Frequency Domain Registration of Computer Tomography Data
Marco Andreetto, Guido M. Cortelazzo
Department of Information Engineering University of Padua, Padua, Italy
Phone: + 39-049-827-7642, Fax: +39-049-827-7826 E-mail:{marco.adreetto,corte}@dei.unipd.it
Luca Lucchese
School of Electrical Engineering and Computer Science Oregon State University, Corvallis, OR 97331
Phone: 541-737-2980, Fax: 541-737-1300 E-mail:[email protected]
Abstract
This paper presents a new method for registering computer tomography (CT) volumetric data of human bone structures relative to observations made at different times. The sys-tem we advance was tested with different kinds of CT data sets. In this paper we report on some representative experi-mental results obtained with the CT data of the hip bones of a patient prior to and after prosthetic surgery aimed at the reconstruction of the hip articulation. The method works with rigid data having arbitrary relative position and orien-tation and proves to be robust with respect to CT acquisi-tion noise and with respect to the segmentaacquisi-tion technique adopted to select the region of interest for registration. The method is capable of registering correctly data sets taken from the same articulation and whose components have un-dergone small relative displacements. The method is also amenable to registration of heteregeneous kinds of volumet-ric data,e.g., CT scans and Magnetic Resonance Imagery (MRI) scans, which show different characteristics in corre-spondence to the same organic structure.
Keywords– 3-D Registration, Motion Compensation, 3-D Fourier Transform, Computer Tomography (CT), Iterative Closest Point (ICP) Algorithm.
1
Introduction
Diagnostic tests based on volumetric data such as CT and MRI have become commonplace in hospitals and health care centers owing to their wealth of information,
supplant-ing, for instance, X-ray based tests where only 2-D informa-tion is available. The examinainforma-tion of a single series of CT slices provides the surgeon or doctor with important infor-mation on a given internal structure of the human body. An-other important application consists in comparing CT mea-surements of a patient made at different times in order to monitor the temporal evolution of a physiological process or to check the success of a surgery operation. Sometimes, it may be necessary to integrate measurements obtained with different acquisition modalities,e.g., CT, MRI, or PET, in order to gain a better diagnostic picture [1]. In all these sce-narios, two or more data sets describing the same anatomi-cal part have to be spatially registered; in the case of mea-surements of a given object taken at different times, this operation is referred to as motion compensation.
In the field of medical imaging, various methods have been advanced for registering volumetric data which are generally extensions of 2-D techniques. Duncan and Ay-ache [2] present a general survey on the algorithms pro-posed for medical imaging applications in the last few years whereas Maintzet al.[3] perform a thorough analysis of the registration methods for a variety of medical data including 2-D imagery, volumetric and surface data.
Among the solutions representing extensions of 2-D techniques, Fei et al. [4] advance a multi-resolution al-gorithm for registering 2-D images obtained through inter-ventional magnetic resonance imaging (iMRI) with respect to a volume acquired through MRI; their method estimates the correct registration by comparing two similarity figures usually adopted for images, mutual information and correla-tion, which are chosen based on the image resolution. Also in [5], the authors resort to the same figures applied to dif-ferent types of images in order to obtain the most robust
Segmentation Segmentation Isosurface Isosurface Extraction Extraction Frequency Registration ICP CT2 CT1 Point Cloud 1 Point Cloud 2 Surface 1 Surface 2 RT RT i f
Figure 1. Registration Scheme.
registration. Hemmendorfet al. [6] suggest a motion com-pensation algorithm for data having arbitrary dimensional-ity (2-D, 3-D, and N-D); their method decomposes the input data in terms of a subband representation and finds the mo-tion relating the two data sets through the phase informamo-tion from the outputs of a filter bank.
Other registration methods exploit the identification of convenient anatomical features in the data to register;e.g., Porter et al. [7] segment the data in order to determine the main blood vessels and align the two vessel networks via a generalized correlation coefficient obtained from the Schwartz inequality. This method, however, requires an ini-tial registration which is provided by human supervision.
Betkeet. al [8] resort to different kinds of anatomical elements, such as vertebrae, trachea, and sternum, to de-termine a first coarse estimate of the registration of thorax CT scans taken at different times; this estimate is then re-fined by means of a variation of the ICP algorithm operating on the segmented surfaces of the lungs [9, 10, 11]. To de-termine the position of the anatomical elements, template matching techniques are utilized; therefore, this method cannot be used in applications where templates cannot be exploited. The two-level architecture of [8] is very similar to the method we propose with one important difference: our method does not rely on the determination of specific el-ements, which makes it more general. The registration tech-nique advanced in this paper exploits the algorithmic frame-work proposed in [13]. The original contribution of this pa-per consists in adding additional stages and pre-processing
steps to make the algorithm of [13], which is tailored to sur-faces, work with volumetric data as well and in assessing its overall performance.
This paper has five sections. Section 2 describes the for-mat of the CT data employed in our experiments and their preliminary processing before registration. The require-ments for the correct operation of our method are discussed as well. Section 3 details the registration algorithm which consists of two steps: in the first, a preliminary alignment is sought after by exploiting the relationships in the frequency domain between the two data sets; in the second, the coarse estimate is refined through the ICP algorithm. Section 4 re-ports on some experimental results relative to both synthetic and actual medical data. Section 5 draws the conclusions and discusses possible extensions of our technique.
2
Preliminary Data Processing
Our algorithm performs the registration of two 3-D data sets which may have been recorded at different times. The alignment is carried out by using all the information en-coded in the CT scans without any external fiducial markers. The only assumption made is that the anatomical structures to register are related by a global 3-D rigid transformation. This constraint can also be relaxed as will be shown in Sec-tion 3. In all our experiments, only CT scans relative to the same patient (intrasubject registration) have been regis-tered (monomodal registration). However, the system can
Figure 2. An example of CT scan.
easily be modified to register different kinds of data,e.g., CT and MRI data (multimodal registration). Furthermore, even though the application scope of the experiments pre-sented in this contribution is hip articulation, our method has also been successfully used to register other anatomical structures where different data sets are reasonably related by rigid transformations.
In order to be aligned with our frequency domain algo-rithm, the CT data sets have to be segmented first. Fig-ure 1 shows the block diagram of our registration system, including the preliminary segmentation stage and the sur-face computation stage for the final application of the ICP algorithm. The data we have used in our experiments are the time series of two CT scans relative to two patients who have undergone prosthetic surgery in the hip-femur socket. For both patients, CT scans have been acquired before and after surgery in order to assess the correct position of the prosthesis.
Every CT scan consists of a series of rectangular slices spatially aligned along theXandY coordinates and taken at different values of the Z coordinate with an adaptive strategy: the spacing alongZincreases where the volume has more articulation. In turn, every slice consists of a reg-ular sampling lattice with even spacing along theXandY
coordinates (the sampling interval is the same for both di-mensions) and the scalar value at each pixel represents the response to radiation of the tissue under investigation (this value will be referred to as the CT intensity). An exam-ple of CT slice is shown in Figure 2. For an in-depth de-scription of this data structure, the reader is referred to the RECTILINEAR GRIDdata structure described in [12].
It is important to observe that CT data sets relative to the
−5000 0 500 1000 1500 2000 2500 3000 1 2 3 4 5 6 7 8x 10 5
Figure 3. Intensity histogram of a CT scan.
same patient have different geometric characteristics (posi-tion, lattice spacing, and dimensions) whereas the intensity histograms are very similar, since they stem from the same acquisition system and protocol. This property is important for the correct operation of our registration system. In fact, if the intensity distributions of the CT data sets are different, they have to be appropriately equalized before processing. As an alternative, only geometric information can be used for registration provided that it is possible to segment the same anatomical structure in different data sets.
From all the information associated with a CT scan, only spatial location within the lattice and intensity are consid-ered, whereas topological information on the lattice is dis-regarded. The points obtained in this way are segmented through thresholding which filters out all the values outside a preselected interval. The goal of this filtering is to keep only the data relative to bone structures that are character-ized by high intensity and that can be assumed to be rather constant during acquisition. The threshold values can be chosen interactively by the user or can be computed with-out supervision by analyzing the CT intensity histograms. Figure 3 displays a typical histogram which shows three modes: one, at low intensities and very close to zero, can be ascribed to acquisition noise. The other two peaks are far away from the first and rather close to each other; they correspond to low-density tissues such as muscles and to high-density tissues such as calcified bones.
Once the CT data sets have been segmented, the baricen-ter of each one is computed and a 3-D orthogonal Cartesian reference system is attached to it. It will be shown in Sec-tion 3 that this provision is beneficial to improve the overall performance of the system.
(a) (b)
Figure 4. (a) A point set; (b) Same data resampled on a regular domain.
3
Registration Algorithm
The registration algorithm is comprised of two steps (see diagram in Figure 1). The first step computes a coarse esti-mate of the rigid transformation relating the pre-processed data sets. The second step performs the fine alignment of the iso-surfaces describing the bone structures.
3.1 Coarse Estimation of the 3-D Rigid
Transfor-mation
The 3-D rigid motion relating the two data sets is com-puted with a variation of the frequency domain algorithm advanced by Luccheseet al.in [13] which operates on sur-faces rather than on volumetric data. The data sets are re-sampled on a regular 3-D lattice in order to apply 3-D Fast Fourier Transform (3-D FFF) algorithms. To this end, the bounding box enclosing both data sets is computed. This is then divided into128×128×128voxels, each of which is assigned an intensity value through a nearest neighbour strategy: each voxel is given the intensity value of the clos-est point to its center. The result of this operation is shown in Figure 4.
Letd1(x) and d2(x), x ∈ R3, be two data sets after this pre-processing stage. We assume that the two sets are
related by a 3-D rigid transformation,i.e.,
d2(x) =d1(R−1x−t), (1) where R ∈ S O(3) is a 3-D rotation matrix1 and t ∈ R3 is a 3-D displacement. Matrix R is expressed as a single rotation ψ about an axis having the direc-tion defined by the unit vector ω = [ωx ωyωz]T =. [cosϕsinϑ sinϕsinϑ cosϑ] ∈ R3, ω = 1, i.e.,
R=eΩψ, where Ω=Ω(ω) = ω0z −0ωz −ωωyx −ωy ωx 0 (2)
is a skew-symmetric matrix associated withω. This rep-resentation is not unique since choosing ω = −ω and
ψ= 2π−ψgives the same rotation asωandψ[19]. The 3-D rotation matrix is thus described by the three anglesψ,
ϑ, andϕ.
By applying the 3-D Fourier transform to Eq. (1), it is
D2(k) =D1(R−1k)e−j2πk TRt
, (3)
whereD1(k)andD2(k),k ∈ R3, are the Fourier trans-forms ofd1(x) andd2(x), respectively. By keeping only
(a) (b)
(c) (d)
Figure 5. (a)-(b) Centered point sets. (c)-(d) Results of the frequency domain registration algorithm of [13].
(a)
(b)
Figure 6. (a) Regular CT slices. (b) Perturba-tion of thezcoordinate of the slices.
the magnitude information of the spectra, Eq. (3) becomes |D2(k)|=|D1(R−1k)|. (4) From Eq. (4), the algorithm of [13] builds an auxiliary func-tion ∆(k)=. |D1(k)| D1(0) − |D2(k)| D2(0) =|D1(k)| D1(0) − |D1(R−1k)| D1(0) , (5) which allows the estimation ofΩin Eq. (2) andψin two steps. From these values the rotation matrix can easily be estimated and, in a third step, the translational displacement
tis eventually computed through phase correlation. For the algorithmic details, the reader is referred to [13].
The algorithm of [13] requires a 3-D lowpass filter-ing operation for removfilter-ing the high frequency components which are typical of the Fourier transforms of surfaces. In our case, however, this kind of filtering is not necessary since the data are already volumes. The only provision taken consists in perturbing thezcoordinate of the slices making up a given CT scan. As an example, Figures 6 (a)
and (b), respectively, show a regular series of CT slices as acquired in a practical scenario and its perturbed counter-part. Since our method works with volumes, the perturba-tion of thezcoordinate transforms a series of 2-D planes containing the data into a 3-D volume. One may then com-pute the 3-D FFT of these volumes and apply the algorithm of [13]. This yields a first coarse estimate of the 3-D rota-tion matrix which allows the alignment of the CT data sets. Figures 5 (a) and 5 (b) show the two CT data sets centered at their baricenters whereas Figures 5 (c) and 5 (d) show the results of their registration with the frequency domain algorithm.
3.2 Fine Estimation of the 3-D Rigid
Transforma-tion
The estimation of the rigid transformation obtained in the frequency domain usually is not accurate enough to be used for registration. It is therefore necessary to resort to a refinement phase. Unlike the first, which operates on vol-umes, the second works with surfaces (see [14]) through the ICP algorithm [9, 10]. The two surfaces are obtained with the isosurface extraction algorithms implemented in the sci-entific visualization library VTK [12, 15]. Rather than us-ing a sus-ingle algorithm, the ICP method determines a class of algorithms which differ in the strategy used to find the solution to various subproblems composing the ICP (com-putation of the closest point, value to estimate, caching, etc.). Among the various implementations available, we have used the version advanced by Pulli [16] for register-ing multiple 3-D views. Other implementative choices can be made as well [17, 18].
4
Experimental Results
In order to assess the performance of our algorithm, we have carried out several series of simulations where various data sets have been used. In each experiment, a known rigid motion has been applied to each point of the selected data set whose geometric position has been corrupted by white Gaussian noise to simulate a real acquisition scenario.
In each experiment a segmented cloud of points is con-sidered. A 3-D rotation is applied to it using as a center of rotation the center of mass of the point set. The three rota-tion anglesψ,ϕ, andϑtake on values between0◦and90◦, for an overall number of 343 rotations. The set of points obtained with these imposed rotations is then perturbed by adding zero-mean white Gaussian noise to their 3-D Carte-sian coordinates. The relative rotation between the two data sets is then estimated with our algorithm. For each rotation, the estimation error has been assessed in two ways: 1) by providing the average Euclidean distance between the orig-inal position of each point and the corresponding point after
ed ψε
File mean std median mean std median
fem post1.vtk 6.016 7.047 3.297 3.916 4.858 2.007 fem post2.vtk 7.691 9.398 4.337 5.194 6.385 2.430 fem pre1.vtk 4.565 5.367 2.822 3.119 3.795 1.677 fem pre2.vtk 5.562 10.355 3.949 6.373 9.53 2.125 hip post1.vtk 0.572 0.400 0.588 0.957 0.670 1.042 hip post2.vtk 0.635 0.445 0.641 1.071 0.740 1.088 hip pre1.vtk 0.547 0.397 0.606 0.918 0.662 1.034 hip pre2.vtk 0.491 0.317 0.518 0.851 0.544 0.905 Table 1. Simulation results. edis measured in millimeters,ψin degrees.
compensation for rotation. Letp1denote the location of a point of the original set and letp2denote the corresponding location after applying a 3-D rotation matrixRo. The two data sets then relate as
p2=Rop1+ε, (6)
whereεthe additive noise. From our algorithm we obtain an estimateRoˆ which leads to an estimated location
p3= ˆRop1 (7)
in lieu ofp2. If the set is comprised ofNpoints, a first error measure can be obtained as
ed= 1
NΣ
N
i=1p2i−p3i. (8)
Another error measure we have considered is
ψ= arccos (0.5(trace(R)−1)) (9)
where R =. RoRˆTo from [13]. For all 343 simulations both error measures have been computed; Table 1 shows the mean value, the standard deviation, and the median for both figures relative to eight data sets. The labels of the files containing these data are displayed in the first column of the table.
The method has also been tested in two real cases (see Figure 5). In both instances, the rigidity assumption is not strictly met. In fact, even though the shape of the bones analyzed is the same (to the exception of the prosthesis), their relative position is different since the patient cannot be placed in the same position during the two examinations.
5
Conclusions
We have presented a new method for registering CT vol-umetric data of bone structures acquired at different times. The system we propose was tested with several CT data sets. In this paper, we have discussed a representative example relative to the CT data of the hip bones of a patient prior to
and after prosthetic surgery aimed at the reconstruction of the hip articulation. The method works with rigid data hav-ing arbitrary relative position and orientation and proves to be robust with respect to CT acquisition noise and to the segmentation technique employed to select the region of in-terest for registration. The method also provides a valu-able support in the registration of data sets relative to the same articulation and whose components have undergone small displacements. The method is also amenable to reg-istration of heteregeneous kinds of volumetric data, such as CT scans and Magnetic Resonance Imagery (MRI) scans, which show different characteristics in correspondence to the same organic structure.
6
Acknowledgments
We would like to acknowledge Cinzia Zanoni and Marco Petrone of CINECA (Bologna, Italy) and Marco Viceconti and Riccardo Lattanzi of “Laboratorio Tecnologia dei Ma-teriali” from Istituto Ortopedico Rizzoli (Bologna, Italy) for kindly supplying the CT data. The research was supported in part by the University of Padova under Progetto Ateneo CPDA 024157/2002 and in part by FIRB-PRIMO.
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