accelerating nmr via nufft algorithms on gpus

Università degli Studi di Napoli “Federico II”

Candidate:

Federico Maria Ariù

matr. M61/000013

CLASSE DELLE LAUREE MAGISTRALI IN INGEGNERIA ELETTRONICA, LM-29

TESI DI LAUREA

Accelerating NMR

via NUFFT algorithms on GPUs

Anno accademico 2012/2013

Supervisors:

Prof. Amedeo Capozzoli

Ing. Claudio Curcio

Ing. Angelo Liseno

MRI

Magnetic Resonance Imaging (MRI) is a medical diagnostic technique based on the physical principle of nuclear magnetic resonance and allows to reconstruct images of the human body.

MRI has become one of the most used medical diagnostic modalities and it is used for many purposes, e.g. brain and orthopedic surveys.

Accelerating NMR via NUFFT algorithms on GPUs

Sampling and processing speed in MRI

Decreasing the required times for the patient immobility

Continuous monitoring of patient’s conditions during surgery


In many recent applications the need for decrease sampling and processing times arises as much as possible.

Monitoring activities and reactions of patients (e.g. functional MRI, fMRI)

1- ACQUISITION METHODS

2- PROCESSING ALGORITHMS

3-PROCESSING HARDWARE


Factors of sampling and processing speed

Thesis statement

The purpose of the thesis was to implement an MRI algorithm based on the

suitable use of parallel computing systems (Graphic Processing Unit, or GPU)

and fast processing algorithms able to handle non-standard acquired data.


MRI reconstruction

MRI system’s data-unknown relation is :

- Cartesian grid discretization m: measured data S: sensitivity map ρ: spin density


t: time parameter - Time discretization

Data are linked to spin density through sensitivity map as a Fourier transform relation. Data are the sampled values of the spectrum on the (Kx(t), Ky(t)) trajectory described by varying time

The relation between ρ and m could be written as a Discrete Fourier Transform (DFT)

Trasformata di Fourier

dxdyeyxyxStktkmytkxtkj

yxyx ])()([2

),(),())(),((

p q

ytkxtkj

qpqpkykxqkypkxeyxyxSyxtktkm

])()([2),(),())(),((

Fast sampling It is possible to use several data sampling method:

Cartesian Radial Spiral

Uniform DFT Non Uniform DFT (NUDFT)


curvetktk yx ))(),((

About cartesian sampling, processing can be attributed to a uniform DFT accelerated by a standard FFT algorithm in order to obtain a time complexity O(NlogN)

elaborate processing

- fast

+ slow

- straightforward processing

+ elaborate processing

- fast

+

Fast data processing

More sophisticated algorithms allowing to use faster sampling methods are required.

From a numerical point of view, Non Uniform Fast Fourier Transform (NUFFT) algorithms restore time complexity of FFT algorithms.

3 algorithms NUFFT types can be characterized: -NUFFT of type 1 – NED (Non-Equispaced Data) -NUFFT of type 2 – NER (Non-Equispaced Results) -NUFFT of type 3 – (Non-Equispaced Data and Results)


Fast data processing

MRI system’s data-unknown relation is :

The relation above can be rewritten as a system of linear equations:

mA mAAA

where

])()([2)( qypx ytkxtkj

i

iemmA

(NER-NUFFT) (NED-NUFFT)

Such system of linear equations has been solved by applying the biconjugate gradient stabilized method (BICGSTAB).


p q

ytkxtkj

qpqpkykx eqkypkxyxyxSyxtktkm

])()([2),(),())(),((

In order to reverse the relation, multiplying the system by the adjoint matrix, depending on the chosen sampling method, matrix-vector multiplications can be more rapidly processed by using NER and NED NUFFT algorithms.

m: measured data S: sensitivity map ρ: spin density t: time parameter

GPU implementation

Matlab C CUDA C

GPU acceleration was executed on Nvidia Tesla Kepler K20c.


The convenience of a GPU-implemented algorithm is the data parallelism, that is the possibility to solve simultaneously arithmetic operations on various data, obtaining better performance in terms of speed and processing.

An MRI algorithm based on a BICGSTAB routine using NER and NED NUFFT was developed both for the GPU and CPU version. Firstly the algorithm was implemented in Matlab language, after in C language and finally in CUDA C for the GPU version.

MRI numerical results

Phantom reconstruction


CPU

196

GPU

3 65

GPU

CPU

Time

Timespeedup

Computation time[sec]

Processing times of MRI application were validated using simulated data from a nonuniform (spyral type) sampling method. The image reconsruction pointed out the same accuracy shown in the literature [“A fast wavelet-based reconstruction method for MRI”, by Guerquin-Kern, IEEE Transaction on Medical Imaging] linked to the simulator.

Sampling grid

Non Uniform Fast Fourier Transform of

type 3 (NUFFT-3)

NUFFT -3 2D steps – Greengard&Lee approach:

1. Projection of the samples on a uniform grid

1

0

2121 ),(),(N

j

jjj yynxxngfynxnfyxyx

2. Precompensation ).(4

1),( 21

)()(

21

22

21 ynxnfeynxnf

yx

yxxyx

yx

nyn

yx

3. FFT t

Xr

Xr

yr

yr

sxyx

yx

yx

yx

ynim

M

Mn

M

Mn

nimeeynxnf

yxtmsmF

22

1 2

11

2

)1(

2

2

)1(

2

2121 ),(2

),(

4. Spectrum reconstruction on irregular frequencies ),(),(2

),( 2

,

121

21

kt

m

mmm

ktkk tmssmgmsmFts

tsFsp

sp

yx

yx

yxyx

5. Postcompensation ),(4

1),(

22

kk

ts

yx

kk tsFeetsFyx

ykxk


NUFFT-3 allows to use a multi resolution approach on the MRI application when the examination with more details of certain areas rather than others is required.

1

0

),(N

j

ytxis

jkkjkjk eeftsF

NUFFT-3 execution times:

CPU vs GPU

2023

GPU

CPUDNUFFT

Time

Timespeedup


Tests on the execution times of the NUFFT-3, for the GPU and CPU version, were carried out by varying the number of samples.

Conclusions


An MRI reconstruction algorithm from simulated data sampled on a spiral grid was implemented.

The convenience of using NER and NED NUFFT algorithms was demonstrated.

A two-dimensional NUFFT-3 for the processing of nonuniform samples for multi resolution imaging applications was implemented.

The performance of the NUFFT-3 implemented algorithm was verified.

MRI processing was entirely developed on GPU.

Thank you for your time


accelerating nmr via nufft algorithms on gpus

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