Wavelet-based Local Tomography

Transcription

1 Inian Society for Non-Destructive Testing Hyeraba Chapter Proc. National Seminar on Non-Destructive Evaluation Dec. 7-9, 006, Hyeraba Wavelet-base Local Tomography M.V. Gopala Rao, E.M.L. Tanua an S. Vathsal 3 Department of Instrumentation an Control Engineering, Narasaraopeta Engineering College, Narasaraopet Department of Electronics an Communication Engineering, Narasaraopeta Engineering College, Narasaraopet 3 Extramural Research & Intellecctual Property Rights, DRDO HQ, New Delhi Abstract In this paper we evelop an algorithm to reconstruct the wavelet coefficients of an image from the Raon transform ata. The propose metho uses the properties of wavelets to localize the Raon transform an can be use to reconstruct a local region of the cross-section of a boy using almost completely local ata which significantly reuces the amount of exposure an computations in X-ray tomography Keywors: Global tomography, Raon transform wavelets, Local tomography T. Introuction The Raon transform was first introuce by J. Raon in 97. Little computational attention was given to it until the avent of computers enable the fast evaluation of Fourier transforms an their corresponing convolution. The Raon transform is now a mainstay of meical imaging as well as many other remote imaging sciences. One problem with the Raon transform is that in -D, where most meical imaging is conucte, the inversion formula is globally epenent upon the line integrals of the obect function f (x, y) [5]. In many situations a physician may only be intereste in images of a very local area of the boy. One woul prefer to expose only that local portion of the patient s boy to whatever raiation is being use. However, the non-locality of the Raon transform forces one to expose to raiation to all the - D slices of the boy which intersect with the region of interest. There have been a number of attempts to alleviate this problem. One approach is to o 3-D tomography, but this requires integrals over full -D hyper planes, which we are trying to avoi. K.T. Smith an F. Keinert [6] was introuce Lamba tomography which oes not attempt to reconstruct the function f (x, y) itself but instea prouces the relate function L f = Λ f + µλ - f. This has an avantage that the reconstruction is strictly local in the sense that computations of L f (x, y) require only integrals over lines passing arbitrarilyclose to (x, y). But it requires knowlege of what kin of useful information about f (x, y) is retaine in L f (x, y). An also this approach is well aapte for ege etection, but is not esigne to recover the original ensity of the image. Wavelets [3] have receive much interest by engineering community in the past ecae, because it has an important property of localization nature in both time an frequency. In [4], Olson an DeStefano implement a irect reconstruction algorithm NDE-006

2 M.V. Gopala Rao, E.M.L. Tanua an S. Vathsal in which the -D wavelet transform of the proection ata is compute for each angle. Delaney an Bresler [7] compute the -D separable wavelet transform of the image irectly from the proection ata. Both algorithms take avantage of the observation, mae in [8], that Hilbert transform of a function with many vanishing movements has rapi ecay. In fact, the Hilbert transform of a compactly supporte wavelet with sufficiently many vanishing movements has essentially the same support as the wavelet itself. Thus in both algorithms, the high resolution parts of the image are obtaine locally, an the low resolution parts are obtaine globally. It has been observe that, in some cases, the Hilbert transform of a compactly supporte scaling function also has essentially the same support as the scaling function itself. We take avantage of this observation to reconstruct the low resolution parts of the image as well as the high resolution parts using almost local ata plus a small margin for the support of the filters. This gives substantial savings in exposure of computations over the methos in [4] an [7]. The propose metho calculates the wavelet coefficients of the reconstructe image with the same complexity as the conventional filtere back proection metho. The wavelet coefficients are obtaine irectly from the proection ata, which saves the computations require to obtain the wavelet coefficients from the reconstructe image. The main features of our algorithm are: () Reuce exposure compare to previous algorithms. () Computatio-nally more efficient than others because we use smaller exposure lengths. (3) Uniform exposure at all angles which allows easier implementation in har ware. (4) Ability to reconstruct off-center or even multiple regions of interest, as well as centere reconstruction. In Section-II escribes the funamentals of Tomography an filtere back proection algorithm briefly. The non-locality nature of the Raon transform also explaine in this section. Section-III escribes the funamentals of wavelets, multi resolution analysis an synthesis an reconstruction formulas using wavelets. In section-iv escribes the local reconstruction technique an algorithm. Section-V is evote for computer simulations an results. Section- VI gives the conclusions, an finally references in Section-VII.. Funamentals of Tomography. Raon Transform In Computerize Tomography (CT), a cross section of the human boy or an in estructive obect is scanne by a noniffracting thin X-ray beam whose intensity loss is recore by a set of etectors. The Raon Transform is a mathematical tool which is use to escribe the recore intensity losses as averages of the tissue ensity functions over hyper planes which, in imension two, are lines []. Let f(x, y) be a two-imensional ensity function of the obect which is usually calle the obect function, or the image function restricte to a isc of raius one. Then the Raon Transform of the function f(x, y) represente by where δ is the one-imensional Dirac eltafunction an t = x cos + y sin is the equation of a line along which the proection has been measure. 46 NDE-006

3 Wavelet-base Local Tomography. Filtere Back Proection In orer to reconstruct the function f(x, y) from the proection ata we use the filtere back proection metho. If proections are known at enough angles, the obect function can be reconstructe by using the formula ω( xcos + ysin ) f (x y) = (ω) ( ) S e ω ω Where S (ω) is Fourier transform of P (t). This algorithm can be implemente in two steps. The first step is the filtering operation q ( t) = H P ( t) which is the IFFT of S (ω) ω. () The secon step is the back proection 0 f ( x y) = Q ( x cos + y sin) Where H is the Hilbert transform an represents the partial ifferentiation. Insertion of an appropriate ban limite winow into the filtering yiels the filtere back proection formula ( ) ω( xcos + ysin ) f (x y) = S (ω) e ( ω H ( ω)) ω = P ( t) * IFFT( ω H ( ω)) ( ) 0 () The winow H(ω) must be chosen to meet two criteria. Eirst the winow shoul be chosen to agree with the essential ban limit of ω, so that the resultant will be a goo approximation to f(x, y). Secon, the winow also represents a mollification of the inversion of the Raon transform, which is an unboune operator without the aition of the winow [6]..3 Nonlocality The problem with the reconstruction formula () is that the inverse Fourier transform of ω H(ω) will not be locally supporte. This stems from the fact that ω is not ifferentiable at the origin. The non locality of this inverse Fourier transform implies that local calculation of the convolution in () will require global values of the Raon transform. The non ifferentiability at the origin can not be significantly altere by the winow, without harming the structure of the image. From (), the ifferentiation is local, but Hilbert transform is not, since it imposes a iscontinuity upon the Fourier trans-form of any function whose average value is not zero, an iscontinuities on the higher erivatives which are not zero at the origin. The imposition of these iscontinuities at the origin in the frequency omain will therefore sprea the support of functions in the time omain. For this reason, a local basis will not remain local after filtering. For the reasons outline above, we woul like a basis of functions which are essentially compactly supporte, an which possess several zero moments. This secon conition will ensure that the basis functions remain essentially compactly supporte after the filtering process, an so will allow a reconstruction from localize ata. Wavelets are generally constructe with as many zero moments as possible, given other constraints such as locality an smoothness [] an [3]. Thus, the local properties of the high resolution components of a wavelet transform will remain local after the filtering in the reconstruction of the image from the Raon transform 3. Funamentals of Wavelets 3. Wavelet Transform The wavelet transform is a tool which is use in many ifferent areas such as signal NDE

4 M.V. Gopala Rao, E.M.L. Tanua an S. Vathsal an image processing, numerical analysis an tomography. Here, we assume that the reaer is alreay some what familiar with the theory of wavelets; hence, we will only highlight some of its results relevant to this work. Wavelets are functions erive by shifts in position an scale from a single function ψ(t) calle the basic (mother) wavelet. It is possible to construct orthonormal wavelet basis, from a so-calle scaling function φ(t), which is the basic wavelet an it can be associate in a unique way. The function φ(t) generates a so-calle multi resolution analysis, represente by the basis functions / φ, k ( t) = φ( t - k), k Z an / ψ, k ( t) = ψ( t - k), k Z, where Ζ is a set of integers. The refinement equations of scaling an wavelet functions are given by The ilations an translations of φ(t) inuce a multi resolution analysis (MRA) of L (R), i.e., a neste chain of close subspaces {V m } whose union is ense in L (R). Here, V is the subspace spanne by / / φ( t - k ), - k. 3. Wavelet Reconstruction In this section we will present an algorithm which can be use to obtain the wavelet coefficients of a function on R from its Raon Transform ata. Given a real value square integral function g on R an let f(x) given in R, then the Wavelet transform of the function f(x) can be represente by - W ( a b) = g (( t - b)/ a)) x = f (- b)* g (- b) a b a 0 From this efinition, the wavelet transform of the Filtere Back Proection formula can be written ½ φ( t) = h[ n] φ(t - n) an n ½ ψ( t) = g[ n] ψ(t - n), n where {h[n] } an {g[n]} are a pair of igital low-pass an high-pass quarature-mirror filters that are relate through g[n] = (-) -n h[-n]. The relationship between the wavelet an scaling function can also be represente in frequency omain as Φ( ω) = Η ( ω/) Φ( ω/) an Ψ ( ω) = G( ω/) Ψ ( ω /), where we have use Φ(ω) an Ψ(ω) are the Fourier transforms of φ(t) an ψ(t): H(ω/) an G(ω/) are the Fourier transforms of the quarature mirror filters an are both perioic functions with perio. ( )* ( ) ( ) P t H P t 0 g W.Tof f ( x y ) = W ( ) = f x y Where P g (ť) is the Raon Transform of g a,0(-b) an ť = a x cos + a y sin In the iscrete form the filtere back proection can be written as W ( n n ) = P ( n)* H P ( n) f ( ) 0 g g, where n = - n cos + - n sin. This algorithm can be implemente in two step process as below. Step: Filtering q - - ( t ) = P ( ) * P ( ) t H t g t Step: The back proection W ( n n ) = Q ( n cos + n sin ) f 0 ( ) 48 NDE-006

5 The filtering part can be implemente in Fourier omain as Q ( ω) = S ( ω) ω G ( ω cos ω sin ) W ( ω), where G (ω cos, ω sin ) is the Fourier Transform of g a,0 (.), S (ω) is the Fourier Transform of P (.), Q, (ω) is the Fourier Transform of q, (.), an W(ω) is a smoothing winow. Therefore the wavelet base filtere back proection can be implemente using the same algorithm as the conventional filtere back proection metho while the ramp filter ω is replace by wavelet ramp filter ω G ( ω cos ω sin ). 4. Multi Resolution Resolution Reconstruction If the wavelet basis is separable, an efining φ(x,y) = φ(x)φ(y), ψ (x,y) = φ(x)ψ(y) ψ (x,y) = ψ(x)φ(y), an ψ 3 (x,y) = ψ(x) ψ(y). Then the approximation coefficient are obtaine by Wavelet-base Local Tomography / A [, ] [( * )(( )cos ( )sin )] f m n = H R P m + n 0 ϕ These co-efficients can be calculate using stanar filtere backproection metho, while the filtering part in the Fourier omain is given by Q ( ω) = S ( ω) ω Φ ( ω cos ω sin ) W ( ω) A where Φ (ω cos, ω sin ) = Φ (ω cos ) Φ (ω sin ). The etail co-efficients can be foun in a similar way as A H = ω Φ ( ω cos, ω sin ) = ω Φ ( ω cos ) Φ ( ω sin ) D H = ω Ψ ( ω cos, ω sin ) = ω Φ ( ω cos ) ψ ( ω sin ) D H = ω Ψ ( ω cos, ω sin ) = ω ψ ( ω cos ) Φ ( ω sin ) D 3 H = ω Ψ ( ω cos, ω sin ) = ω ψ ( ω cos ) ψ ( ω sin ) To reconstruct the image from these coefficients, the iscrete approximation at resolution + can obtaine by combining the etail an approximation at resolution, [3],[9] + k = l= A ( m n ) = h( m - k ) h( n - l ) A ( k l ) + h ( m - k ) g ( n - l ) D ( k l) k = l= + g ( m - k ) h( n - l ) D ( k l ) k = l= + g ( m - k ) g ( n - l ) D ( k l ) k = l= 5. Local Reconstruction 3 It can be shown that if the wavelet function has sufficient number of zero moments, the Hilbert transform oes not essentially increase the support of basis. We have note that this is the case even for scaling functions when we use wavelet basis with sufficient number of vanishing moments. Fig. shows the Daubechies biorthogonal wavelet an scaling function an also the Ramp filtere version of these functions. Therefore the Ramp filtere scaling function oes not sprea either for the wavelet basis. Using this fact we have evise an algorithm to reconstruct local regions using l / l D [, ] [( g * )(( )cos ( )sin )] f m n = H t Pψ P m + n only local ata plus a small margin of the 0 support of the reconstruction filters. In local l=,an 3. This means that the wavelet an reconstruction artifacts are common close to scaling co-efficients of the image can be the bounary of the ROI. In orer to avoi obtaine by filtere back proection metho these artifacts, we have extene the P while the ramp filter is replace by scaling (t) continuously to be constant on the missing an wavelet ramp filters, given below. proections. The algorithm propose is escribe briefly in the following steps. NDE

6 Step : Obtain the Sinogram of the Hea Phantom. M.V. Gopala Rao, E.M.L. Tanua an S. Vathsal Fig.: Top, the scaling an wavelet basis in Daubechies biorthogonal wavelet basis. Bottom, the ramp filtere version of the scaling an Wavelet functions Fig. 5: Blowup of the centere region of interest Left: Reconstruction using wavelet metho using local ata. Right: Reconstruction using stanar FBP using global ata Step : Perform the -D multi resolution analysis on these proections, to get approximation an etail coefficients at appropriate level. Step 3: Select the require region of interest in each approximation an etail coefficients obtaine in step, an reconstruction the region of interest proections. Fig.: Left: the shepp-logan hea Phantom Right: The reconstruction using FBP Fig. 3: Left: Wavelet reconstruction Right: Reconstruction from the wavelet coefficients Fig.4: Blowup of the centere region of interest Left: Reconstruction using local ata Right: Reconstruction using stanar FBP metho using global ata Step 4: Perform filtere back proection to get the require region of interest image. 6. Computer Simulations an Results We have obtaine the one level approximation an etail coefficients of the 56x56 pixel image of the Shepp-Logan hea phantom using global ata. (Fig.4). In this ecomposition we use Daubechies biorthogonal filters (Fig ). The quality of the reconstructe image is the same as with the filtere back proection metho (Fig 3). Figs. 5 an 6 show two examples in which two regions of interest are reconstructe using the local reconstruction metho propose in this paper. In Fig.5 an 6 the blow up of the ROI using both local reconstruction an stanar filtere back proection using global ata is shown for comparison. We have reconstructe the offcenter isk of raius 3 pixels locate 80 pixels from the center of the image. (Fig 6).in both Fig 5&6 the proections are collecte from a isk of raius 44 pixels, 50 NDE-006

7 Wavelet-base Local Tomography therefore the amount of exposure is 7% of the conventional filtere back proection metho. 7. Conclusions In this paper we have explaine the non locality nature of the Raon transform an how the local ata can be extracte using wavelet transform of the proection ata. The algorithm presente here uses the properties of wavelets to localize the Raon transform an can be use to reconstruct a local region of a cross section of a boy with significantly reuce exposure in X- ray tomography. The results presente here shows, the ability to reconstruct off-center or even multiple regions of interest, as well as centere reconstruction more efficiently. 8. References. Kak. A.C. & Slaney. M., Principles of Computerize Tomographic Imaging, IEEE press, New York, 988. Rao.R. M. Wavelet Transforms: Introuction to theory an applications : Pearson eucation, E Mallat. S.G., A theory for multi-resolution signal ecomposition: The wavelet representation by IEEE Trans. on Pattern. Ana. an Machine. Intel. vol-, pp , Olson.T, an Stephano, Wavelet localization of the Raon Transform, IEEE Tran.on Signal Processing, Aug Ael Fariani, Introuction to the Mathematics of Compute Tomography, MSRI Publications, Volume 47, K.T. Smith an F. Keinert, Mathematical founations of compute tomography, Appl. Optics 4, pp , A.H. Delaney an Y. Bresler, Multiresolution tomographic image reconstruction using wavelets, IEEE Intrn. Conf. Image Proc.Vol ICIP- 94, pp , Nov D. Walnut, Application of Gabor an wavelet expansions to the Raon transform, Prob. An Stoch. Methos in Anal., with applications, Kluwer Acaemic publishers, Inc., 87-05, F. Rashi farrokhi, K.J.R. Liu, C.A.Berenstein.D. Walnut. Localize Wavelet base Computerize Tomography, Electrr Engg. Dept an Institute for System s research, University of Marylan. USA 99. NDE-006 5