On canonical parameterizations of 2D-shapes

This paper is devoted to the study of unparameterized simple curves in the plane. We propose diverse canonical parameterizations of a 2D-curve. For instance, the arc-length parameterization is canonical, but we consider other natural parameterizations like the parameterization proportionnal to the curvature of the curve. Both aforementionned parameterizations are very natural and correspond to a natural physical movement: the arc-length parameterization corresponds to travelling along the curve at constant speed, whereas parameterization proportionnal to curvature corresponds to a constant-speed moving frame. Since the curvature function of a curve is a geometric invariant of the unparameterized curve, a parameterization using the curvature function is a canonical parameterization. The main idea is that to any physically meaningful stricktly increasing function is associated a natural parameterization of 2D-curves, which gives an optimal sampling, and which can be used to compare unparameterized curves in a efficient and pertinent way. An application to point correspondance in medical imaging is given

Adding Curvature to Minimum Description Length Shape Models

The Minimum Description Length (MDL) approach to shape modelling seeks a compact description of a set of shapes in terms of the coordinates of marks on the shapes. It has been shown that the mark positions resulting from this optimisation to a large extent solve the so-called point correspondence problem: How to select points on shapes defined as curves so that the points correspond across a data set. However, this MDL approach does not capture important shape characteristics related to the curvature of the curves, and occasionally it places marks in obvious conflict with the human notion of point correspondence. This paper shows how the MDL approach can be fine-tuned by adding a term to the cost function expressing the mismatch of curvature features across the data set. The method is illustrated on silhouettes of adult heads. The MDL method is able to solve the point correspondence problem and a classification of the heads into male and female improves dramatically when using the MDL-generated marks

Analysis of the three-dimensional anatomical variance of the distal radius using 3D shape models

BACKGROUND: Various medical fields rely on detailed anatomical knowledge of the distal radius. Current studies are limited to two-dimensional analysis and biased by varying measurement locations. The aims were to 1) generate 3D shape models of the distal radius and investigate variations in the 3D shape, 2) generate and assess morphometrics in standardized cut planes, and 3) test the model's classification accuracy. METHODS: The local radiographic database was screened for CT-scans of intact radii. 1) The data sets were segmented and 3D surface models generated. Statistical 3D shape models were computed (overall, gender and side separate) and the 3D shape variation assessed by evaluating the number of modes. 2) Anatomical landmarks were assigned and used to define three standardized cross-sectional cut planes perpendicular to the main axis. Cut planes were generated for the mean shape models and each individual radius. For each cut plane, the following morphometric parameters were calculated and compared: maximum width and depth, perimeter and area. 3) The overall shape model was utilized to evaluate the predictive value (leave one out cross validation) for gender and side identification within the study population. RESULTS: Eighty-six radii (45 left, 44% female, 40 +/- 18 years) were included. 1) Overall, side and gender specific statistical 3D models were successfully generated. The first mode explained 37% of the overall variance. Left radii had a higher shape variance (number of modes: 20 female / 23 male) compared to right radii (number of modes: 6 female / 6 male). 2) Standardized cut planes could be defined using anatomical landmarks. All morphometric parameters decreased from distal to proximal. Male radii were larger than female radii with no significant side difference. 3) The overall shape model had a combined median classification probability for side and gender of 80%. CONCLUSIONS: Statistical 3D shape models of the distal radius can be generated using clinical CT-data sets. These models can be used to assess overall bone variance, define and analyze standardized cut-planes, and identify the gender of an unknown sample. These data highlight the potential of shape models to assess the 3D anatomy and anatomical variance of human bones

Multilevel principal component analysis (mPCA) in shape analysis: a feasibility study in medical and dental imaging

Background and objective Methods used in image processing should reflect any multilevel structures inherent in the image dataset or they run the risk of functioning inadequately. We wish to test the feasibility of multilevel principal components analysis (PCA) to build active shape models (ASMs) for cases relevant to medical and dental imaging. Methods Multilevel PCA was used to carry out model fitting to sets of landmark points and it was compared to the results of “standard” (single-level) PCA. Proof of principle was tested by applying mPCA to model basic peri-oral expressions (happy, neutral, sad) approximated to the junction between the mouth/lips. Monte Carlo simulations were used to create this data which allowed exploration of practical implementation issues such as the number of landmark points, number of images, and number of groups (i.e., “expressions” for this example). To further test the robustness of the method, mPCA was subsequently applied to a dental imaging dataset utilising landmark points (placed by different clinicians) along the boundary of mandibular cortical bone in panoramic radiographs of the face. Results Changes of expression that varied between groups were modelled correctly at one level of the model and changes in lip width that varied within groups at another for the Monte Carlo dataset. Extreme cases in the test dataset were modelled adequately by mPCA but not by standard PCA. Similarly, variations in the shape of the cortical bone were modelled by one level of mPCA and variations between the experts at another for the panoramic radiographs dataset. Results for mPCA were found to be comparable to those of standard PCA for point-to-point errors via miss-one-out testing for this dataset. These errors reduce with increasing number of eigenvectors/values retained, as expected. Conclusions We have shown that mPCA can be used in shape models for dental and medical image processing. mPCA was found to provide more control and flexibility when compared to standard “single-level” PCA. Specifically, mPCA is preferable to “standard” PCA when multiple levels occur naturally in the dataset