Fall 2007

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IMA Outcomes

Outcomes, impacts, highlights, nuggets... whatever name they go by, these short, novice-friendly research descriptions are just one of the ways the IMA shares the achievements of its visitors with the scientific community and strives to increase public appreciation of applied mathematics. The links to some of our most intriguing stories can be found on our homepage to give visitors a glimpse of the fascinating research connected with the IMA. If your research has been influenced by a visit to the IMA, whether as a workshop participant, long-term visitor, or postdoc, please share your story with us!

Limited data tomography

Tomography, which enables imaging of cross sections of the body without dissection, is among the most important tools in diagnostic medicine. Tomographic reconstruction algorithms, which reconstruct a function from linear views or line integrals, are the basis of tomography. Moreover, the same underlying mathematical principles can be used to study the structure of objects of all sizes, ranging from huge supernova remnants to tiny viruses. Ideally, to obtain an accurate reconstruction, data should be collected along as many views as possible and distributed in a full angular range. However, due to physical constraints or to minimize costs, either a limited angular range of views is imaged or only a few views are available, resulting in notorious errors in the reconstructed images.

IMA postdoc Hstau Y. Liao created an algorithm which produces images with high accuracy even when the projection data are not finely sampled in a full angular range. One application of his work is to dental surgery, where limited range tomographic reconstructions can assist with the restoration of the jaw bone for the subsequent tooth implantation, as well as enhance the root canal therapy. Lesions or concavities in the supporting bone can be identified in the reconstructed images, in order to place grafts and artificial titanium roots. Liao presented his work at the 2007 International Symposium on Biomedical Imaging.

Mathematics like Liao's can help put the smile back on a patient's face.


Tomographic reconstruction of a third mandibular molar from only 23 views. Left, with Liao's algorithm. Right, with an Algebraic Reconstruction Technique, widely used in commercial scanners. Data from Maaria Rantala of PaloDEx Group.

Related material

Stratification Learning


Data in high dimensions is becoming ubiquitous, from image analysis and finance to computational biology and neuroscience. This data is often given or represented as samples embedded in a high dimensional Euclidean space, point cloud data, though it is assumed to belong to lower dimensional manifolds. Thus, in recent years, there have been significant efforts in the development of methods to analyze these point clouds and their underlying manifolds. These include numerous techniques for the estimation of the intrinsic dimension of the data and also its projection onto lower dimensional representations. These disciplines are often called manifold learning and dimensionality reduction.

The vast majority of the techniques developed in the literature assume, either explicitly or implicitly, that the given point cloud are samples of a unique manifold. It is very easy to realize that a significant part of the interesting data can have mixed dimensionality and complexity. That is, we can have samples not of a manifold but of a stratification.

In these cases it is useful to cluster the data according to the complexity (dimensionality) of the underlying possible multiple manifolds (see example in right Figure). Such clustering can be used both to better understand the varying dimensionality and complexity of the data, e.g., states in neural recordings or different human activities for video analysis, or as a pre-processing step for some manifold learning and dimensionality reduction and dimensionality reduction techniques.

IMA postdoc Gloria Haro together with IMA long term visitors Gregory Randall and Guillermo Sapiro have proposed a technique for "stratification learning." The method is based on a mixture of Poisson distributions that locally model the counting process of points in each different manifold. This technique automatically gives a soft clustering of the point cloud according to dimensionality and density, with an estimation of both quantities for each class. The Figure below shows two examples in computer vision where this technique allows to separate different digits and activities in video according to their dimensionality.