Alexey Koloydenko

NON-EUCLIDEAN STATISTICS FOR COVARIANCE MATRICES, WITH APPLICATIONS TO DIFFUSION TENSOR IMAGING

The statistical analysis of covariance matrices occurs in m any important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. We consider the situation where it is of interest to estimate an average covariance matrix, describe its anisotropy and to carry out principal geodesic analysis of covariance matrices. In medical image analysis a particular type of covariance matrix arises in diffusion weighted imaging called a diffusion tensor. The diffusion tensor is a 3 × 3 covariance matrix which is estimated at each voxel in the brain, and is obtained by fittin g a physically-motivated model on measurements from the Fourier transform of the molecule displacement density (Basser et al., 1994). A strongly anisotropic diffusion tensor indicates a strong direction of white matter fibre tracts, and plots of measures of anisotropy are very useful t o neurologists. A measure that is very commonly used in diffusion tensor imaging is Fractional Anisotropy

Diagnosis of tumors during tissue-conserving surgery with integrated autofluorescence and Raman scattering microscopy.

Significance Histopathology is the standard method for diagnosis of cancer. However, this method requires time-consuming procedures for sectioning and staining of tissues, making histopathology impractical for use during surgery for most cancer types. We report a unique method based on two optical spectroscopy techniques—autofluorescence imaging and Raman scattering—that can accurately measure molecular differences between tumor cells and healthy tissue and allows diagnosis of tumors faster than histopathology, without requiring tissue sectioning or staining. Our study demonstrates the potential of this technique for diagnosis of tissues during cancer surgery, providing a quick and objective way to determine whether the tissue layers removed by the surgeon are clear of tumor. Tissue-conserving surgery is used increasingly in cancer treatment. However, one of the main challenges in this type of surgery is the detection of tumor margins. Histopathology based on tissue sectioning and staining has been the gold standard for cancer diagnosis for more than a century. However, its use during tissue-conserving surgery is limited by time-consuming tissue preparation steps (1–2 h) and the diagnostic variability inherent in subjective image interpretation. Here, we demonstrate an integrated optical technique based on tissue autofluorescence imaging (high sensitivity and high speed but low specificity) and Raman scattering (high sensitivity and high specificity but low speed) that can overcome these limitations. Automated segmentation of autofluorescence images was used to select and prioritize the sampling points for Raman spectroscopy, which then was used to establish the diagnosis based on a spectral classification model (100% sensitivity, 92% specificity per spectrum). This automated sampling strategy allowed objective diagnosis of basal cell carcinoma in skin tissue samples excised during Mohs micrographic surgery faster than frozen section histopathology, and one or two orders of magnitude faster than previous techniques based on infrared or Raman microscopy. We also show that this technique can diagnose the presence or absence of tumors in unsectioned tissue layers, thus eliminating the need for tissue sectioning. This study demonstrates the potential of this technique to provide a rapid and objective intraoperative method to spare healthy tissue and reduce unnecessary surgery by determining whether tumor cells have been removed.

Towards intra-operative diagnosis of tumours during breast conserving surgery by selective-sampling Raman micro-spectroscopy

Breast-conserving surgery (BCS) is increasingly employed for the treatment of early stage breast cancer. One of the key challenges in BCS is to ensure complete removal of the tumour while conserving as much healthy tissue as possible. In this study we have investigated the potential of Raman micro-spectroscopy (RMS) for automated intra-operative evaluation of tumour excision. First, a multivariate classification model based on Raman spectra of normal and malignant breast tissue samples was built and achieved diagnosis of mammary ductal carcinoma (DC) with 95.6% sensitivity and 96.2% specificity (5-fold cross-validation). The tumour regions were discriminated from the healthy tissue structures based on increased concentration of nucleic acids and reduced concentration of collagen and fat. The multivariate classification model was then applied to sections from fresh tissue of new patients to produce diagnosis images for DC. The diagnosis images obtained by raster scanning RMS were in agreement with the conventional histopathology diagnosis but were limited to long data acquisition times (typically 10,000 spectra mm(-2), which is equivalent to ~5 h mm(-2)). Selective-sampling based on integrated auto-fluorescence imaging and Raman spectroscopy was used to reduce the number of Raman spectra to ~20 spectra mm(-2), which is equivalent to an acquisition time of ~15 min for 5 × 5 mm(2) tissue samples. This study suggests that selective-sampling Raman microscopy has the potential to provide a rapid and objective intra-operative method to detect mammary carcinoma in tissue and assess resection margins.

The adjusted Viterbi training for hidden Markov models

To estimate the emission parameters in hidden Markov models one commonly uses the EM algorithm or its variation. Our primary motivation, however, is the Philips speech recognition system wherein the EM algorithm is replaced by the Viterbi training algorithm. Viterbi training is faster and computationally less involved than EM, but it is also biased and need not even be consistent. We propose an alternative to the Viterbi training -- adjusted Viterbi training -- that has the same order of computational complexity as Viterbi training but gives more accurate estimators. Elsewhere, we studied the adjusted Viterbi training for a special case of mixtures, supporting the theory by simulations. This paper proves the adjusted Viterbi training to be also possible for more general hidden Markov models.

Robust acoustic object detection

We consider a novel approach to the problem of detecting phonological objects like phonemes, syllables, or words, directly from the speech signal. We begin by defining local features in the time-frequency plane with built in robustness to intensity variations and time warping. Global templates of phonological objects correspond to the coincidence in time and frequency of patterns of the local features. These global templates are constructed by using the statistics of the local features in a principled way. The templates have clear phonetic interpretability, are easily adaptable, have built in invariances, and display considerable robustness in the face of additive noise and clutter from competing speakers. We provide a detailed evaluation of the performance of some diphone detectors and a word detector based on this approach. We also perform some phonetic classification experiments based on the edge-based features suggested here.

A Constructive Proof of the Existence of Viterbi Processes

Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM <i>(X</i> _t,<i>Y</i> _t)_{t ¿ 1}, observations <i>X</i> ₁,<i>X</i> ₂,... are assumed to be conditionally independent given a Markov process <i>Y</i> ₁,<i>Y</i> ₂,..., which itself is not observed; moreover, the conditional distribution of <i>X</i> _t depends solely on <i>Y</i> _t. Central to the theory and applications of HMM is the Viterbi algorithm to find a maximum <i>a posteriori</i> probability (MAP) estimate <i>v</i>(<i>x</i> _1: <i>T</i>)=(<i>v</i> ₁,<i>v</i> ₂,...,<i>vT</i>) of <i>Y</i> _1: <i>T</i> given observed data <i>x</i> _1: <i>T</i>. Maximum <i>a posteriori</i> paths are also known as the Viterbi paths, or alignments. Recently, attempts have been made to study behavior of the Viterbi alignments when <i>T</i>¿ ¿. Thus, it has been shown that in some cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more constructive manner and for a very general class of HMMs.

Adjusted Viterbi Training

Viterbi training (VT) provides a fast but inconsistent estimator of hidden Markov models (HMM). The inconsistency is alleviated with a little extra computation when we enable VT to asymptotically fix the true values of the parameters. This relies on infinite Viterbi alignments and associated with them limiting probability distributions. First in a sequel, this article is a proof of concept; it focuses on mixture models, an important but special case of HMM where the limiting distributions can be calculated exactly. A simulated Gaussian mixture shows that our central algorithm (VA1) can significantly improve the accuracy of VT with little extra cost. Next in the sequel, we present elsewhere a theory of the adjusted VT for the general HMMs, where the limiting distributions are more challenging to find. Here, we also present another, more advanced correction to VT and verify its fast convergence and high accuracy; its computational feasibility requires additional investigation.

Theory of Segmentation

1.1 Preliminaries In this chapter we focus on what Rabiner in his popular tutorial (Rabiner, 1989) calls “uncovering the hidden part of the model” or “Problem 2”, that is, hidden path inference. We consider a hidden Markov model (X,Y) = {(Xt,Yt)}t∈Z, where Y = {Yt}t∈Z is an unobservable, or hidden, homogeneous Markov chain with a finite state space S = {1, . . . ,K}, transition matrix P = (pi,j)i,j∈S and, whenever relevant, the initial probabilities πs = P(Y1 = s), s ∈ S. A reader interested in extensions to the continuous case is referred to (Cappe et al., 2005; Chigansky & Ritov, 2010). The Markov chain will be further assumed to be of the first order, bearing in mind that a higher order chain can always be converted to a first order one by expanding the state space. To simplify the mathematics, we assume that the Markov chain Y is stationary and ergodic. This assumption is needed for the asymptotic results in Section 3, but not for the finite time-horizon in Section 2. In fact, Section 2 does not even require the assumption of homogeneity. The second component X = {Xt}t∈Z is an observable process with Xt taking values in X that is typically a subspace of the Euclidean space, i.e. X ⊂ Rd. The process X can be thought of as a noisy version of Y. In order for (X,Y) to be a hidden Markov model, the following properties need to be satisfied:

Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics.

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First, we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical considerations are discussed, with some examples. Second, we discuss weighted Procrustes methods for diffusion tensor imaging interpolation and smoothing, and we compare methods based on different metrics on a set of examples as well as analytically. We establish a key relationship between the principal-square-root-Euclidean metric and the size-and-shape Procrustes metric on the space of symmetric positive semi-definite tensors. We explain, both analytically and by experiments, why the size-and-shape Procrustes metric may be preferred in practical tasks of interpolation, extrapolation and smoothing, especially when observed tensors are degenerate or when a moderate degree of tensor swelling is desirable. Third, we introduce regularisation methodology, which is demonstrated to be useful for highlighting features of prior interest and potentially for segmentation. Finally, we compare several metrics in a data set of human brain diffusion-weighted magnetic resonance imaging, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric.

A Bayesian method with reparameterization for diffusion tensor imaging

A multi-tensor model with identifiable parameters is developed for diffusion weighted MR images. A new parameterization method guarantees the symmetric positive-definiteness of the diffusion tensor. We set up a Bayesian method for parameter estimation. To investigate properties of the method, Monte Carlo simulated data from three distinct DTI direction schemes have been analyzed. The multi-tensor model with automatic model selection has also been applied to a healthy human brain dataset. Standard tensor-derived maps are obtained when the single-tensor model is fitted to a region of interest with a single dominant fiber direction. High anisotropy diffusion flows and main diffusion directions can be shown clearly in the FA map and diffusion ellipsoid map. For another region containing crossing fiber bundles, we estimate and display the ellipsoid map under the single tensor and double-tensor regimes of the multi-tensor model, suitably thresholding the Bayes factor for model selection.