An efficient dynamic approach for exact reconstruction for CT image application

An Efficient Dynamic Approach for Exact

Reconstruction for CT Image Application

S.Asif Hussain, P.Syamala devi, C.Hemanth

Department of ECE, A.I.T.S, Rajampet, Andhra Pradesh, India

Abstract— For reconstructing volume of filtered projection data a parallel computing algorithm is proposed which suits spiral cone beam CT access for improvement of speed with high end processor. This work uses phantom data to achieve higher speed by accessing memory many times here the work is carried frame by frame as the pixel data arrives from filtered back projection; Interpolation is used for dedicated volume data which is decided by the coordinates of neighborhood pixels. Image quality is achieved in real time to refine artifacts by using parallel computing algorithm, experiment results shows the reduction of memory accessing times from many to one.

Keywords- Filtered back projection, Parallel computing Spiral cone beam CT, Artifacts

IINTRODUCTION

Digital image reconstruction is a robust by means by which the underlying images hidden in blurry and noisy data can be reveled [14]. The main challenge is sensitivity to measure noise to the input data, which can be magnified strongly, resulting in large artifacts in the reconstructed image .Many imaging techniques are based on reconstructing an image from data that can be interpreted, either directly or after some preprocessing, as a set of projections of the imaged object. The mathematical foundation is provided by the Radon transform (RT) which computes 1-D projections of a 2-D data at different view angles. Different from X-ray radiograph, the inner structure of the FOV can be detected with CT, or the CT image has much better space resolution than tradition X-ray radiograph [14].

 

CT has gain widely use as a tool not only in medical as medical imaging, noninvasive diagnostics and surgical planning but also in industry for nondestructive inspection[15]. In CT imaging, for example, the data is obtained by passing a set of narrow X-ray beams through the scanned object and collecting their intensities using an array of sensors. The acquired data represents the Radon transform of the cross-sectional absorption densities that form the image [1]. Munson et al. [2] showed that the data collected by the SAR, after demodulation and low pass filtering, represents the Fourier transform of the projections obtained from the reflectivity density of the targeted ground patch.

X-ray computed tomography is a technique closely combined with the X-ray radiation. With the technique, the inner structure of an N dimensional object can be reconstructed from the N -1 dimensional projections [4] Different from X-ray radiograph, the inner structure of the FOV can be detected with CT, or the CT image has much better space resolution than tradition X-ray radiograph[12].

However most popular in practice are methods based on the back projection (BP) operation which reduces the distortion by avoiding the interpolation step. The approach is also more suitable to handle other problems such as wave front curvature effects in SAR imaging the image reconstruction algorithm for the spiral cone beam CT can be divided into two categories, the approximate and the exact. Typically, the former one contains various kinds of the FDK-type algorithm transformed from the classic FDK-algorithm for circular cone beam CT which is widely used in the clinic now a days . The advantage of the FDK algorithm includes its fast computing speed and good image quality. But the disadvantage is also obvious, that is the cone beam angle could not be too big, or the artifact becomes very serious. Nowadays, the spiral cone-beam CT is not only the mainstream for its really fast scanning speed and the ability to produce truly 3D image and [5,6]. In 2002, katsevich proposed the first theoretical exact reconstruction formula for spiral; cone beam CT [7,8]. It is of truly FBP type with one dimensional shift invariant filtering, as a result the computation is more efficient than the random transform based reconstruction algorithm .It has feature of solving the long object problem and uses the data inside the Tam-Danielson window[9] and some more outside the window. Though it’s numerous advantages, the bottle neck is still obvious for its intensive computation. After the formula came out, several numerical study and implementation for the formula has been reported

In 2007, Jiang proposed a fast algorithm for katsevichs formula, the cone beam cover method [10] different from the pi –line method; the new method adopts the new concept cone beam cover. The method can update the voxel defined in the cone beam cover for each projection frame, so a parallel computing can be achieved .then, and then implement the new method with a Linux cluster [12] .Each frame occupies a computation node

In this paper, we focus the back projection (BP) operation the computational bottle neck [13] of the FBP algorithm.

II . BACKGROUND

p in F O c c tr

o

c

p a th o w is s w to n c A O p n th o f υ o th W p th in F im f f projection step nto all elemen

Fig 1. Back pr array o

One might as collected proje case due to th

ransform. In O

of the Fourie

center and le

perform back and examine t he contributio one contributi way of unders s with the for simply back p would be heav o the entire L needs to be re contributions t

A. filtered-ba

Our goal is to projection, the needs to be

hroughout the of the back pr filter

υ=

of the Fourier he Fourier spa

F ( ,

Where B ( projected imag

he back proj nverse Fourie F(x,y) This is

mage reconst first back proj filter, and then

p, the best we nts along the L

rojection b(x,y f all values P(

ssume that str ections will re he oversampl Other words, e

er space res

ess

sampling projections at the Fourier tr on at the orig ion at the edg standing this o rward projecti project the po vily blurred sin LOR from wh e-weighted, or throughout the

ck projection

o compute f ( e oversamplin filtered in e Fourier spac

rojected imag

this cone fil space and de ace. This oper

, , is the 2 ge and F( ,

jection-filtere er transform

known as th truction metho

jected, filtere n inverse Four

e can do is pl LOR.

y,Ø), into an i (S,Ø) for a fix

raight back p eturn the Imag ling in the ce each

projectio

sulting in ov

at the edges. t only two an ransform of th gin is doubled ges of the fie oversampling ion of a single oint source pr

nce the projec hich they cam r ‘filtered’, in e field of view

(fbp) reconst

(x, y) from p ng in the cen

order to ha e. Basically, t ge must be fi

ter accentuate accentuates v ration is summ

2-d Fourier tr

is the 2-D d image. Th

of F( , t

he back proje od, where the

d in Fourier rier transform

lace a constan

image reconst xed value of Ø

projection of ge, but this is enter of the

on fills in on

versampling

For example gles, say Ø1 he result we s d while there eld of view. A

in the space e point source rojections, the ctions are adde e. The oversa n order to hav w

truction

p(s, Ø) . Afte nter of Fourie

ave equal sa the Fourier tra iltered with a

es values at th values at the ce marized in

ansform of th Fourier transf he final step

to obtain the ction-filtering e projection d space with th med. Alternativ nt value ruction Ø all the not the Fourier

ne slice

in the

e, if we and Ø2 see that is only Another domain e. If we e image ed back ampling ve equal er back r space ampling ansform a ‘cone’ he edge enter of he back form of is the e image g (BPF) data are he cone vely the filt of tha to gra Th usi for int dis B. If pro (FB W Ca Fo | dim filt fin filt vie cal can pa A. Th int an F W Th Sin (FB Ba f(x   He tra pro Th |ωr

B. In dis an the [0,

tering can be f b(x, y) with at the function

the convolu adually decay hus any nume ing a signific r the final re terchanging t scussed next.

Reconstruct

we intercha ojection steps BP) image rec F(x, y) = here the 'filter

an be regarded ourier transfor

| is a sectio mensional con ter is applied nite support i tered projectio ew. This mean lculated with n be used with art of the reaso

FILTERED

he Radon tran tegral project ngles θ. The co

,

here r and θ a he projections nograms. In BP) algorithm ased on the we

x,y)= ,

( ,

ere

ansform in th ojection opera

F(x,y) = β

he projections r| and then bac

DISCRETE

n practice, the stribution are nd the reconstr e projection a , π) and that

performed in

h A d

n b(x, y) has a ution with the ying values o

erical proced antly larger im esult. This di the filtering

tion by filtere

ange the ord ,we obtain th construction m

, Ø

red' projection

,

d as pre-corre rm of f(x,y). T

on through th ne filter. An ad d to each me in s, and we

ons for | | le ns that with F

a much sma h BPF, for the on for the popu

III. M

BACKPROJE

nsform (RT) tions of a 2-ontinuous Rad

,

are polar coord s ˆ f(r, θ) are

their original ms are

ell-known inv

,

| | (r

,

he variable r, ator

,

s f(r, θ) are fi ck projected t

DIRECT BA

e number of determined b ructed image angles θ are e all images ar

image space disadvantage o

a larger suppo e filter term outside the su dure must firs mage matrix isadvantage c

and back p

ed-back projec

der of the f he Useful filter method:

n, given by

| |

ected for the o The one dimen

he rotationall dvantage of F easured projec

only need to ss than the ra FBP the image

aller reconstru e same level o ularity of the F

METHODS ECTION ALG

represents a D function f don transform

dinates and δ

e also referred l form, filtere

version formul

r, ).

, represen and B is th

irst filtered us to reconstruct

CK PROJEC

projections P by the data ac is discrete. W evenly distribu e square with

via the convo of this approa ort than f (x, y , which resu upport of f (x st compute b size than is n can be avoide projection step ction (FBP) filtering and red-back proje

,

oversampling nsional 'ramp ly symmetric FBP is that the ction, which o back projec adius of the fi

e can be effic uction matrix of accuracy. T FBP algorithm

GORITHMS

set of paralle f(x, y) at dif is defined by

is the unit im d to as the da ed back proje

la for the RT:

nts a 1-D F he continuous

sing the ramp

the image.

CTION.

P and the sam cquiring equip We will assum uted in the in h N × N pixe

olution ach is y) due ults in

im p w p o

F

I r a a m o k

A

T H c p

W o T

N w T d

  

 

 

  

mplement th projection ope windowed an performed for over all angles

F (m, n) = ∑ (

nterpolation w required to com approximation achieved by model the phy our implemen kernel.

IV. PRACT A. THE SPIR

Figure The geometry Here, is t coordinate sy position which

Where s is the of the support Then, the unit

Now, a rotatin with its two ax To the detecto detector plane,

       

  

 

e FBP algor eration is dis nd sampled. T

each pixel f(m s θ:

(mcos +nsin

with a kernel mpute sample n to the con introducing t ysical propertie ntation, howe

TICAL RECO RAL CONE-B

1. The geome of the spiral c the voxel to b ystem O-XYZ h can be denot

cos .

e rotation ange cylinder. vector pointin

,

ng coordinate xis’s ̂ and or plane while , defined by:

      

̂ sin

̂ cos

̂

rithm on a c scretized and The discrete m, n) as a sum

, θ).       l φ (ρ) in th ed at non-in ntinuous back

the image sa es of the sensi ever, we use

ONSTRUCTI BEAM CT G

etry of spiral c cone-beam CT be reconstruct

Z. Denotes

ted as

. sin ,

el of the sourc

ng from (S)

is adopted at d parallel

the axis ˆw e

  

n , cos ,

s , sin

0,0, 1  

computer, th d the ramp f

back projec m of projected

he radial direc ntegral values k projection ampling oper ing equipment

the ideal sa

ION APPRO GEOMETRY

cone beam CT T is shown in ted. In the Ca s the X-ray

ce and R is the

to is :

t source positi

e perpendicula

, 0  

, 0  

he back filter is ction is d values

ction is s. Better

can be rator to t [3]. In ampling

OACH

T n Fig. 1.

artesian source

e radius

ion y(s)

ar to the On ori pla the

B.

Pa Le

Ob po Fo

Th F(

H we ex f(

As co the

δf( Ac the Pa ho

δf( e

n the detecto

igin at the pr ane and the tw e detector plan

Proposed al

arallel back pr et’s make the f

Df

bviously, s F osition s . Furth ollowing defin

f( =

hen,

Here is the -e can s-e-e tha

ists a δf( . w is updated a

s shown in F nnecting sour e following eq (

ccording to th e projection no arametric in olds

(

0

lse

or plane, a th rojection of th wo axis’s para ne is expanded

lgorithm:

rojection following defi

f( (s), ,

FP is the filt hermore, let’s nition: F

- line decided at for each so we say that

at source posi

Fig. 3., let rce s and voxe quation holds: he sufficient co

ot at the sourc nterval to be

0 ,

hird coordina he X-ray sour allel to . ̂ an d by the two v

finition

tered projecti s make the

… by the voxel ource position

tions s by δf(

to be the el . Obviou

ondition of π

ce on the 0.Then, the

if s

ate is defined rce on the de nd In this vectors

.

ion data at s

…….δf( .

ns s on the

.

voxels on th usly,

-line, we can

following equ d with etector

s way,

source

there

he ray

make

This indicates that if the projection data of a ray is given, then all the voxels on the ray can be updated.

Here, we define the voxels on a ray to be:

X= , ,

Where connects sourse s and projection F . so X is the set of the voxels on

Now

Δf(x)=

Then the exact reconstruction formula can be expressed as:

f(X) = +……δ f(x ) =

Eqn.updates the voxels on the ray connecting source s and projection F If we define FP to

be the set of all the projection at source s , s Ray to be all the ray pass Ra _. to be all the voxels on

s Ray . Then we come to the parallel reconstruction formula:

F( … … .

We can see from the equation that the voxels in s X can be updated if one frame of filtered projection arrives. When the next frame of filtered projection arrives, another set of s X can be updated.

The voxel to be updated has the following features: (1) The voxel is in the support cylinder

(2) the source s is on the parametic interval decided by voxel

V. THE EXACT RECONSTRUCTION FORMULA The exact formula proposed by Katsevich can be expressed as following:

f( =- , dγ

Here Df( , is the projection of the reconstructed object . is the π parametric interval.

From the equation we can see that the projection data is first filtered, then the filtered projection data is Back projected to form the attenuation image. It’s of truly FBP type.

The formula can be solved by the following five steps: (1) Derive the projection data Df(y,θ) using chain rule

, , , , ,

1

(2) Length weight correction

, , (s,u,w)

(3) Rebinning

Define r to be the maximum[15] radius of the object arcsin(r/R), using linear interpolation to all the

[ -π/2- ,

, , , , ,

Here , = (Ψ+ )

(4) IDfiltering

  

, , =   , , ,  

 

(5) Rebinning

  

, , , , ,  

Here  ,    is th one with the least resolution value , 

satisfyig  

).

 

(6) BACK PROJECTION

f(

x

, ,_,

ds

VI.SIMULATIONRESULTS

Fig1:Shows the CT image

       

 

Fig 2:The sinogram image acquired from CT

Fig 3 : The simple projection image

T s c m th th d u p e r m r a w m a g Fig 4

Fig 5: T

F

The implemen spiral cone-bea computation. memory is qui

his paper, we hat can reduce down to once updated once pipeline can b experiment re reconstructed method may researchers int algorithms in works to be do more interpola and find some greatly reduce

4 : The first pro

The constructio

Fig 6: The back

VII. CO ntation of the

am CT is quit Among them ite a bottle nec e propose a p

e the memory e. As some c

a filtered b be achieved esults, we c by the new a provide som tending to im

hardware. T one. The most

ation methods e new paralle

d if proper int

ojection on th

on of back pro

k projection in

ONCLUSION exact reconstr te time consum m, the frequ

ck to improve parallel back y accessing tim

certain attenu back projecti for the recon an see that algorithm is sa me suggestion mplement the

Though, ther t important on s for the filte l algorithm a terpolation can he sonogram ojection image n action N ruction algorit ming for its in uent access e the performa projection alg mes from man

uation value ion data arr nstruction. Fr the image atisfactory. T ns to engine

exact reconst e are some ne is to develo

ered projectio as the artifact n be employed

e thm for ntensive of the ance. In gorithm ny times can be rives, a rom the quality The new eers or truction further op some on data, can be d. Th De the [1] [2] [3] [4] [5] [6] [7] [8] [9] [10 [11 [12 [13 [14 [15

his work was elhi at AITS , e successful co

F. Natterer, The

Wiley, 1986. D. C. Munson, J

of spotlight-m vol. 71, pp. 91 S. Basu and

reconstruction

Processing, vo G. T. Herman Academic Pre Kalender, W. A

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AUTHORS P

a Hemanth recei Communication Eng ndia. Presently he hnology & Science nd pursuing his M.T eries Analysis and Im

n. Shaik received B & Communicati Hyderabad, India. D Degree in Biome ersity, Anantapur, rya Institute of A.P., India. He n Dept. of ECE. H l Conferences & jo (India), BMESI (In E (Hongkong).His alysis and Image Pr

T

of A.I.C.T.E nks to my guid

Tomography.New

omographic form

roceedingsof the

filtered backpro

IEEE Trans. on

00.

projections. New

y”, Physics in M

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04), “Cone beam ojection in Kats urnal of X-Ray S 04.

, Jiang, M., “P International Jou

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D. Satya Naraya econstruction “in

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PROFILE

ived B.Tech Deg gg. from J.N.T.Un e is with Annam es, Rajampet, A.P., Tech. His research mage Processing.

B.Tech & M.Tech D ion Engg. from

He is currently edical Image Proce India. Presently he Technology & S is working as A e presented many ournals. He is a me ndia), IACSIT (Sin research interests ocessing. , New de for w York: mulation

e IEEE,