Project Overview

Reconstruction in Diffracted Reconstruction in Diffracted Ultrasound TomographyUltrasound Tomography

Tali Meiri & Tali SaulTali Meiri & Tali SaulSupervised by:Supervised by:

Dr. Michael ZibulevskyDr. Michael Zibulevsky

Dr. Haim AzhariDr. Haim Azhari

Alexander & Michael BronsteinAlexander & Michael Bronstein

Project Goals Different methods of image reconstruction for

diffraction ultrasound tomography. Computing an image of a slice of an object

from projections. More specifically :

In acoustic imaging, diffraction must be taken into consideration when modeling the interaction of radiation with matter. Diffraction is caused by the wave nature of the radiation.

Terminology & Definition Tomographic Reconstruction: A method of computing a

sliced image of an object from data collected from projections.

Projection: Data acquired from a single illumination at a specific angle. The illumination is by electromagnetic radiation or acoustic waves, and the intensity of the radiation traversing the object is measured.

Sinogram: A picture of a set of projections at different angles placed one on top of the other.

Straight-ray tomography: Radiation illuminating the object has the nature of straight rays.

Diffraction tomography: Radiation illuminating the object has the nature of waves. In such case, wave phenomena such as diffraction should not be overlooked.

Input image

System Overview

Incident plane wave

Measured forward

scattered field

Sample data in frequency

domain

Make necessary interpolation

Reconstruct the picture by

Fourier inversion

System overview

Straight-Ray tomographyIn straight-ray tomography the radiation illuminating the object has the nature of straight rays. As such, the wavelength of the radiation is infinitely small compared to the dimensions of the illuminated object.

Radon transform:The Radon function is the projection of the image intensity along a radial line oriented at a specific angle. The Radon transform of an object represents the image intensity along many radial lines oriented at different angles.

Tomographic scan: It would be fair to define a tomographic scan as a discretization of the Radon transform of an object since the projections are taken at discrete angles around the object.

2

sincos,R

dxdyryxyxfrp

r is the radial axis oriented at angle

Straight-Ray Tomography Fourier Slice Theorem:

This theorem connects the Radon transform with the Fourier transform: 1D Fourier transform of the Radon transform equals the 2D Fourier transform of the object.

rP yxf ,

2,1 wwFF D1

Radon

F D2

Straight-Ray Tomography

w1

w2

Fourier transform

DFT of the discrete tomographic projections gives the values of the 2D Fourier transform of the objects on radial straight lines oriented at the same angles as the projections. Reconstruction is usually using Filtered Back-Projection.

x

y

r

Space domainFrequency domain

Diffraction Tomography Fourier Diffraction Theorem:

DFT of the discrete tomographic projections gives the values of the 2D Fourier transform of the object along a semi-circular arc in the frequency domain.

2

0K

Diffraction tomography

Straight-ray tomography

0

The radius of the arc is proportional to the frequency of

the incident wave :

Diffraction Tomography Monochromatic Illumination:

The object is rotated and the scattered field for different orientations is measured. For each orientation the object is illuminated with a monochromatic wave. This produces an estimate of the object’s Fourier transform along a circular arc rotated at the same angle as the object.

Each arc contains 32 sampling points.

Diffraction Tomography Broadband Illumination:

Transmitting a superposition of monochromatic waves at different frequencies. This allows obtaining more information from a single projection since the frequency domain is sampled on several arcs simultaneously. Consequently, a smaller amount of projections should suffice for covering the entire frequency domain.

Each arc contains 32 sampling points.

Methods of reconstructionMethods used in straight-ray tomography are not applicable to tomography with diffractive sources. In this work we introduce and compare the performances between three different methods of reconstruction which will include the following:

Reconstruction using inverse NUFT:Straightforward computation of the forward and the inverse NUFT by creating the transform matrix and applying it to the picture in column stack.

Reconstruction using frequency domain interpolation: Frequency interpolation of the non-uniform data to a uniform Cartesian grid using bilinear interpolation.

Reconstruction using Non-uniform Fast Fourier Transform:A method equivalent to a convolution regridding method on an over sampled grid using an optimal selection of a Gaussian kernel.

Inverse NUFT (2D) 2D Non Uniform Fourier Transform:

Our goal is to move from non uniform frequency samples in frequency domain to uniform samples in space domain.

The 2D NUFT matrix will be built in the following way:

for every frequency sample which will be represented in frequency domain

as , the basis pictures will be built such that:

and will represent the columns of . (n,m) are the reconstructed samples in space domain.

Kk ,...,1

),( ][),( mnj

kkkemn

Xx TT 1)(Pseudo inverse

Inverse NUFT (2D)Since this is the straight forward computation without using any approximations, it is the most accurate reconstruction technique but is also computationally extensive and requires operations as it is equivalent to matrix multiplication. When the signal is large (typically 64 X 64 = 4096 and above), straightforward inversion is practically impossible.

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Example of applying the transformation on a 2D sinc function in Frequency domain to get a 2D step function in space domain. In this example the frequency samples where uniformly spaced.

Inverse NUFT (2D)

Example of applying the transformation on a 2D sinc function in Frequency domain to get a 2D step function in space domain. In this example the frequency samples where randomly spaced.

Inverse NUFT (2D)

Freq. domain interpolation The algorithm by Kak and Slaney:1. Start from a Cartezian grid.2. For each point (w1,w2) find its representation in (w, ) coordinates.3. Use bilinear interpolation to find the most accurate value of the signal at the

new location.

where:

4. After computing at each point on the rectangular grid, the object is obtained by a simple 2-D inverse FFT.

Complexity:

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else

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Freq. domain interpolation Bilinear interpolation:

F(m,n)

F(m+1,n) F(m+1,n+1)

F(m,n+1)

F(x,y)

Freq. domain interpolation Shepp-Logan phantom:

In order to avoid forward-projection errors, and analytic Shepp-Logan phantom was used. This phantom is a superposition of ellipses representing features of the human brain. The advantage of such a phantom is that its Fourier Transform has an analytical expression. The Fourier Transform of an ellipse is given by:

where is the first order Bessel function of the first kind, is the center of the ellipse, its intensity, its orientation and lastly A and B are the lengths of horizontal and vertical semi-axes respectively.

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sincos'

')'(

')'((),(

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yx

yx

yKxKiyx

KK

KKu

ABu

ABuBAJeKKE yx

1J ),( 00 yx

Freq. domain interpolation

Monochromatic illumination using a 64X64 phantom picture with 64 projections.

Freq. domain interpolation

Monochromatic illumination using a 64X64 phantom picture with 64 projections and with different addition of Gaussian noise.

Freq. domain interpolation

Calculated mean squared error and the max error between reconstructed picture and original picture as function of the number of projections.

Freq. domain interpolation

Less coverage of samples in the high frequency region between 17 projections and 18 projections due to multiple equal samples.

Freq. domain interpolation

Broadband illumination using a 64X64 phantom picture with 15 different angles and 5 different frequencies.

Freq. domain interpolation

Calculated mean squared error and the max error between reconstructed picture and original picture as function of the number of different frequencies in each projection

The NUFFT method (1-D) Definition of the problem:

The input parameters is a vector of samples of a signal sampled in non-uniformly distributed frequencies

. Our objective is to reconstruct the signal from its non-uniform frequency samples using a method which takes a non-uniformed data in frequency domain and transforms it to a uniform data in space domain.

Method:Using the method of Fast Fourier Transform approximation for non-equispaced data suggested by A. Dutt and V. Rokhlin. This method uses interpolation of the data on some over-sampled Cartesian grid using a Gaussian kernel. Once the data is uniformly spaced on the rectangular grid, the signal

can be obtained by a simple inverse FFT. Complexity of algorithm:

O(NlogN+Nq) where q is a constant.

CkN :,...,1

,:,...,0 kN www

Cffff N :,...,1

The NUFFT method (1-D) The algorithm:

For a given signal in frequency domain, the inverse transformation is defined by the formula:

where is the non-uniformed frequency. The algorithm approximates this formula by finding a suitable approximation for any expression of the form

using a q number of expressions of the form where . It is proven that the error between the reconstructed signal and the original signal obeys the following inequality:

where

CkN :,...,1

N

k

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/2

kw

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The NUFFT method (1-D)

w

w

w

w q

DFT of a non-uniformly sampled set of N data points may be computed with an ordinary FFT of length m*N with a precision that depends on the selection of m. Usually a choice of m=2 is sufficient for most practical applications.

The NUFFT (1-D) – results

NUFFT using an analytic sinc function.

Red dots: non-uniformed samples of the sinc function.

The NUFFT (1-D) – results

UFFT using an analytic sinc function.

Red dots: uniformed samples of the sinc function.

The Sarty NUFFT (2-D) Definition of the problem:

The input parameters is a CS vector of samples of a 2_D signal sampled in non-uniformly distributed frequencies on the 2_D range.

Method:Voronoi areas.Extension of the 1-D NUFFT algorithm to 2-D.

The NUFFT method (2-D) Direct computation using Voronoi areas:

In this case the equation is:

where S(p) is the vector of the samples in CS of the signal, W(p) is the CS vector of the correlated weights derived from the Voronoi areas associated with each sample point and are the non-uniformed frequencies in CS.

This straight forward computation requires multiplications and additions. It may take hours of computational time for a typical 256X256 signal picture. The fast algorithm requires only while using the FFT algorithm.

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p

ypixpi yx eepWpSyxP0

)()()()(),(

)(),( pp yx

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21loglog mNmNmNO

The NUFFT method (2-D) Data representation using Voronoi area:

The Voronoi areas associated with a k-space point is the area of the set whose points are closer to the given point than all the other k-spaced sample points.

The NUFFT method (2-D) Efficient implementation using the D-R algorithm:

Let andFurthermore we use:

The image reconstruction is computed as:

where

)( pm yp )( pm xp

Mp

qqj

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Q

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bjpmjp

bjpmjp

py

px

,...,02

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2

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The NUFFT (2-D) – results

Monochromatic illumination using a 64X64 phantom picture with 64 projections.

The NUFFT (2-D) – results

Monochromatic illumination using a 64X64 phantom picture with 64 projections and with different addition of Gaussian noise.

Conclusion

   

 

ComplexityTime of calculation(seconds)

Mean squared error

Infinite Error

Gridding 17.9256.4572

7.8607

Direct transformation

81.070.28120.096

NUFFT15.80203.84458.8118

21loglog mNmNmNO

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For 32X32 pictures

222 log4 NNNO