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Automatic Generation of Shape Models Using Nonrigid Registration
with a Single Segmented Template Mesh
Geremy Heitz, Torsten Rohlfing, and Calvin R. Maurer, Jr.
300 Pasteur Drive, Stanford, CA, 94305-5327, USA Abstract
beling of the training shapes used to build the model[2].
In addition to requiring that the landmarks Statistical Shape modeling using point distribu- accurately represent each object individually, there tion models (PDMs) has been studied extensively is a requirement of identifying which landmarks for segmentation and other image analysis tasks.
across the training set correspond to each other. The Methods investigated in the literature begin with problem of landmark designation across the training a set of segmented training images and attempt set is called the correspondence problem, and is one to find point correspondences between the seg- of the primary directions of research in this area.
mented shapes before performing the statistical Because manual landmark selection is a long and tedious process, particularly in 3D, an automatic processing stage where each shape must be manu- mechanism of labeling the corresponding points on ally or semi-automatically segmented by an expert.
the various shapes is required. Additionally, man- In this paper we present a method for PDM genera- ually determining landmarks correspondences be- tion requiring only one shape to be segmented prior tween two shapes is observer dependent; an auto- to the training phase. The mesh representation gen- matic method can remove this extra source of vari- erated from the single template shape is then prop- agated to the other training shapes using a nonrigid In this paper we present a procedure for automat- registration process. This automatically produces a ically generating a statistical shape model of bio- set of meshes with correspondences between them.
logical structures from a set of training images. Our The resulting meshes are combined through Pro- method produces a surface model for the shape in crustes analysis and principal component analysis question, and requires hand segmentation of only into a statistical model. A model of the C7 vertebra one training image. In Section 2 we review work was created and evaluated for accuracy and com- by other groups towards solving the correspondence of our method, followed by an evaluation of ourmethod applied to the C7 vertebra in Section 4. Sec- Introduction
Statistical models of shape in general, and point dis-tribution models (PDMs) in particular, have seen Prior Work
wide use in the medical image analysis commu-nity [1]. These models contain a priori information Several groups have approached the correspon- about an object that can be used in a wide variety dence problem from a number of directions. Work of tasks, including recognition, classification, and by Davies et al. [3] optimizes an information the- oretic cost function of the landmark positions used As many authors have pointed out, however, ac- to create the shape model. The motivation for this curate models require a reliable and consistent la- approach comes from the hypothesis that the best Stanford (California), USA, November 16–18, 2004 model is the one that is simplest and hence general- shape in the training set, as opposed to every one.
izes the best. Towards this end they find a param- eterization of the shape surface that minimizes the input, and the technique for shape model generation information required to encode the model plus the information in the training set, given that model.
1. Selection of a template image and initial seg- Another approach was championed by Lorenz and Krahnst¨over [4]. They seek to solve the prob- 2. Generation of a template surface mesh from lem by creating a mesh from a template shape, and “coating” the mesh vertices onto the other shapes in 3. Intensity-based nonrigid registration of the the training set. The vertices are coated onto the tar- template image to the remaining training set get shape in a two-step process that involves an ini- tial affine registration, followed by an active surface 4. Warping of the template mesh to the training technique patterned after that described by McIner- set shapes using the transformation produced Paulsen and Hilger [6] use a similar strategy.
5. Alignment and statistical analysis of the train- However, they cast the problem in a Bayesian ing set meshes resulting from warping of the framework, seeking a deformation field that max- imizes an a posteriori probability measure.
This model creation process is shown schemati- Methods proposed by Frangi, Rueckert and col- cally in Fig. 1, and each step of the process is de- leagues [7] are the most similar to the one de- scribed in this paper. Their scheme relies on a non-rigid registration of the manually segmented train- Template Selection and Segmentation
ing shapes. Work on so-called active deformationmodels (ADMs) [8] foreshadows the tools used in The first step in the creation pipeline is to select this paper, where a dense set of correspondences (a a template image from the training set, and seg- deformation field) between images is used to prop- ment the template shape in the selected image. In agate surface landmarks from a template mesh to our case, we interactively segmented the template each sample in the training set. They suggest that shape in each slice of the image using an adaptive the deformation fields can be used to propagate a boundary segmentation method known as “intelli- shape representation to the shapes in a training set.
gent scissors” or “Live-wire” [9]. This produces a They construct a model of the ventricles and cau- label image that describes pixels as either inside or date nuclei with a single surface segmentation, but provide no details or quantitative assessment of it.
Choosing a good template shape is important for Our work explores this possibility further, creating the results of this process, as will be discussed later.
a vertebral shape model in order to evaluate the vi- The quality of a template can be judged by the qual- ity of the registrations obtained between the tem-plate and all the other images in the set.
Shape Model Generation
Template Mesh Generation
The goal of our work is to create a statistical From the template shape label image produced in shape model of the surface of an object given a the first step, the marching cubes algorithm is uti- set of three-dimensional (3D) images containing in- lized to generate a triangulated surface mesh. Given stances of the object. We do not assume that any of this initial fine mesh, we smooth and decimate it to the images are previously segmented, and there is a desired mesh resolution. These operations were no requirement of uniform resolution or size across performed using the Visualization Toolkit (VTK) the training set. The method we developed is semi- software [10], which provides many tools for mesh automatic, in that it requires user initialization, but is fully automatic after this beginning phase. The It is important to note that this process of seg- primary advantage of this method over those pre- mentation followed by marching cubes with mesh viously described is the need to segment only one refinement is performed only for the template Training
Shape Set
Template Image
Template Mesh
Figure 1: Statistical shape model creation process. First, a template image is selected from the training set.
The template shape is segmented in the selected image, which produces a binary label image. A templatemesh is generated from the label image using marching cubes and decimated and smoothed to a desiredmesh resolution. The template mesh is warped using nonrigid transformations produced by intensity-basednonrigid registration of the template image to the training images. This produces a set of training setmeshes, which are aligned using generalized Procrustes analysis. Finally, principal component analysis(PCA) is performed to obtain the statistical shape model.
shape. The number of vertices and triangles cho- sen for the template shape will then be fixed across target image independently, and the transformed lo- the training set, with a one-to-one correspondence cation of a source point ( , , ) in the target image between vertices in each of the training surfaces.
Nonrigid Registration of Template Im-
age to Training Images
In order to warp the template mesh onto the other shapes in the training set, we must determine the appropriate transformation to use. This deforma-tion is computed using an initial affine registra- tion step, followed by an independent implemen- control point cell that contains ( , , ), and (✩ , ✪ , tation [11] of an intensity-based nonrigid registra- ) represents the relative position of ( , , ) within tion technique developed by Rueckert et al. [12].
the cell. These quantities can be computed using: The 3D grayscale template image is registered toeach of the remaining training set images contain- ing the anatomical object of interest. The deforma- tion is defined on a uniformly-space grid, and usesB-splines for interpolation between control points.
age, and should lie on the surface of the structure ifthe nonrigid transformation is accurate. The appli-cation of the transformation corresponding to eachtraining sample is shown in Fig. 1, resulting in a setof such surfaces with vertex correspondences.
Alignment and Statistical Analysis
Figure 2: Mapping of landmarks from the source or template shape (left) onto the target or training training shapes, the next step is to align the shapes shape(right). The mapping is the transformation into a common reference frame and remove the produced by the intensity-based nonrigid registra- rotational, scaling, and translational components, tion of the template image to the training image.
which we do not wish to include in the model ofshape. This is done with an iterative method knownas generalized Procrustes analysis (GPA) [14].
Although our registration technique begins with an affine registration step, we still must perform the Procrustes analysis to determine the affine transfor- mation that relates the training mesh to the template the control points in each dimension in the source mesh for two reasons. The first is that the elastic registration phase may add some additional global order approximating spline polynomials. The opti- affine components. This must be included in the fi- mization step finds the control point displacements nal alignment of the training shapes. The second that maximize the image similarity measure, nor- reason is that we are concerned with the optimal malized mutual information (NMI) [13].
transformation between the object surfaces. While The output of this step provides a dense cor- the global transformation is optimal across the en- respondence between points in the template and tire volumetric image, a slightly different affine transformation may more accurately relate the sur- thought of as a lookup function that specifies where face representations. For these reasons, we perform in the target (training) image to look to find the pixel the GPA step once all the surface meshes have been that corresponds to a given pixel in the source (tem- obtained. The result is a set of triangulated meshes with vertex correspondences that are aligned in acommon reference frame with the affine compo-nents of their variations removed.
Warping of the Template Mesh
The final step in construction of the statistical We compute a non-rigid transformation for every model is to perform principal component analysis shape in the training set. Note that the warp for the shape used as the template is the identity trans- as a vector of the vertex positions, i.e., a vector The target mesh is produced by finding the warped mesh. Modeling this point cloud as a multivari- location of every vertex in the template mesh. In ate Gaussian distribution, we can then compute the An example of this mapping from source to target The locations of the mesh vertices are now de- scribed in the local coordinates of the target im- The principal components (or principal axes) of this distribution are then given by the eigenvectors of the sponding to the largest eigenvalues as columns, we Training Sample
can approximate any shape in the training set using Figure 3: Distance between the vertices of the trans- formed (automatically-generated) mesh used to rep-resent the training sample surface and the ground truth surface mesh (manually segmented). This dis- In addition, we hope that this decomposition of the tance is computed for each vertex on the mesh, surfaces generalizes well, i.e., that any viable in- and the mean distance is denoted by the black cir- stance of the shape is described using appropriate cle. For each training sample, the graph also shows values in the vector of mode coefficients, In other words, 10% of the vertices had distances smaller than the bottom hash mark, and 10% haddistances larger than the top hash mark.
The method described in the preceding section wasused to create a surface model for the C7 vertebra.
the marching cubes algorithm on this segmentation As training data, we used ten CT image volumes of produced a mesh that we then used as a ground truth the neck and upper spine. The in-slice image reso- with which to compare our automatic results.
lution varied from 0.5 to 1.0 mm; the slice thickness We chose to measure the distance between the was 1.25 mm for all images. To isolate the C7 ver- mesh vertices and the closest point on the ground tebra, each image was manually cropped to include truth mesh surface as a quantitative description of the vertebra in question and approximately half of mesh accuracy. This is a measure of how closely the automatically-generated mesh surface approxi-mates the true surface according to the manual seg- Registration Accuracy
mentation. Figure 3 shows the results of apply- As mentioned above, the accuracy of the training ing this measurement to each mesh in the training mesh as a representation for the surface in the train- set. For each training shape, the mean surface-to- ing image is directly determined by the quality of surface distance is reported. In addition, the graph the nonrigid registration relating the template image also shows the range over which the distances vary, to the training image. If the surfaces of the anatom- ical structures can be put in one-to-one correspon- The mean distance was typically about 1 mm, dence, and the registration is perfect (i.e., the map- which is comparable to the in-plane resolution of ping is correct and exact), then the vertices trans- the images. The mean error is thus on the order of formed by the warp will lie on the surface, and the one pixel dimension, and less than the slice thick- In order to measure this accuracy, we manually segmented each vertebra (Section 3.1). This pro- Model Evaluation
duced a label image where the foreground label isapplied to pixels inside the vertebra, and the back- To further evaluate the scheme described above, ground label is applied to all other pixels. Running we performed the alignment step and the statisti- Mean - 3SD
Mean + 3SD
Figure 4: Instances generated from the model of the C7 vertebra. The first three modes are varied acrossthree standard deviations. The modes are organized in rows, with the left vertebra for each pair being themean shape minus three standard deviations (SD), and the right vertebra being the mean plus three SD. Themean shape appears in the middle of the figure.
cal analysis on the ten resulting meshes, and inves- would not be desirable, as it would mean that the tigated the model that was created from this train- correspondences are incorrect, and would lead to ing set. Figure 4 shows some instances generated non-viable instances for certain values of the mode from the model. Here the first three modes are var- ied across three standard deviations. The modes are Because one of the primary reasons for using a organized in rows, showing the mean shape minus PDM is the ability of this model to describe the data three standard deviations (SD), and the mean plus with a small number of degrees of freedom, we in- three SD. The mean shape appears in the middle of vestigated how much variability was described by each mode of variation. Figure 5 shows the per- The modes correspond to physical aspects of the centage of shape variance described by the first ✟ vertebra. The third mode, for instance, corresponds modes. Ideally, if there is little variation in the train- to a variation in length of the transverse processes.
ing set, we should be able to explain most of it with This fits well with our intuition that the variation a smaller number of modes. As can be seen, we can across the training set should show variations in describe 88% of the variance using only five modes.
widths, lengths, etc. In addition, because the struc- The final aspect of the model that we investigated tures are symmetric, we expect the variations to was the accuracy of the mesh representation of the also be symmetric, which we observe in this figure.
training set shapes using various numbers of modes.
There is not a noticeable twisting effect among the modes, which demonstrates that the vertices on the fully reconstruct each training sample, and then re- meshes did not slide across the surfaces. Sliding Number of Modes
Number Of Modes
Figure 5: The cumulative shape variance described Figure 6: The residual error using the first of variation to describe the training set. The error with zero modes we are using the mean mesh to here measures the distance from each vertex recon- represent each surface, so none of the variance is ex- plained. With all modes, however, the entire train- in the true mesh. These distances are then averaged ing set can be described perfectly. With five modes, we can explain 88% of the variation.
over similar existing methods, which require a pre- segmentation of every shape in the training set.
Results from the creation of a C7 vertebral mal method for compressing the shape representa- model showed that an accurate representation can tion, i.e., causes the least loss in accuracy. The met- be achieved (with a mean surface distance of about ric used to measure accuracy was the residual er- 1 mm) by warping the template mesh to correspond ror between the reconstructed mesh vertex and the to the appropriate training image. In addition, the true mesh vertex (the mesh produced by warping the shape model can describe the mesh to an accuracy template mesh), measured in millimeters.
of about 1 mm using only five modes of variation.
Figure 6 shows the mean and maximum residual The performance of our method is comparable to a study done by Kaus et al. [2]. They used images each shape in the training set. Notice that even if with a slice thickness of 2.0 mm, and a similar range we use only the mean shape (no modes of varia- of in-plane resolutions. They were able to obtain a tion) to reconstruct each mesh, we still see only 2 mean surface distance of 0.8 mm, and a mesh accu- mm of mean residual error. This means that across racy of 1.6 mm with ten modes (they used a set of 32 most of the vertices in a given mesh, most of them vertebrae, including L1-L4). Their results required do not deviate very far from their mean value. We a prior segmentation of every shape, however, and can also see that, as expected, the error goes to zero we believe the savings in time using our method are when we use all nine modes. It is a mathematical worth the slight decrease in surface representation consequence of PCA that we can reconstruct all ten accuracy. In addition, for most applications, it is training vectors perfectly if we use all nine modes.
likely that an accuracy of 1 mm should be sufficient.
Their method treats the affine-transformed mesh as an active surface, and allows the mesh to deform Discussion and Conclusion
so as to push the mesh vertices towards the bound-aries in the volumetric image. While their input im- In this paper we have presented an automatic pro- ages consist of white foreground pixels on a black cedure for the construction of 3D statistical shape background, this process would likely work with models. The method requires the manual segmen- the original grayscale image. As a result, we be- tation of only one image in the training set, and gen- lieve that this technique could also be applied to our erates the surfaces and provides correspondences warped meshes, and work is underway to refine the mesh after the non-rigid warping step to improve its [5] T. McInerney and D. Terzopoulos.
accuracy. Additional refinement of the mesh should formable models in medical images analysis: help to push the vertices closer to the boundaries, a survey. Medical Image Analysis, 1(2):91– creating a more accurate surface representation.
The experiments described above were per- [6] R. R. Paulsen and K. B. Hilger. Shape mod- formed on a small set of ten CT images, and we elling using markov random field restoration believe that it is important to evaluate the method of point correspondences. In Information Pro- on a larger set. We hope to obtain a larger set of cessing in Medical Imaging, volume 2732 of images and build a model from these in the future.
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