Original Contribution - acil.med.harvard.edu · for the best viewing direction (also called...

doi:10.1016/S0301-5629(02)00762-7

● Original Contribution

A THEORETICAL FRAMEWORK TO THREE-DIMENSIONALULTRASOUND RECONSTRUCTION FROM IRREGULARLY

SAMPLED DATA

RAUL SAN JOSE-ESTEPAR,* MARCOS MARTIN-FERNANDEZ,*P. PABLO CABALLERO-MARTINEZ,* CARLOS ALBEROLA-LOPEZ* and JUAN RUIZ-ALZOLA†

*Department of Teorıa de la Senal y Comunicaciones e Ingenierıa Telematica, University of Valladolid, Valladolid,Spain; and †University of Las Palmas de Gran Canaria, Las Palmas, Spain

(Received 15 April 2002; in final form 20 November 2002)

Abstract—Several techniques have been described in the literature in recent years for the reconstruction of aregular volume out of a series of ultrasound (US) slices with arbitrary orientations, typically scanned by meansof US freehand systems. However, a systematic approach to such a problem is still missing. This paper focuseson proposing a theoretical framework for the 3-D US volume reconstruction problem. We introduce a statisticalmethod for the construction and trimming of the sampling grid where the reconstruction will be carried out. Theresults using in vivo US data demonstrate that the computed reconstruction grid that encloses the region-of-interest (ROI) is smaller than those obtained from other reconstruction methods in those cases where thescanning trajectory deviates from a pure straight line. In addition, an adaptive Gaussian interpolation techniqueis studied and compared with well-known interpolation methods that have been applied to the reconstructionproblem in the past. We find that the proposed method numerically outperforms former proposals in severalcontrol studies; subjective visual results also support this conclusion and highlight some potential deficiencies ofmethods previously proposed. (E-mail: [email protected]) © 2003 World Federation for Ultrasound in Medi-cine & Biology.

Key Words: 3-D ultrasound imaging, Irregularly sampled ultrasound data, Reconstruction grid extraction, PCA,Adaptive Gaussian interpolation.

INTRODUCTION

Three-dimensional (3-D) ultrasonic imaging is becominga widespread practice in clinical environments due to thepotential of applications based on 3-D representation.Basically, the major drawback that physicians have tocope with when traditional single 2-D B-scans are usedthe need of mentally reconstructing the 3-D anatomy canbe naturally overcome using 3-D ultrasound (US) imag-ing and appropriate further processing. But we may alsoenumerate other interesting benefits of 3-D echography:the spatial relationships among 2-D slices are preservedin the 3-D volume, allowing an off-line examination ofscans previously recorded even by another clinician;slices that cannot be acquired because of the geometricalconstraints imposed by other structures of the patient cannow be readily rendered by the so-called any-plane slic-

ing; and volume visualization and accurate volume esti-mation may greatly enhance the diagnosis task.

Several 3-D US techniques have been reported inthe literature in the past few years. These techniquescan be coarsely classified as those derived from a 2-Dphased-array probe, and those that obtain a 3-D dataset from 2-D B-scans acquired in rapid successionwhile the probe is in motion. The former techniqueemploys a bidimensional array of piezoelectric ele-ments and the volume is scanned by electronicallysteering the array elements. The main drawback of thispromising technique is the limited field of view of theexisting probes. The latter technique makes use ofconventional 2-D US systems and a positioning sys-tem. This technique includes both the freehand and themechanically-swept volume acquisition techniques.Freehand has received increasing attention, especiallysince the second half of the 1990s (Nelson and Preto-rius 1997; Detmer et al. 1994), probably due to itsinherent flexibility and low cost compared with the

Address correspondence to: Carlos Alberola-Lopez, ETSI Tele-comunicacion, Campus Miguel Delibes, 47011 Valladolid, Spain. E-mail: [email protected]

Ultrasound in Med. & Biol., Vol. 29, No. 2, pp. 255–269, 2003Copyright © 2003 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/03/$–see front matter

255

3-D US probes, and it can be now considered as awell-trusted technique.

In freehand imaging, a 3-D positioning sensor pro-vides a measure of the position and orientation of thecoordinate system of a receiver (typically attached to theprobe) with respect to a fixed coordinate system locatedat the transmitter. Figure 1 sketches the elements in-volved in a freehand system. Each B-scan pixel, which isinitially measured in the coordinate system of the probe,can be spatially located with respect to the fixed coordi-nate system by means of a vector of position and a vectorof orientation that jointly comprise the six degrees offreedom between two 3-D coordinate systems. A simpleaffine transformation allows us to convert each positionin the image plane to a position in 3-D space. A criticalstep is the calibration of the freehand system. Calibrationdeals with the problem of finding the transformationbetween the image plane and the receiver coordinatesystem. The accuracy of the overall system criticallydepends on this calibration stage, which, if successfullycarried out, provides an accurate registration of US im-ages to 3-D space. Interesting papers have addressed thisissue in detail (Detmer et al. 1994; Prager et al. 1998).

The acquired data can be seen as a scattered distri-bution of planes in 3-D space due to the total freedom ofthe physician to perform the scanning procedure. Recon-struction of a volume out of a number of planes witharbitrary orientations (hereafter referred to as irregularlysampled data) to end up with data in a regular grid is notan obvious processing task. Although a real-time visual-ization based on raw B-scans (without any prior recon-struction) is possible and allows online diagnosis (Prageret al. 1999) and, despite the existence of efforts ofauthors (Treece et al. 1999, 2000, 2001) to do signalprocessing directly on the raw data, a regular grid is stillneeded to apply further processing with off-the-shelfalgorithms.

Consequently, this problem of volume reconstruc-tion of US has been tackled by several authors (Barry et

al. 1997; Rohling et al. 1997; Sanches and Marques2000) in the past. All these studies focus on incrementalinterpolation of the gaps between the slices, as well as onthe determination of the final intensity of a voxel whenseveral B-scans overlap on this voxel (this latter proce-dure is known as compounding). Specifically, Rohling etal. (1997) focus on spatial compounding with image-based registration. The regular grid is filled using anearest neighbor (NN) approach and the solution theypropose for compounding is an average operation. Theirmost outstanding contribution is the registration of B-scans related to a baseline that is built from a quick passover the region-of-interest (ROI). The registration step isintended to avoid blurring of misregistered structures dueto spatial compounding. Barry et al. (1997) follow aninverse distance weighting (DW) scheme; each pixel thatfalls within a circular area of a fixed radius (centered atthe voxel) contributes to the voxel value inversely to thedistance from the voxel to the pixel. Rohling et al. (1999)provide a survey of well-known reconstruction methodslike voxel nearest neighbor (VNN), pixel nearest neigh-bor (PNN) and DW interpolation; they also introduce anew interpolation method based on radial basis function(RBF). Finally, Meairs et al. (2000) apply a weightedellipsoid Gaussian convolution kernel to tackle the re-construction of irregularly-sampled data.

All the above-mentioned studies, despite their unob-jectionable quality, share the same pitfall: no attention ispaid to the details of the construction of the regular grid(cuberille). However, we have found that this problem isof paramount importance for further data processing andstorage. It is obvious that, before applying any interpo-lation/compounding technique, the volume where thedata are going to be resampled must be defined. Thedefinition of the volume grid implies the selection of anorientation and extent for the volume and the selection ofa voxel size. With respect to the former, on one sideclinicians (ultrasonographers) often prefer that the recon-struction volume resembles as much as possible eitherthe original sweep or some anatomical predefined plane.But, on the other side, it is needless to say that one of themost outstanding difficulties of US imaging is that astandard scanning policy does not exist; clinicians lookfor the best viewing direction (also called insonationangles) at will for each patient, attempting to avoidannoying effects like shadowing. Therefore, the overallscan may consist of several subscans from which a singlescanning direction (say, principal direction in the exam-ination) may not be clearly defined.

The resulting situation can be made clearer with anexample: assume the physician has made a scan, consist-ing of two subscans, the trajectories of which draw theletter “x”. Assuming that two subscans suffice, then theintersection of the segments of the “x” will comprise the

Fig. 1. Schematic description of a freehand system. The posi-tion parameters define the transformation between the receivercoordinate system and the transmitter coordinate system.

256 Ultrasound in Medicine and Biology Volume 29, Number 2, 2003

ROI. However, no slice in any of these two subscansgives a representative scanning direction. Hence, how tocreate a regular volume in this example is not straight-forward, due to the lack of a clear scanning direction,unless an expert does provides additional information.Moreover, for the example, any reconstructed regularvolume that contains all the pixels from all the B-scanswill probably be very large and it will contain a lot ofinformation-lacking voxels (i.e., voxels with an intensityvalue imposed by the reconstruction process, and typi-caly equal to zero). In this situation, we understand that,if the direct destination of the data set is a computer, to,for instance, perform a posterior registration with, per-haps, other imaging modalities, the need for a compactregular volume is high, so that common software can beused to that end and the volume can be efficiently dealtwith by avoiding processing on a large number of irrel-evant voxels. If, on the contrary, the direct destination ofthe volume is a physician, that person will have topatiently move through a large amount of slices withonly a few relevant pixels in two areas of the slices (thosethat result from the intersection of the two tails of the “x”with the slices), until the ROI is reached. That makes theprocess of data analysis very cumbersome.

This paper aims at providing a theoretical frame-work to the problem of US volume reconstruction fromfreehand systems (i.e., at facing the problem of theconstruction of the regular grid that best fits the data).This problem comprises three subproblems, namely, 1.the selection of the coordinate system of the reconstruc-tion grid, 2. the selection of the extent of the recon-structed volume, and 3. the determination of the voxelsize.

Principal component analysis (PCA) has been usedin this study to deal with the first of the aforementionedsubproblems. Considering the 3-D positions of the pixelsas samples of a population, we look for the coordinatesystem that achieves the largest data variance in eachdirection while being uncorrelated with the others. Thesize of the reconstructed volume is pruned using theeigenvalues information provided by PCA. Furthermore,an adaptive Gaussian convolution kernel for dealing withirregularly-sampled data is introduced and comparedwith some well-known reconstruction techniques. Wewill demonstrate the application of this reconstructionmethod to in vivo data.

STATE OF THE ART

Notation and terminologyThe position of a B-scan pixel is computed as a set

of homogeneous coordinate system transformations. Fol-lowing the notation provided by Prager et al. (1998), theoverall transformation can be written as:

Cx ! CTTTTRRTPPx. (1)

The position of every pixel in every B-scan, Px, istransformed to a 3-D position into the C coordinatesystem, Cx. P is the coordinate system attached to everyB-scan; R and T are the coordinate systems of the posi-tion sensor receiver and transmitter respectively and,finally, C is the coordinate system of the reconstructedvolume. The transformation between P and R stems fromthe calibration process. The second transformation be-tween R and T is given by the position sensor’s records.The last transformation is a custom transformation thatallows us to choose the best representation of the recon-structed volume. The sub- and superscripts P, R, T and Cwill be used in each variable/transformation to denote thecoordinate system that the variable/transformation is re-ferred to.

Reconstruction: a surveyFigure 2 sketches the main steps of the reconstruc-

tion. A critical issue of every reconstruction technique isthe accuracy of the position measurements. This accu-racy would ideally depend only on the finite resolution ofthe position sensor but, in reality, electromagnetic con-tamination in the environment where the measurementsare taken may distort the transmitter’s magnetic fields(Leotta et al. 1997). The main drawback of electromag-netic position devices is the susceptibility to interferencefrom metals and electronic devices. These circumstances

Fig. 2. Reconstruction process.

Theoretical framework to 3-D ultrasound reconstruction ● R. SAN JOSE-ESTEPAR et al. 257

can be alleviated with a proper setup of the inspectionroom; however, it is almost impossible to completely getrid of them. Assuming that the probe position variesslowly between consecutive slices, smoothing the mea-sures given by the position sensor contributes to thesuppression of high-frequency artefacts and, conse-quently, the system accuracy can be enhanced.

The second step in the reconstruction process is theselection of a coordinate system for the reconstructedvolume (i.e., the definition of CTT) as well as the extentof the volume and the grid spacing (voxel size). Al-though every reconstruction process has to deal with thisissue somehow, to our knowledge, it has not been ex-plicitly treated in the literature. Barry et al. (1997) pro-pose the use of a Key US frame to define the axes of thereconstruction volume. This Key frame is chosen by auser and, typically, turns out to be one that is centrallylocated and depicts a complete cross-section of the tissueof interest. The volume axes are parallel to those of theKey US frame and the origin of the volume is at thecenter of the Key frame. Clearly this approach workscorrectly when the B-scans are quasiparallel; however,under other circumstances, for example, when the vol-ume consists of several sweeps in different directions,the selection of a representative Key frame may bedifficult to do, or such a representative frame may evennot exist; in addition, it will usually not be the bestsolution in terms of minimizing the amount of dataneeded to represent the scanned volume. As far as thegrid spacing is concerned, several authors pose the prob-lem as a trade-off between resolution and size of thereconstructed volume (Barry et al. 1997; Rohling et al.1998; Leotta and Martin 2000) and the specific value ischosen a priori.

The third step deals with the reconstruction itself(i.e., resampling the information into a regular grid). Wehave to cope with a well-known problem of scattereddata interpolation. A good survey about scattered datainterpolation was provided by Franke (1982). In ourdomain, the reconstruction cannot be merely seen as aninterpolation problem; the process can be divided intotwo stages, as suggested by Rohling et al. (1999) abin-filling stage and a hole-filling stage. The former is acompounding operation that combines the information ofplanes that intersect each voxel. The latter is an interpo-lation operation because a value has to be inferred for thevoxels that have not been filled during the former step.Although the interpolation can be carried out by severalmethods (Franke 1982), the use of global methods1 is notfeasible, due to the high dimensionality that the interpo-lation procedure would suffer. Hence, it is necessary to

pose simple solutions to minimize the time and memoryrequirements. Some well-known approaches are VNN,PNN and DW (Rohling et al. 1999).

Now, we elaborate on each step shown in Fig. 2.

METHODS

Hardware and softwareFor probe tracking, an electromagnetic sensor

(miniBird, Ascension Technologies, Burlington, VT)was used. This position sensor provides position (x, y, z)and orientation (", #, $) information between the re-ceiver and the transmitter.

US imaging was performed with a Hitachi EUB-515. Images were digitized in RGB PAL format at 25frames/s. Both the position sensor and US system videooutput were attached to a Silicon Graphics O2 worksta-tion. The US slices and position sensor measures werematched using Stradx freehand 3-D US system (Cam-bridge University 3-D Ultrasound Research Group, Cam-bridge, UK; Prager et al. 1999). The calibration of thesystem was performed using a single-wall phantom asdescribed by Prager et al. (1998). After the images andposition were recorded, the volume reconstruction wasperformed with custom software on a 800-MHz proces-sor Pentium III. The custom software was implementedas a VTK (visualization toolkit) (Schroeder et al. 1998)class. VTK libraries were also used for data visualiza-tion.

PreprocessingIn this study, we have performed a non-causal FIR

filtering of the position and orientation data obtainedfrom the positioning system (after been synchronized byStradx). This simple procedure has drawn results at anacceptable quality. The patient, however, was asked tominimize motion and to refrain from breathing for theshort period of the scanning so as to minimize motionartefacts. This paper does not focus on this issue; there-fore, a simple procedure has been adopted. More sophis-ticated procedures to assure the accuracy of the positionsensor measures (mainly based on registration ideas)(Rohling et al. 1997; Rohling et al. 1998) have beendescribed in the literature.

Volume coordinate systemThe selection of the coordinate system of the recon-

structed volume is one of the most critical steps in theoverall process of volume reconstruction. Under thisdenomination, we include the following design issues:the volume axes, the volume axes origin, the volumeextent and the size of each voxel. The elements involvedin this stage are depicted in Fig. 3.

The optimum coordinate system often depends on1 By global method we mean that the interpolation is dependent

on all data points. They are theoretically the optimum.


the application domain of the reconstructed volume;however, two general requirements can be highlighted:1. the volume grid should enclose as large an informationdensity as possible 2. the volume grid should have ashomogeneous an information density as possiblethroughout the whole volume.

The first requirement accounts for the need forhaving a small volume that depicts all the tissue infor-mation presented in the original B-scans. The secondrequirement attempts to reduce the number of interpo-lated voxels; it would be risky to try to reconstruct areasof a volume where there is a great lack of informationfrom the original B-scans, because such a reconstructionmay lead a clinician to confusion if a diagnosis was madeon a large area of interpolated data. Therefore, having aformal method that deals with these concepts would helpto choose the main features of a volume reconstructionmethod that best fits the ROI while retaining only therelevant data from the original B-scans.

Volume axes. Principal component analysis (PCA)is a statistical tool that is used to find the linear combi-nation with the largest variance of a data population.Typically, this tool has been used to reduce the numberof variables to be treated by discarding the linear com-binations with smaller variances and retaining only thosewith larger variances. In our case, the data population isthe 3-D position of every B-scan pixel. As is well-known, if the purpose is to find the best way to get acompressed volume, the Karhunen–Loeve transform, or,in its discrete equivalent formulation, the Hotelling trans-form or PCA, is the optimum solution (Jain 1989).

Formally speaking, let us suppose that the 3-Dposition of each pixel in the transmitter coordinate sys-tem, Tx, is a random vector TX of three components withcovariance matrix !!. The actual distribution of TX isirrelevant except for the covariance matrix; however, ifTX is supposed to be normally distributed, more meaningcan be given to the principal components. The key ideaof PCA is to determine the orthogonal linear transforma-

tion, "", that transforms the original 3-D position into anew space:

CXPCA ! ""TX (2)

so that the covariance matrix of CXPCA is a diagonalmatrix, the components of which are the eigenvalues ofthe data covariance matrix !!. The columns of the trans-formation "" turn out to be the eigenvectors of !!. Thestraight meaning of the aforementioned transformation isthat the components of the transformed position, CXPCA,are the ones that have maximum variance being mutuallyuncorrelated. The fact of being uncorrelated will allow usto trim the volume in one dimension with minimumimpact on the others. A complete formulation of themathematical background can be found in Anderson(1984).

The covariance matrix !! of our position data hasbeen estimated from the sample positions, Txp, for eachpixel in the transmitter coordinate system:

!! !1

N % 1 !p#1

N

$Txp % Tx! %$Txp % Tx! )" (3)

where Tx! is the sample mean vector.

Tx! !1N !

p#1

NTxp (4)

p indexes the set of pixels that contribute to the PCAanalysis.

Volume axes origin. The transformation "" onlydefines a rotation to the transmitter coordinate system.As a matter of fact, the columns of matrix "" are thenormalized axes of the reconstructed volume. In order tocompletely define the transformation CTT, a center forthe volume axes is needed. Although several approachescan be undertaken, in this study, the origin has beenchosen to ensure that the coordinates of all the pixels arenonnegative. The position of the origin Txo, is deter-mined component-wise as follows:

Txi0 & Txip, (5)

where i # 1, 2, 3 indexes the vector components and p #1 . . . N indexes every pixel.

Finally, the affine transformation CTT sought at thisstage is:

CTT ! " "" Tx00 0 0 1 # . (6)

Fig. 3. Volume construction.


Volume extent. The extent of the reconstructedvolume turns out to be an important issue that has notbeen previously addressed in the literature. Some au-thors propose a volume size that encloses all theB-scans (Rohling et al. 1997; Meairs et al. 2000);however, this approach is clearly inefficient becauseoutermost zones of the volume typically have a lowinformation density, and any interpolation attempt in-creases the computation time and generates poor re-sults. Henceforth, we will refer to this approach as thebounding box approach.

PCA provides a powerful tool to trim the volume sizeoptimally without losing important information. Two factsare essential: the position components are uncorrelated andthe pixel position variance for each dimension is known(the eigenvalues of the covariance matrix !!). A simpleapproach (hereafter referred to as the eigenvalue-drivenapproach) is to look for the volume size that fits into thebounding box and that retains the aspect ratio given by theeigenvalues in each direction. As we said before, the eig-envalues give an idea of how scattered the pixel positionsare with respect to the mean value of the population; there-fore, volume dimensions should preserve this ratio.

Assuming that the pixel position is normally dis-tributed, TX # N (0, !!), an expression for the volumesize can be worked out. For a normal distribution, sur-faces of constant probability density are ellipsoids:

x&!'1x ! C. (7)

The principal axes of this ellipsoid correspond to thePCA of TX (Anderson 1984). It can be found that thelength of the ith (i # 1, 2, 3) principal axis of theellipsoid with density C is:

li ! 2$'iC, (8)

where 'i is the ith eigenvalue of the covariance matrix !.The value of C can be calculated by integrating thedensity function within a prism that contains the ellipsoiddefined by eqn (7) for a given probability r:

r !23

$2(%3/ 2$'1'2'3%1/ 2

)0

$'1C

0

$'2C

0

$'3C e'12%x12'1

(x22

'2(x32

'3&dx1dx2dx3. (9)

Therefore, the length of the trimmed volume along co-ordinate i that confines a given information density, sayr, is:

li ! 2$2'i · erfinv $r1/3%, (10)

where erfinv is the inverse error function. Figure 4 showsthe variation of the volume length related to r and '. It isclear that, for each coordinate, the greater the density ofinformation (r) and the data dispersion ('), the larger thereconstructed volume length along this coordinate.

Voxel size. Voxel size computation aims at deter-mining the grid spacing. On one side, it is obvious thatthe greater the grid spacing, the lower the interpolationrequirements. On the other, the greater the grid spacing,the greater the loss of resolution in the reconstructedvolume. Therefore, a satisfactory trade-off between thetwo tendencies must be achieved.

The original B-scans have a resolution given by theprobe parameters; consequently, a reasonable criterionwould be to choose the greatest voxel size so that after theB-scans are resampled into the volume, the planes of theB-scans do not suffer any loss of resolution (aliasing).

For the sake of clarity in the exposition, let us reducethe dimension of our problem. Instead of a volume, think ofa plane grid and, instead of the B-scans, think of linessampled with some sampling interval. Each sample wouldbe a pixel intensity value. Figure 5 depicts the situation thatis being posed; a 2-D object is sampled by means of lines.Suppose that we have a single line parallel to the x-axis; inthis case, the maximum x-axis spacing would be the sameas the line spacing ()). With only this line, no informationwould exist to find the spacing in the y-axis. As the linerotates around the x-axis counterclockwise, the optimumspacing would be reduced by a cos * factor, where * is theangle between the line and the x-axis. If we continueddecreasing the spacing in the x-axis we would eventuallyreach a null spacing (i.e., an infinite sampling frequency).This is obviously nonsense, so we take, as a threshold forfinding the spacing in the x-axis, the value * # (/4. Similarconsiderations can be made about the y-axis. In this case,

Fig. 4. Variation of volume length (l) with information density(r) and data dispersion (').


the spacing increases with sin *, with (/4 * * * (/2. Anactual situation encompasses several lines (indexed by i #1, . . . ,m) with orientations *i. The above procedure can becontinued to each line, so the spacing, )x and )y, is finallygiven by:

')x ! mini

$)(cos*i(% if *i & (/4 @i ! 1, . . . , m)y ! min

i$)(sin*i(% if *i + (/4 @i ! 1, . . . , m.

(11)

The * # (/4 threshold has not been chosen at random.Assuming that the scanned body space variability is fullycaptured by the line resolution and that we have a highenough line density, the worst case, as far as spacing isconcerned, arises when the lines lie parallel to the griddiagonal.

The 3-D case has a similar expression from theformer discussion. It is more convenient to use a vecto-rial notation though. The ith B-scan (out of, say, Mscans) is represented by of si and qi (i.e., unitary vectorsthat define the B-scan coordinate system). The volumeaxes are given by x, y and z (see Fig. 3). )si # )s and )qi# )q are the B-scan resolutions for directions si and qi,respectively, @i # {1, . . . ,M}. The grid spacing in eachdirection can be written as:

)l ! mini $)ki( , ki,1 + (%

with ki ! ' si if ( , si,1 + ( - cos(/4qi if ( , qi,1 + ( + cos(/4)

@i ! 1, . . . , M (12)

where 1 denotes x, y or z direction and *, + denotes theinner product.

3-D reconstructionMost 3-D freehand systems use similar algorithms

to construct a regular volume after a volume grid hasbeen established. Basically, all these methods transformeach pixel position to a voxel position using eqn (1). Twosituations can arise; 1) several pixels fall onto the samevoxel; 2) several voxels are not intersected by any B-scan and are empty.

To face these two situations, the reconstruction pro-cess is divided into two subprocedures, namely, datafusion (bin filling) and hole filling.

Data fusion. A single voxel may contain severalpixels due to different reasons. One of them is the factthat the voxel size may be larger than the B-scan pixelsize. Another possibility is that each voxel may be inter-sected by several B-scans. Both situations generate re-dundant information that we have to manage somehow.As is stated by Rohling et al. (1997), every freehandsystem has to deal with compounding in some manner,because it is almost unavoidable that the scan planesintersect. A great activity has been carried out regardingspatial US compounding (Burckhardt 1978; Wagner etal. 1988) and, more specifically, in 3-D freehand ultra-sound (Detmer et al. 1994; Rohling et al. 1998; Leottaand Martin 2000).

Averaging has been traditionally the compoundingoperation, because the speckle signal may be partiallyfiltered out; theoretically, the improvement in the signalto noise ratio (SNR) may reach proportionality with thenumber of samples that are averaged. However, when thenumber of overlapping samples is small, the maximumsample-compounding technique seems preferable(Burckhardt 1978). Specifically, compounding pursuesthree objectives: speckle reduction, tissue boundary en-hancement and shadowing reduction. It has been proventhat averaging is an optimal way for speckle reduction;nonetheless, its optimality is not so clear as far as bound-ary enhancement is concerned. A tissue boundary exhib-its different brightnesses depending on the insonationangle, so the averaging operation may dramatically blurthe results. Burckhardt (1978) has shown that the maxi-mum sample provides a similar improvement in SNR fora small number of pixels falling into the same voxels,and allows a clear boundary enhancement and shadow-ing reduction.

Hole filling. The hole-filling step aims at finding avalue for those voxels that have not been filled in thedata-fusion step. This present stage is an interpolationstep and many of the attempts have focused on low-orderinterpolation. Given the great amount of data available,

Fig. 5. Voxel size calculation. A volume is represented by aplane grid. A body is scanned by means of slices represented bylines and intensity pixels by dots. The spacing between dots is

).


NN approaches seem to efficiently tackle the problem.The NN approach involves sequencing through the reg-ular volume (i.e., through the original B-scans) and as-signing to the empty voxel the value of the closest pixel.

In this paper, however, we have used an approachbased on a basis function interpolation (Franke 1982)similar to the one proposed by Meairs et al. (2000). AGaussian convolution kernel is placed on each voxelafter the data-fusion step and the voxel value is com-puted as a weighted sum of all the voxels filled in theformer step. Meairs et al. (2000) propose an ellipsoidalkernel, the main axis of which is oriented along the framenormal of the current image plane to avoid a smoothingof the image data in the horizontal direction; however,nothing is said about the selection of the variance of theGaussian kernel. Moreover, it is not clear what the ori-entation of the image plane is for those voxels which turnout to be unfilled in the data fusion step.

Our approach is based on a spherical Gaussiankernel, the variance of which is a function of the varianceof the intensity of the nearby pixels. This is an adaptationto the US case of what has been called sigma filterelsewhere (Lohmann, 1998). The variance of the inten-sity grey–scale images carries information about the ex-tent of speckle formation (Dutt and Greenleaf, 1996). Aspeckle formation-driven interpolation allows us to de-tect likely structures out of the completely resolvedspeckle zones, reducing the smoothing effect of theGaussian interpolation kernel by means of sharpeningthe kernel shape. Dutt and Greenleaf (1996) came upwith a normalized variance-dependent parameter, f, thattracks the statistic of the speckle image:

f !(2

24D2

.I2, (13)

where D is a compression constant and .I2 is the image

variance. The statistical analysis carried out by the au-thors states that f parameterize scatterer densities; lowdensities result in fully resolved scatterers (f # 0) andlarge densities result in fully formed speckle (f # 1). Theimage variance, .I

2, can be estimated using the voxelsfilled during the bin-filling stage. The compression con-stant D stems from the logarithmic compression that theultrasonic signal envelope must undergo to fit into thedisplay’s dynamic range. The authors propose a deter-ministic approach to resolve this parameter, althoughnew blind inverse methods for gamma correction can bealso applied (Farid 2001).

In this work, we have used f as a mapping parameterbetween the image variance and the kernel variance.Assuming that an interval of acceptable variances (.min2 ,

.max2 ) can be defined, the adaptive variance of the Gauss-ian kernel, .K

2 , is worked out as:

.K2 ! .min

2 / $.max2 % .min

2 % f, (14)

The image variance, .I2, is estimated using the vox-

els filled during the bin-filling step. This could be inter-preted as an unsharp masking filtering over the kernelvariance driven by the underlying speckle pattern. Figure6 shows the variation of the kernel variance as the imagevariance increases. For computational purposes, insteadof building the exact kernel for each variance, a loga-rithmic quantification of the former plot is devised.Hence, a bank of precomputed kernels is used to speedup the interpolation process. As a rule of thumb, wechoose the size of the Gaussian kernel, k, so that itencloses 98.76% of the area within the Gaussian curve(i.e., Trucco and Verri 1998).

k - 5.K. (15)

The kernel function is then sampled at unitary distancesand the values are normalized to maintain the DC com-ponent after the filtering.

RESULTS

A set of four US sweeps has been used to test thereconstruction process. US volumes from a thyroid and aliver were available from the Cambridge University 3-D

Fig. 6. Variation of Gaussian kernel variance vs. image vari-ance. To preserve the structures in the reconstructed volume,the kernel variance decreases as the image variance increases.The intensity grey-scale value has been normalized between [0,1] and a compression constant D # 0.22 was assumed. Thedesired kernel variances were set to lie within the interval .min

2

# 0.796 and .max2 # 10.


Ultrasound Research group. A kidney and a fetus2 havebeen scanned under our supervision. All the scannedvolumes are single-sweeps, except for the fetus case,which is formed by two overlapping sweeps.

Evaluation of the volume grid construction procedureEvaluating the PCA process capability to define a

coordinate system is a difficult task, due to the lack of aground truth orientation; it is our opinion that there couldbe as many optimum orientations as ultrasonographerswere asked to provide one, so the orientation given byother methods, like the Key-frame approach, could beequally valid. However, more objective differences be-tween methods may arise when a trimming of the outputvolume is carried out to process only the relevant areas ofthe scanning. In this case, as we will show, the effec-tiveness of the trimming process is highly dependent onthe chosen coordinate system.

Evaluation methodology. We will analyze five dif-ferent coordinate systems and two different trimmingprocedures. About coordinate systems, these will be thePCA proposed in this paper, three choices of the Key-frame approach proposed by Barry et al. (1997) and thetransmitter coordinate system given by the positioningsystem (hereafter denoted by T). The three choices ofKey frames will be the B-scan in the middle of therecorded sequence, the B-scan located right at the end ofthe first quarter of the sequence starting from the begin-ning, and the B-scan located at the end of the thirdquarter from the beginning. These methods will be re-ferred to as KF#2Q, KF#1Q and KF#3Q, respectively. Inthe examples, the numbers after the # sign will be theactual slice number of the Key frame within each dataset.About trimming procedures, Key frame and T will betrimmed with a deterministic method described later inthis subsection, and PCA will be trimmed both with theeigenvalue-driven procedure and with the deterministicmethod. PCA methods will be referred to, respectively,as PCA ' eig and PCA ' det.

For each data set, a careful delineation of the ROIhas been carried out, assisted by the supervision of anexpert using the 3-D slicer (Gering et al. 1999). Theresulting ROIs will be taken as our ground truth (i.e., asthe target to be preserved in the reconstructed volume).The best procedure (both in terms of coordinate systemand of trimming procedure) will be that that most com-pactly comprises the whole ROI (i.e., the method withwhich, as soon as the whole ROI is captured within thereconstructed grid, the physical volume3 of such grid is

the least). For illustration purposes, Fig. 10 shows a 3-Dmodel of the ROI with the original B-scans superim-posed for each of the four datasets that will be used in thepaper.

To monitor the trimming procedure, we will obtainROC-like curves (Metz 2000), in which the magnitude inthe horizontal axis will be the physical volume of thereconstructed grid (Vgrid), and the magnitude in the ver-tical axis will be the portion of the ROI captured by thereconstructed grid. Such a magnitude will be hereafterreferred to as captured ROI and denoted by VROIc . Bothaxes will be normalized by the physical volume of theROI (VROI), for the curves of the four datasets to becomparable irrespective of the actual size of the ROI.Such curves will be referred to as ROC curves, forsimplicity. The physical volume Vgrid for which VROIc #VROI will be hereafter referred to as optimal grid physicalvolume and denoted by VOGPV.

To apply a trimming algorithm to those coordinatesystems that are not based on an eigenvalue computation,a deterministic approach has been applied. This deter-ministic trimming approach has been introduced to com-pare the PCA with the Key frame. The chosen determin-istic method follows an ad hoc scheme, where the lengthof each axis is reduced based on the deviation of thecenter of the volume bounding box from the center ofmass of the data. Departing from the bounding boxlimits, each dimension of the bounding box is trimmedaccording to a term, dj, which controls the extent of thetrimming process in this direction as a function of thedeviation of the center of mass with respect to the bound-ing box limits for the axis. Everything is weighted by aterm p that is used to control the whole process. In ourparticular implementation, the length of the trimmedvolume along the jth coordinate (j # x,y,z) is given by:

lj ! ,$BBj % mcj%$dj / 0.5%

% $bbj % mcj%$1 % dj%- ! p, (16)

where BBj and bbj are the respective jth coordinates ofthe lower and upper bounding box extremes, and mcj isthe jth coordinate of the center of mass. p is a coefficientthat is used to control the trimming strength, and dj is aparameter that accounts for the deviation from the centerof mass following an arctangent shape. Specifically:

dj !( / 2 arctan "5%mcj % bbj

BBj % bbj%12&#

4(. (17)

dj is bounded over the interval (0, 0.5) as mcj tends to bbjor BBj.

2 The corresponding author’s first son!3 We will used interchangeably the terms reconstructed grid and

reconstructed volume. When we need to refer to the size (in cubic units)of the reconstructed volume, we will use the term physical volume.


Coordinate system performance. Figure 7 shows theROC curves that result from the different methods for thefour datasets. These plots depict how fast the recon-structed volume comprises the ROI as a function of thegrowing physical volume of the reconstructed grid, as weincrease either r or p, as convenient for the correspond-ing trimming method. Needless to say that the bestprocedure will grow faster in the vertical axis than theothers, as they all move rightwards along the horizontalaxis.

It can be seen that the PCA with eigenvaluesmethod clearly outperforms the others in three of the fourdatasets (Figs. 7a, c, and d). It is only for the liver (Fig.7b) that the proposed method it is not the best, eventhough differences are almost negligible. This can beexplained very easily by noticing that, for this dataset,the ROI comprises almost the whole sequence (see Fig.10b). If this is so, PCA does not add any particularadvantage. On the other hand, if we focus on the fetuscase (Fig. 7d), where two overlapping sequences (withdifferent scanning directions) have been acquired tocover the whole fetus (see Fig. 10d), our methodachieves the largest difference with respect to the othermethods. The fetus case reveals the Key frame difficultyto choose the proper B-scan that clearly compiles thebest trajectory. Moreover, the advantage of the eigen-

value with respect to the deterministic trimming methodsis revealed, for this case, in terms of least physicalvolume enclosing the ROI, given the large amount ofdata that can be considered outlier (i.e., outside the ROI).Similar conclusions can be drawn for the thyroid case(Fig. 7a), where the ROI is defined only within a subsetof the dataset (see Fig. 10a), making the eigenvaluetrimming method the best choice.

A deterministic trimming turns out to be a fairlypoor solution, given that, though we are able to reducethe volume, there is no knowledge about how to performthis reduction depending on the axes and pixel distribu-tion in the 3-D space (i.e., how to trim the volume,preserving the ROI as much as possible for a giventrimming ratio).

Another issue that we have addressed in our study isthe relation between the least physical volume of thereconstructed grid Vgrid that comprises the bounding box(say, Vgrid # VBB) and VOGPV. Fig. 8 shows this relationfor each case and each coordinate system considered.The bargraphs reflect how worthy it is to use a trimmingmethod in terms of the amount of data needed to enclosethe ROI. Also, it is worth noting that PCA is not themethod that achieves the least physical volume when thewhole bounding box is required. That makes sense if wetake into account that PCA tries to choose the bestorientation to be as insensitive as possible to portions ofB-scans not within the ROI; consequently, any attempt toenclose those B-scans will result in a greater volume thanwith other orientations that implicitly use the knowledgeof the scanning policy (if it exists) as, for instance, theKey-frame approach. As we stated in the introduction, as

Fig. 7. Study of the evolution of the VROIc with Vgrid, bothnormalized by VROI (see main text for definitions). Four caseshave been analyzed: (a) thyroid gland, (b) liver, (c) kidney, and(d) fetus. For each case, six methods have been studied,namely, PCA with eigenvalue trimming method, PCA withdeterministic trimming method, Key frame with deterministictrimming method (the chosen Key-frame sequence number isused to label each case), and T coordinate system with deter-ministic trimming method. The vertical lines show the point atwhich each chosen coordinate system starts to enclose the ROI.

Fig. 8. Comparison between the VBB and VOGPV (see main textfor definitions) for (a) thyroid, (b) liver, (c) kidney, and (d)

fetus.


far as a human observer is concerned, PCA is not thepreferable option if the intention is to preserve all thedata and the observer is willing to scroll through irrele-vant B-slices. Our PCA approach becomes relevant whenthe trimming is carried out; in that case, as we haveshown, our method outperforms the Key-frame approachin terms of physical size of the volume.

The evolution of the eigenvalue-driven physicalvolume with r is shown in Fig. 9. Figure 9a shows theevolution of the ratio VROIc /VROI for each case. From thisplot, it can be seen that an interval ranging from r # 0.7to r # 0.9 covers the ROI for the cases under studywithout adding information-lacking regions. For illustra-tive purposes, Fig. 10 shows the resulting reconstructedvolume for the eigenvalue method. The original freehandB-scans are shown, as well as the boxes corresponding tothe bounding box, r # 0.9, r # 0.8 and r # 0.7 (seefigure caption for details). It is also worth analyzing theevolution of the Vgrid normalized by the VBB. This mea-sure sheds light upon the sensitivity of the trimmingapproach with respect to the density of information r thatwe want to retain. Figure 9b plots this measure. Thethyroid case turns out to be the most sensitive because itsnormalized volume quickly increases when r gets closerto 1. On the other hand, the fetus is the least sensitivecase. The conclusion that can be drawn is that, the greaterthe deviation of the scanning trajectory with respect to astraight line (either by several sweeps or misalignmentsof the original B-scans), the greater the bounding boxand, consequently, the proposed trimming approach be-haves more effectively.

Therefore, the PCA coordinate system and the eig-envalue driven approach constitute the basis of a formalframework for building and trimming at controlled dis-tortion levels a regular volume out of irregularly sampleddata. Results clearly show that the volume sizes so ob-

tained are considerably smaller than those obtained byreported methods when the scanning trajectory is notcorrectly represented by a single slice within the dataset.

Voxel size. The selection of the voxel size is basedon a criterion that retains the B-scans resolution into thevolume grid. Table 1 shows the voxel sizes for differentB-scan sequences and for different coordinate systems.The original B-scan resolution is also shown. For all thecases, it can be shown that the output resolution isbounded by the input resolution and a 1/2 factor of theinput resolution. The upper bound is almost achieved inuniform scans like the thyroid case and the lower boundis more likely in several sweeps like the fetus case.Although the lower bound could be proposed as thegeneral solution that guarantees our criterion, the pro-posed method tries to relax this lower bound by choosingthe optimum for each particular case.

Evaluation of the resampling procedureDifferent approaches can be used to evaluate the

quality of the reconstruction process (bin- filling andhole-filling). For instance Meairs et al. (2000) state thatone of the best ways to judge the quality of 3-D USvolume acquisition and reconstruction is through volumevisualization using rendering techniques. However, we

Fig. 9. (a) VROIc normalized by VROI vs. r for each case. (b) Vgridnormalized by VBB vs. r. The horizontal dashed line representsthe level where the trimming algorithm reaches VBB. Note that,the closer the scanning trajectory to a straight line, the faster(with respect to r) VBB is reached and, from that value r on, thetheoretical trimming box is larger than the bounding box. It isobvious that the tails of the assumed Gaussian distribution does

not hold any real voxel information.

Fig. 10. Eigenvalue-driven trimming performance. The originalB-scan planes, bounding box (outer blue) and boxes for r #0.9, r # 0.8 and r # 0.7 (black, green and inner blue, respec-tively) are shown for the volumes (a) thyroid gland, (b) liver,(c) kidney, and (d) fetus. The difference between the bounding

box volume and the trimmed volume are fairly clear.


believe that this approach remains subjective and that thevisual cues associated to each particular rendering tech-nique can potentially disguise pitfalls of the reconstruc-tion method. One can even change the spatial position ofthe triangles in a triangle mesh according to differentcriteria to get a better visualization (San Jose et al. 2001),even though the original volume remains unaffected.

Another possibility could be to build a phantom inwhich the material had known acoustic properties so thatvalues of the B-scan could be anticipated by analyticalcalculations and compared with the actual B-scans. Thispossibility, however, requires an effort that can be cir-cumvented by other methods.

In this study, we have followed the testing approachintroduced by Rohling et al. (1999). The main idea is toevaluate the ability of the reconstruction method to pre-dict the intensity values at the locations where the orig-inal data have been removed. We will use the thyroidgland case as a figure of merit. This examination presentsa regular enough sweep to allow an easy interpretationwithout any lack of generality. A B-scan near the middleof the sweep is selected. A Key-frame approach is usedto align the volume coordinate system using the chosenB-scan. The voxel size is set so that x and y axes have aresolution equal to the B-scans. Therefore, the selectedB-scan falls perfectly into voxels in the volume grid. Topreserve this property, a voxel size in z should be prop-erly chosen. The distance between the centers of theadjacent B-scan from the key B-scan is used as the voxelsize in the z direction.

A percentage of pixels is randomly removed fromthe B-scan, creating holes of various sizes that should be

interpolated. The rest of the pixels are used in the inter-polation process to fill in all the voxels in the voxel array.The interpolated value is compared with the removedpixel value to assess the reconstruction accuracy. Theaverage of the absolute difference between the interpo-lated and the original values over all missing values isused as a goodness measure

V !1L !

i#1

L

(pi % ci(, (18)

where pi is the original pixel intensity value and ci is thevoxel interpolated intensity value aligned with pi. L is thenumber of discarded pixels that need an interpolationvalue. V also includes the quantization error, because theoutput values are stored with integer precision.

The tests have been carried out with six differentpercentages of removed data: 25%, 50%, 75%, 100%,300% and 500%. The tests corresponding to 25%, 50%,75% and 100% remove data from the chosen B-scan.Percentages over 100% mean that additional pixels fromthe two adjacent B-scans are removed. Specifically, forthe 300% test, all the pixels from the chosen B-scan andthe two closest adjacent scans to it have been removed.Evaluating the 300% and 500% accurately requires anexamination with slices almost parallel and with onesweep so that the missing voxels are mostly aligned withthe ground truth removed and a reliable comparison canbe carried out. This is the reason for the choice of thethyroid case to perform the test. A kernel size of 3 . 3. 3 voxels has been used for the smallest percentages.

Table 1. Volume voxel sizes using the proposed method for the four datasets used in the paper

B-scan pixel size (mm/pixel)(s q)T

Key-frame PCA

Vol. voxel size(mm/pixel) Vol. dimensions

Vol. voxel size(mm/pixel) Vol. dimensions

(x y z)T (x y z)T)

Thyroid 0.08713 0.08703 368 0.08713 6720.08913 0.08713 522 0.07923 519

0.08713 595 0.08701 373

Liver 0.27986 0.27804 549 0.27658 5780.28526 0.27353 502 0.27499 527

0.27986 391 0.27986 376

Kidney 0.26138 0.25967 635 0.25774 6740.28978 0.26138 556 0.26138 586

0.26138 319 0.26138 297

Fetus 0.32335 0.22937 1127 0.25066 12160.35377 0.25140 1043 0.22899 1433

0.25140 1223 0.22946 1159

The output voxel size and the volume dimensions are shown both for the PCA and the Key-frame methods. The Key-frame corresponds to themiddle B-scan of the recorded sequence. The volume dimensions correspond to VBB for the given coordinate system.


For the 300% and 500% tests, kernel sizes were 5. 5 .5 and 7. 7 . 7 voxels, respectively. A 7. 7 . 7 maskhas been used to estimate the image variance that drivesthe adaptive Gaussian interpolation procedure.

Figure 11 plots the results for some well-knownmethods and the one proposed in this study. The resultsshow that a Gaussian approach performs much betterthan well-known methods, as long as the percentage ofdata removed is below a certain threshold. Focusing onresults below 300%, the method introduced in this paperclearly outperforms former proposals, closely followedby a traditional Gaussian interpolation. For percentagesover 300%, the great lack of information makes thosetechniques that do not underestimate farther pixels, likePNN, behave better in terms of the error metric consid-ered.

For illustrative purposes, Fig. 12 shows the set ofinterpolated images for the 100% case, (i.e., when thewhole slice has been interpolated using out-of-planeinformation for the methods discussed in the paper)together with the original image. The figure shows thatboth DW and PNN tend to oversmooth the data. Aboutthe other three methods, no important differences canbe found; only minor comments can be made. Forinstance, VNN seems slightly to overemphasize thetexture patterns. This is the case within the upperrectangle drawn in the original image. With respect tothe Gaussian interpolators, as we said, their behavioris very similar; however, the nonadaptive versiontends slightly to smooth the texture patterns. This isthe case for the area within the rectangle located in themiddle of the original image. These comments, how-

ever, do not lead to a clear preference for one partic-ular method.

The second example highlights the main differ-ences of the methods; Fig. 13 shows a slice of thereconstructed volume for the case of the kidney data-set. The PCA-based proposed method has been used towork out the reconstruction volume grid and the trim-ming algorithm has been set up with r # 0.70. TheVNN method computes the nearest pixel in a spheri-cally-shaped local neighborhood with radius R # 5voxels. For the PNN and DW methods, the kernel sizehas been set up to 7 . 7 . 7. For the adaptiveGaussian method, we have chosen .min # 0.8 and .max# 1.4 as the desired marginal deviations. The kernel

Fig. 11. Grey-level error for the thyroid case. Four well-knowninterpolation methods (VNN, PNN, DW, and Gaussian convo-lution kernel) are compared with the proposed method (adap-

tive Gaussian convolution kernel).Fig. 12. Thyroid B-scans interpolated by several methods. Allthe images show the result of the interpolation for a 100%percentage of data removed. From left to right, row 1: theoriginal slice (before data removal) and it is the ground truthVNN, PNN; row 2: DW, Gaussian and adaptive Gaussian.

Fig. 13. Kidney B-scans interpolated by several methods. Fromleft to right, row 1: PNN, VNN, DW; row 2: Gaussian, adaptive

Gaussian.


variance will be bounded between these two values,(see eqn (14)). Then, the adaptive Gaussian kernel sizeis computed using eqn (15). As before, PNN and DWtend to oversmooth data. Both Gaussian kernels showa smoother behavior than VNN, especially the non-adaptive version, which creates a blurry image ascompared with the other two. VNN shows a moretextured pattern, as in the previous example. However,the price to pay is the presence of misalignments in theresampled image, due to the overlap of the originalslices. This is particularly clear in the lower contour ofthe kidney, where, for an important part of it (enclosedwithin the rectangle shown), a textured diagonal pat-tern has been superimposed, distorting its presenceand making it difficult to be traced. The adaptiveGaussian method turns out to be the one that shows thebest trade-off between hole-filling capabilities andblurring. Misalignment artefacts are not visible. Con-sequently, although these misalignments are commonto all the methods, kernel-based methods can moreefficiently deal with them.

CONCLUSIONS

In this paper, a theoretical framework for thereconstruction of a regular US volume out of irregu-larly-sampled volume data has been proposed. Wehave focused on two main problems, namely, how tocreate a regular volume grid and how to find the valueof each of the voxels in the grid. With respect to theformer, the main contribution of the paper is theproposal of a well-known statistical tool to find theoptimum coordinate system; in our experiments, wehave shown that the method proposed in the paperencloses the ROI with a smaller overall physical vol-ume than other methods previously proposed in thosecases that the scanning trajectory differs from a purestraight line; as a matter of fact, we have shown thatthe greater the reduction of the volume size of ourmethod, the greater the deviation of the scanningtrajectory with respect to a straight line. In addition,we have proposed a method to find the size of eachdimension of the voxel so as to reduce aliasing effectwhile maintaining low storage requirements. With re-spect to the second problem, we have proposed anadaptive Gaussian kernel to carry out the interpolationprocedures, and numerical results show that this pro-cedure achieves the minimum distortion for the data-sets used in the paper. In addition, graphical resultsshow that this method achieves a moderate blur whileavoiding misalignments artefacts that stem from thesuperposition of slices. Although more extensive dataanalyses should be done to infer possible correlationsbetween the scanning trajectories and the appropriate

value of the parameter r to perform the trimming, it isour belief that this paper shows a formal methodologyto reconstruct volumes from freehand US scanningwhile maintaining moderately low processing andstorage needs. The graphic tools here developed (anexample of which is shown in Fig. 10) allow practi-tioners to fine-tune the final volume size at will veryeasily.

Acknowledgment—The authors acknowledge the Comision Interminis-terial de Ciencia y Tecnologıa for research grants (IFD97-0881-C02and TIC01-3881-C02) and Junta de Castilla y Leon for research grants(VA78/99 and VA91/01). The authors appreciate the support given byJulio Diaz, M.D., Hospital Rıo Carrion, Palencia, Spain in performingthe scanning, by James Ellsmere, M.D., Brighman and Womens Hos-pital, Boston, USA, in delineating the ROI, and by Miguel Martın-Fernandez, University of Valladolid, Spain, in Vtk and Tcl/Tk issues.The first author acknowledges the useful discussions with Dr. Carl-Fredrik Westin, Brighman and Womens Hospital, Boston, USA, aboutcontents and presentation of the paper. The authors also thank theanonymous reviewers for their help in improving the quality of thepaper.

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