Univ Seminar

8/3/2019 Univ Seminar

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Submitted by:Prathibha Saseedharan

EKAHEBM046S7 BME

Guided by: Jibin Jose

Parallel Image Reconstruction in MRIUsing Wavelet Transform


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CONTENTS.Introduction.Image formation.Introduction to k-space.

Parallel image reconstructionWAVELET-REGULARIZED RECONSTRUCTION FOR

RAPID MRIAUTOCALIBRATED REGULARIZED PARALLEL MRI

RECONSTRUCTION IN THE WAVELET DOMAINConclusion


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INTRODUCTION

MRI is a non-invasive medical procedure.Nothing is inserted in a patients body, no dyes are

swallowed, and no contrast agents are injected, except under

special circumstances.Moreover, patients are not exposed to ionizing radiation,as

is the case with X-ray Computed Tomography (CT)

imaging.


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A powerful magnetic field aligns the nuclear

magnetization of hydrogen atoms in water in the body.

Radio frequency (RF) fields are used to systematicallyalter the alignment of this magnetization.


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This causes the hydrogen nuclei to produce a rotating

magnetic field detectable by the scanner.

This signal can be manipulated by additional magnetic fields

to build up enough information to construct an image of the

body.

The magnet first aligns hydrogen atoms that come within its

field, and then a radio-frequency pulse is applied to jostle

them momentarily.


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As they realign, special receivers pick up their signals and

transmit that information into computers in which special

programs convert those signals into vivid images.

The various tissues and fluids are distinguishable from

one-another largely because the concentration of hydrogen

varies within different tissues and bodily fluids


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Instrumentation.MRI scanner consists of a magnet, 3 magnetic field gradient

coils and an RF coil.

Magnet polarizes the protons in the patient, produces a

homogeneous magnetic field within patient.

Either permanent or resistive magnets are used for low

magnetic fields like 0.35T.Permanent magnet systems are made of cobalt-samarium.

Resistive magnets are created by passage of current through

copper.


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Current MRI scanners produce a field of 1.5T or more

by using superconducting magnets .

Made out of a set of 4 or 6 solenoidally woundsuperconducting wires.

Wires are made of niobium-titanium alloy.

Field is homogenised by using ferromagnetic blocks

and electrically fed resistive coils.


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Gradient coils- generation of magnetic field gradient so that

resonant frequencies within patient are spatially dependent.

Fitted directly inside the bore of cylindrical magnet.

3 separate gradients are required to encode x,y,z dimension

of image.

Maxwell pair produces a linear variation in Bo along z-axis.Production of linear gradients along x and y axes are

performed by Golay coils.(saddle coils)


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RF Coils : produces oscillating magnetic field necessary for

creating phase coherence b/w protons.

Also receives MRI signal via faraday induction.

Volume coils irradiate the whole body or just one specific

anatomical region.

Surface coils especially phased array coils are good in

imaging organs which lie close to the surface as a good SNR

is achieved.


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MR image is a 2-dimensional signal composed of

many distinct coded elements called pixels.

Arrangement of pixels in a planar slice of tissue gives

an array called matrix.

Volume of a pixel is called as a voxel.To accomplish MR imaging the 3 gradient coil pairs

must be timed for i.) slice selection , ii.) phase

encoding, iii.) freq encoding and k-space formalism.


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Frequency and PhaseFrequency and Phase

== tt

The spatial information of the proton pools contributing MR signal isThe spatial information of the proton pools contributing MR signal is

determined by the spatial frequency and phase of their magnetization.determined by the spatial frequency and phase of their magnetization.


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Slice selection. Clinical MRI studies acquire a series of slices through anatomical area

of interest.

Slice selection is accomplished using a freq selective RF pulse applied

simultaneously with one of magnetic field gradients denoted as Gslice.

Coronal, axial or sagittal slice selections corresponds to sections in y, z

or x directions.

If the selective RF pulse is applied at a freq ws with an excitationbandwidth of ws, then protons precessing at frequencies b/w

ws+ws and ws-ws are rotated into the transverse plane and those

with resonant frequencies are not affected and remain in the z-

direction.


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The thickness T of the slice corresponding to protonsthat are affected by the RF pulse is determined by T = 2ws/ Gslice.Slice thickness can be increased either by decreasing

the strength of Gslice or increasing the freq bandwidthof excitation pulse.

A longer RF pulse results in a narrower freq spectrumand therfore a thinner slice for a given value of Gslice.If the direction of Gslice is denoted by z,then

sl(z) = Gz.z. /2

where = duration of pulsez= proton position within slice


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Mathematically, signal from precessing magnetizationafter slice selection can be represented as

S slice slice p (x,y) dxdy

Where p(x,y) is the no: of protons at positions (x,y)within the body and is called the proton density.


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Phase- encodingHaving selected a slice, the other 2 dimensions must

be encoded to produce a 2-dimensional image.One of these directions is encoded by imposing a

spatially dependent phase on the signal fromprecessing protons.Other by creating a spatially dependent precessional

freq during signal acquisition.

Phase is encoded by a gradient turned on or off beforedata acquisition begins. S(Gy, pe) = slice slice p(x,y) e-j Gy pe dxdy


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Frequency encodingFreq encoding gradient Gfreq is turned on during data

acquisition.Assuming that Gfreq is applied in the x direction and

considering only the effect of this freq-encodinggradient the acquired signal is

s(Gx,t) sl sl p(x,y) e-jwxt dx dy =

slsl p(x,y) e-j Gx.xt dx dy


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k-space is a formalism widely used in magnetic resonance

imaging independently introduced in 1983 by Ljunggren and

Twieg.

Simply speaking, k-space is the temporary image space in

which data from digitized MR signals are stored during data

acquisition.

When k-space is full (at the end of the scan), the data are

mathematically processed to produce a final image.Thus k-space holds raw data before reconstruction.


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k-space is in spatial frequency domain. Thus if wedefine kFE and kPE such that


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where FE refers tofrequency encoding, PE tophase encoding, tis the sampling time (the reciprocal of sampling frequency), is

the duration ofGPE, (gamma bar) is the gyromagnetic ratio, m is

the sample number in the FE direction and n is the sample numberin the PE direction (also known aspartition number),

the 2D-Fourier Transform of this encoded signal results in a

representation of the spin density distribution in two dimensions.


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k-space has the same number of rows and columns as the final image.

During the scan, k-space is filled with raw data one line per TR

(Repetition Time). Although a strict mathematical proof does not exist and

counterexamples can be provided, in most cases it is safe to say that

data in the middle ofk-space contain the signal to noise and contrast

information for the image, while data around the outside of the image

contain all the information about the image resolution.


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k-space information is somewhat redundant

An image can be reconstructed using only one half of the k-

space,

Either in the PE (Phase Encode) direction saving scan time

(such a technique is known as half Fourier or half scan)

Or in the FE (Frequency Encode) direction, allowing forlower sampling frequencies and/or shorter echo times (such

a technique is known as half echo).


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Two Spaces

FTFT

IFTIFT

k-spacek-space

kkxx

kkyy

Acquired DataAcquired Data

Image spaceImage space

xx

yy

Final ImageFinal Image


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Parallel MRI (pMRI) is a way to increase the speed of the MRI

acquisition by skipping a number of phase-encoding lines in the k-

space during the MRI acquisition.

Data received simultaneously by several receiver coils with distinct

spatial sensitivities are used to reconstruct the values in the missing k-

space lines.We focus on the minimizing of the presence of noise in the

reconstructed image and also on removing of the aliasing artifacts from

the reconstructed image (artifacts caused by skipping some phase-

encoding lines in the k-space during the acquisition).


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A k-space image is formed by measuring the retransmitted signal.

The k-space image corresponds to the image in the Fourier space.

The real image of the object is obtained by Fourier transform of the k-

space image (it resolves the correspondence of the frequency and

spatial position of the signal).


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2D FFT

====>

k-space image final image


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In MRI, signal is usually received by a single receiver coil with an

approximately homogeneous sensitivity over the whole imaged object.

In pMRI, MRI signal is received simultaneously by several receiver coils with

varying spatial sensitivity -> This brings more information about the spatial

position of the MRI signal.


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The task of pMRI is to speed up the acquisition in order to:

be able to image dynamic processes without major movement artifacts (i.e. reduce the

speed of the acquisition so the movement during the acquisition time does not cause

significant artifacts),

shorten the MRI acquisition time that could be very long.

The bottleneck of the MRI acquisition is the number of retrieved lines in k-space and the

time needed to acquire one line in k-space. In pMRI, only a fraction 1/M of k-space lines is acquired while preserving spatial

resolution.

The acquisition is M times faster.

It causes an aliasing in the images - M points from the original image overlaps over

themselves in the image with aliasing.


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Linear combination of at least M images with aliasing retrieved by

the coils with varying sensitivity is used to reconstruct the original

image

(the coil configuration is supposed to be suitable for pMRI

reconstruction - the coil sensitivities should be distinct, all parts of

the imaged slice should be covered by at least one coil with

reasonable SNR in this part of the slice).

The parameters of the reconstruction are estimated using the exact

knowledge of the coil sensitivities.


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Aliasing

=====>

+

Reconstruction

==========>

+

Aliasing

=====>


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WAVELET-REGULARIZED

RECONSTRUCTION FOR RAPID MRI


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What is wavelet analysis?A wavelet is a waveform of effectively limited duration that

has an average value of zero and is of varying freq.Fourier analysis decomposes a signal into sine waves of

various frequencies.

Similarly, wavelet analysis breaks up a signal into shiftedand scaled versions of the original wavelet.

Signals with sharp changes might be better analyzed with anirregular wavelet than with a smooth sinusoid.

Wavelet analysis can be applied to one-dimensional data(signals), two-dimensional data (images) and, in principle, tohigher dimensional data.


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Advantages of this method.

By this method artifacts are significantly reduced

compared to conventional reconstruction methods.

Capable of recovering the missing k-space regions.Employs a non-linear approach and therefore blurring,

noise propagation, undersampling,aliasing are reduced.

Speeds up the reconstruction process.


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All problems related to reconstruction can be solvedby Daubechies Tl algorithm.Potential difficulty arises when forward model is

poorly reconditioned.This paper describes a TL algorithm specially tailored

to solve this problem.


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Method Consider a single receiving coil with homogeneous sensitivity. Thecorresponding model for the complex time-varying MR signal is

where " is the unknown proton density map to be recovered and k(t)denotes the so-called k-space trajectory.

In order to perform a numerical reconstruction, we must provide a

discretized version of the forward model .Time is sampled at N instantsresulting in the k-space samples {kn}. The signal to be reconstructed is represented as a linear combination of

basis functions that are shifted versions of a generator # on a finiteCartesian 2-D grid Cs:


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The signal is thereby parametrized by a set of Mcoefficients {c[p]}, represented as a vector c.

The term bn is introduced to represent both themeasurement noise and model mismatch. This model

is linear; thus there exists a N M matrix E such that:


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Variational formulation. The solution "c is defined as the minimizer of a cost function that

involves two terms: the data fidelity F and the regularization R that

favors solutions according to given prior knowledge. This is

summarized as

where the tuning parameter balances the effects of the two terms. F is chosen

as the square of the l-norm :

#m Ex#2 , which is justified when the noise is Gaussian.

The ill-conditioning, inherent to undersampled trajectories, imposes the choice of

an adequate regularization term R.

TV reconstruction is related to the l-norm of the modulus of the gradient and is

an optimal regularization for piecewise-constant solutions.


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Wavelet regularization

The underlying idea of wavelet regularization is that naturalimages tend to be sparse in the wavelet domain.Based on the property that a small l-norm promotes sparsity,

this solution is defined as:

where W and W1 are the wavelet decomposition andsynthesis matrices, respectively.


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Principle of TL algorithmSolution of the simpler wavelet denoising problem (E is the

identity matrix and W is orthonormal) is a single-stepthresholding:

Daubechies algorithm can then be explained by

iteratively bounding the initial reconstruction problemby a simpler denoising problem.Specifically, at iteration step n, one defines the

auxiliary variable

Where the wavelet vector "cn specifies the currentestimate of the solution.


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A fundamental point for our algorithm is that the multiplicationwith the matrix EHE corresponds to a 2-D convolution (Block-Toeplitz matrix). Indeed, by defining the kernel

Note that the kernel G has a support twice as large as Cs in eachdimension.The matrix-vector multiplication with EHE correspondsto the

most computer-intensive part of Algorithm. Based on the aboveproperty, we implement this operation by a pointwisemultiplication in the frequency domain using FFTs on a gridtwice as large as Cs .

The advantage is that this computation is exact while it avoidsthe use of regridding.


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For the linear reconstruction, a Conjugate Gradient (CG) loopis used,

with a tolerance fixed to 1e 8.

The TV reconstruction was implemented using the Iterative Re-

weighted Least Square (IRLS) method with 10 outer iterations. For the linear solver, CG with a tolerance 1e 8 is applied

For the wavelet reconstruction, we chose the Haar basis, which is the

simplest and fastest wavelet transform. 3 decomposition levels wereconsidered and cycle-spinning is used as in to avoid blocking artifacts.


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AUTOCALIBRATED REGULARIZED PARALLEL MRI

RECONSTRUCTION IN THE WAVELET DOMAIN


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IntroductionTo reduce scanning time in MRI parallel acquisition

techniques with multiple coils were used.This is usually done using the SENSE reconstruction

method.SENSE (Sensitivity Encoding) have been developed in order

to unfold the aliased registerd images in the k-space and inthe image domain, respectively.This method is supposed to achieve an exact reconstruction

in the absence of noise.

An array of multiple, simultaneously operated receiver coilsis used for signal acquisition.The array elements are usually surface coils, which exhibit

strongly inhomogeneous, mutually distinct spatial

sensitivity.

Sensitivity encoding makes it possible to reduce the density


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Sensitivity encoding makes it possible to reduce the density

and, consequently, the number of these steps.

In the widely used k-space view, reducing phase encodinginthis fashion means that the same k-space area is sampled by

fewer, more widely spaced readout lines.

The factor by which the number of readout lines is reducedis referred to as the reduction factorR.

In conventional image reconstruction, such reduced phase

encoding approach with multiple-coil acquisition, however,

permits the reconstruction of a full-FOV image without the

aliasing effect.


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Consequently, we propose to look for an image representation where

these localized transitions can be easily detected and hence attenuated.

The WT has been recognized as a powerful tool that enables a good

space and frequency localization. The statistics of the wavelet coefficients can also be easily modelled

allowing us to efficiently employ a Bayesian framework for the

estimation procedure.

W d fi th lti l t ffi i t fi ld f th


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We define the resulting wavelet coefficient field of thetarget image by = (a,m)m, (h,j,m, v,j,m, d,j,m)1jjmax,m

where a,m denotes an approximation coefficient atresolution level jmax and location m and o,j,m with o

{h, v, d} denotes a detail coefficient at resolution

level j, location m and orientation o which may bevertical, diagnol or horizontal.

We aim at building an estimate of from d. Then, anestimate of the objective image is easily derived by justapplying the inverse WT operatorT to .


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O ti i ti l ith


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Optimization algorithmThe goal of this algorithm is to iteratively compute a field of

coefficients that minimizes J .For doing so, we will use the concept of proximity operator which

was found to be fundamental in a number of recent works in convex

optimization.

Note that blurring effects in the Tikhonov regularized image are no

longer present in the WT regularized image.

Moreover, the aliasing artifacts in the basic-SENSE reconstructed

image are significantly smoothed with the proposed wavelet-based

approach but they are completely removed only if they were not


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CONCLUSION.Both these methods of reconstruction reduces aliasing

artifacts in data and improves the image .


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