (using 722 real samples):


Standard Image

Standard Image (using 1662 real samples ):

Recovery Method

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Sub - Sample

• Using 100 out of 361 coefficients:

Can a smaller number of samples be used?

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8x 10


f [MHz]

Am

plitu

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Number of Real samples:200 (per image line per transducer)Recovery Method : ProjectionsRecovery Runtime : 0.060768 [Sec]


Recovery Method

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Artifact• Using 100 out of 361 coefficients:



Recovery Method

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Artifact: Solution

0 20 40 60 80 1000.2

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1Weight Vector

Fourier Coeff Index

We

ight

Non-Ideal Band Pass:



• Using 100 weighted coefficients:

Recovery Method

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Proposed Solution , with weights (using 200 real samples):

OMP (using 200 real samples):

Number of Real samples:200 (per image line per transducer)Recovery Method : OMP-NoamRecovery Runtime : 2.4111 [Sec]



Recovery Method

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Proposed Solution , with weights (using 200 real samples):



Standard Image

Standard Image (using 1662 real samples ):

Recovery Method

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ImageConstruction

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Sampling Scheme

Receiver Elements

Low Rate Samples

RecoveryImage

Construction

Block Diagram

𝒄𝑡 𝑙 ,𝑏𝑙

35Image Construction

Image Construction1. Signal Creation: For each image line (angle), create

signal from estimated parameters2. Interpolation: Interpolate Polar data to full Cartesian grid

Image Construction 36

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

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Signal Creation

• Standard method – Use Hilbert transform to cancel modulation

• In signal creation, pulse envelope can be used beforehand

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Signal Creation

• Convolution with pulse envelope• Problem: Image is blurred• Estimated Phase is needed for a clear image

𝑠 [𝑛 ]=¿ �̂� [𝑛 ]∨∗h [𝑛 ]




Alternative Imaging - Not Using PhasePSNR: 12.9517 [dB]Imaging Runtime: 0.70984 [Sec]


𝑠 [𝒏 ]=𝑹𝒆 {�̂� [𝒏 ]∗ (𝒈 [𝒏 ]𝒆𝒊 (𝒏∆𝑠𝝎𝟎− 𝜷))}

Signal Creation

𝑠 [𝑛 ]=∑𝑙=1

𝐿

|𝑏𝑙|𝑔 [𝑛−¿ql ]cos (𝜔0∆𝑠 (𝑛−𝑞𝑙 )+𝛽𝑙− 𝛽)¿Signal Model:

}

Using:

Convolution Form:


Image Construction1. Signal Creation: For each image line (angle), create

signal from estimated parameters2. Interpolation: Interpolate Polar data to full Cartesian grid


2D Interpolation

Number of Real samples:440 (per image line per transducer)Recovery Method : ProjectionsRecovery Runtime : 0.1852 [Sec]Filter: : None

Standard ImagingImaging Runtime: 8.4127 [Sec]PSNR: 14.3835 [dB]

• 2D Linear interpolation• High quality image, but very slow


Nearest Neighbor Interpolation

Number of Real samples:440 (per image line per transducer)Recovery Method : ProjectionsRecovery Runtime : 0.19201 [Sec]Filter: : None

Standard ImagingImaging Runtime: 4.314 [Sec]PSNR: 13.4968 [dB]

• Each Cartesian gets the value of the nearest polar data point

• Lower quality image, but fast


My method

• Interpolate only in the angle axis (1D interpolation)• Place each polar data point in the nearest point on the

Cartesian grid


Alternative Imaging - Using PhaseImaging Runtime: 0.66153 [Sec]PSNR: 13.9718 [dB]


Image Construction - Results

• Almost identical images• Significant runtime reduction




Standard ImagingPSNR: 14.3814 [dB]Imaging Runtime: 8.4543 [Sec]

My method: Standard Imaging:

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Summary

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• New recovery method• Significantly faster recovery runtime• Very simple hardware implementation• Much better image quality• Significantly faster image construction runtime

Achievements:

Summary

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Future Goals

• Improve the simplified sampling scheme

• Cooperation with GE Healthcare

• Build a demo which shows the efficiency of the

Sub- Nyquist method

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References:[1] N. Wagner, Y. C. Eldar and Z. Friedman, "Compressed Beamforming in Ultrasound Imaging", IEEE Transactions on Signal Processing, vol. 60, issue 9, pp.4643-4657, Sept. 2012.

[2] Ronen Tur, Y.C. Eldar and Zvi Friedman, “Innovation Rate Sampling of Pulse Streams With Application to Ultrasound Imaging”, IEEE Trans. Signal Process., vol. 59, no. 4, pp. 1827-1842, 2011

[3] K. Gedalyahu, R. Tur and Y.C. Eldar, “Multichannel Sampling of Pulse Streams at the Rate of Innovation”, IEEE Trans. Signal Process., vol. 59, no. 4, pp. 1491-1504, 2011

http://webee.technion.ac.il/people/YoninaEldar/journals/137compressed.pdf

performed by: ron amit supervisor: tanya chernyakova in cooperation with: prof. yonina eldar

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