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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
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10-6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1h(t), Known Pulse Shape
t [sec]
Am
plitu
de0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10-6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1h(t), Known Pulse Shape
t [sec]
Am
plit
ud
e
⢠Define:
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Sampling Scheme
Receiver Elements
Low Rate Samples
RecoveryImage
Construction
Background
Block Diagram
đđĄ đ ,đđ
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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 signalâs 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
Compressed Beamforming
Background 16
Simplified Sampling Scheme :
⢠Based on approximation⢠One simple analog filter per receiver⢠Linear transformation applied on samples
Compressed Beamforming
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Recovery Method
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Sampling Scheme
Receiver Elements
Low Rate Samples
RecoveryđĄ đ ,đđ
Image Constructio
n
Block Diagram
đ
19Recovery Method
Parameter Recovery⢠Problem : Recover and from samples
⢠Time delay : ⢠Amplitude and phase:
⢠Complex samples: - Partial group of the Beamformed signalâs Fourier Coefficients
⢠The relation shown in [1]:
Compressed Sensing Formulation
Time quantization:
đ=âđâđ
âNumber of times samples:
Equation Set:
, j=1,..,K
Recovery Method 20
21Recovery Method
(~đ1
âŽ~đđž
)=( eâ đ
2đđâđ đ1 1
⯠⯠⯠eâ đ
2đđâđ đ1 đ
⎠⯠⯠⯠âŽ
eâ đ 2đ
đâ đ đđž 1
⯠⯠⯠eâđ 2đ
đâ đ đđž đ )(
đĽ1
âŽâŽâŽđĽđ
)
Define :1â¤nâ¤đ
Matrix Form:
[KxN] â 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
1
2
3
4
5
6
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8x 10
-7 H(f), Fourier Transform of Known Pulse
f [MHz]
Am
plitu
de
Proposed Solution Solve: = V
ďż˝Ěďż˝= 1N
VđŻ
~đPossible 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:
-8 -6 -4 -2 0 2 4 6 80
1
2
3
4
5
6
7
8x 10
-7 H(f), Fourier Transform of Known Pulse
f [MHz]
Am
plitu
de
Number of Real samples:722 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.30741 [Sec]
Alternative Imaging - Using PhasePSNR: 14.5923 [dB]Imaging Runtime: 0.65576 [Sec]
Proposed Solution - Result
Recovery Method
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Proposed Solution - Result
Number of Real samples:722 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.30741 [Sec]
Alternative Imaging - Using PhasePSNR: 14.5923 [dB]Imaging Runtime: 0.65576 [Sec]
Proposed Solution (using 722 real samples):
Number of Real samples:1662 (per image line per transducer)
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?
-8 -6 -4 -2 0 2 4 6 80
1
2
3
4
5
6
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8x 10
-7 H(f), Fourier Transform of Known Pulse
f [MHz]
Am
plitu
de
Number of Real samples:200 (per image line per transducer)Recovery Method : ProjectionsRecovery Runtime : 0.060768 [Sec]
Alternative Imaging - Using PhasePSNR: 13.8379 [dB]Imaging Runtime: 0.61273 [Sec]
Recovery Method
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Artifact⢠Using 100 out of 361 coefficients:
Number of Real samples:200 (per image line per transducer)Recovery Method : ProjectionsRecovery Runtime : 0.060768 [Sec]
Alternative Imaging - Using PhasePSNR: 13.8379 [dB]Imaging Runtime: 0.61273 [Sec]
Recovery Method
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Artifact: Solution
0 20 40 60 80 1000.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Weight Vector
Fourier Coeff Index
We
ight
Non-Ideal Band Pass:
Number of Real samples:200 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.06322 [Sec]
Alternative Imaging - Using PhasePSNR: 13.5536 [dB]Imaging Runtime: 0.59366 [Sec]
⢠Using 100 weighted coefficients:
Recovery Method
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Number of Real samples:200 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.06322 [Sec]
Alternative Imaging - Using PhasePSNR: 13.5536 [dB]Imaging Runtime: 0.59366 [Sec]
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]
Alternative Imaging - Using PhasePSNR: 10.9991 [dB]Imaging Runtime: 0.49933 [Sec]
Proposed Solution - Result
Recovery Method
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Number of Real samples:200 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.06322 [Sec]
Alternative Imaging - Using PhasePSNR: 13.5536 [dB]Imaging Runtime: 0.59366 [Sec]
Proposed Solution , with weights (using 200 real samples):
Proposed Solution - Result
Number of Real samples:1662 (per image line per transducer)
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
đđĄ đ ,đđ
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
x 10-6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1h(t), Known Pulse Shape
t [sec]
Am
plitu
de
Signal Creation
⢠Standard method â Use Hilbert transform to cancel modulation
⢠In signal creation, pulse envelope can be used beforehand
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10-6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1h(t), Known Pulse Shape
t [sec]
Am
plit
ud
e
37Image Construction
Signal Creation
⢠Convolution with pulse envelope⢠Problem: Image is blurred⢠Estimated Phase is needed for a clear image
đ [đ ]=Âż ďż˝Ěďż˝ [đ ]â¨âh [đ ]
Number of Real samples:440 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.19118 [Sec]
Alternative Imaging - Using PhasePSNR: 14.5382 [dB]Imaging Runtime: 0.66487 [Sec]
Number of Real samples:440 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.18607 [Sec]
Alternative Imaging - Not Using PhasePSNR: 12.9517 [dB]Imaging Runtime: 0.70984 [Sec]
Image Construction 38
đ [đ ]=đšđ {ďż˝Ěďż˝ [đ ]â (đ [đ ]đđ (đâđ đđâ đˇ))}
Signal Creation
đ [đ ]=âđ=1
đż
|đđ|đ [đâÂżql ]cos (đ0âđ (đâđđ )+đ˝đâ đ˝)ÂżSignal Model:
}
Using:
Convolution Form:
39Image 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 40
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
Image Construction 41
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
Image Construction 42
My method
⢠Interpolate only in the angle axis (1D interpolation)⢠Place each polar data point in the nearest point on the
Cartesian grid
Number of Real samples:440 (per image line per transducer)Recovery Method : ProjectionsRecovery Runtime : 0.17816 [Sec]
Alternative Imaging - Using PhaseImaging Runtime: 0.66153 [Sec]PSNR: 13.9718 [dB]
Image Construction 43
Image Construction - Results
⢠Almost identical images⢠Significant runtime reduction
Number of Real samples:440 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.19118 [Sec]
Alternative Imaging - Using PhasePSNR: 14.5382 [dB]Imaging Runtime: 0.66487 [Sec]
Number of Real samples:440 (per image line per transducer)Recovery Method : Min L2 NormRecovery Runtime : 0.18752 [Sec]
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, âInnovation Rate Sampling of Pulse Streams With Application to Ultrasound Imagingâ, IEEE Trans. Signal Process., vol. 59, no. 4, pp. 1827-1842, 2011
[3] K. Gedalyahu, R. Tur and Y.C. Eldar, âMultichannel Sampling of Pulse Streams at the Rate of Innovationâ, IEEE Trans. Signal Process., vol. 59, no. 4, pp. 1491-1504, 2011