Image segmentation is fundamentally a discrete problem. It consists of finding a partition of the image domain such that the pixels in each element of the partition exhibit some kind of similarity. The solution is often obtained by minimizing an objective function containing terms measuring the consistency of the candidate partition with respect to the observed image, and regularization terms promoting solutions with desired properties. This formulation ends up being an integer optimization problem that, apart from a few exceptions, is NP-hard and thus impossible to solve exactly. This roadblock has stimulated active research aimed at computing “good” approximations to the solutions of those integer optimization problems. Relevant lines of attack have focused on the representation of the regions (i.e., the partition elements) in terms of functions, instead of subsets, and on convex relaxations which can be solved in polynomial time.
In this paper, inspired by the “hidden Markov measure field” introduced by Marroquin et al. in 2003, we sidestep the discrete nature of image segmentation by formulating the problem in the Bayesian framework and introducing a hidden set of real-valued random fields determining the probability of a given partition. Armed with this model, the original discrete optimization is converted into a convex program. To infer the hidden fields, we introduce the Segmentation via the Constrained Split Augmented Lagrangian Shrinkage Algorithm (SegSALSA). The effectiveness of the proposed methodology is illustrated with simulated and real hyperspectral and medical images.
We present the current state of our work on a mathematical framework for identification and delineation of
histopathology images-local histograms and occlusion models. Local histograms are histograms computed over
defined spatial neighborhoods whose purpose is to characterize an image locally. This unit of description is
augmented by our occlusion models that describe a methodology for image formation. In the context of this
image formation model, the power of local histograms with respect to appropriate families of images will be shown
through various proved statements about expected performance. We conclude by presenting a preliminary study
to demonstrate the power of the framework in the context of histopathology image classification tasks that, while
differing greatly in application, both originate from what is considered an appropriate class of images for this
We propose a design procedure for the real, equal-norm, lapped tight frame transforms (LTFTs). These transforms
have been recently proposed as both a redundant counterpart to lapped orthogonal transforms and an
infinite-dimensional counterpart to harmonic tight frames. In addition, LTFTs can be efficiently implemented
with filter banks. The procedure consists of two steps. First, we construct new lapped orthogonal transforms
designed from submatrices of the DFT matrix. Then we specify the seeding procedure that yields real equal-norm
LTFTs. Among them we identify the subclass of maximally robust to erasures LTFTs.
We propose an active mask segmentation framework that combines the advantages of statistical modeling,
smoothing, speed and flexibility offered by the traditional methods of region-growing, multiscale, multiresolution
and active contours respectively. At the crux of this framework is a paradigm shift from evolving
contours in the continuous domain to evolving multiple masks in the discrete domain. Thus, the active
mask framework is particularly suited to segment digital images. We demonstrate the use of the framework
in practice through the segmentation of punctate patterns in fluorescence microscope images. Experiments
reveal that statistical modeling helps the multiple masks converge from a random initial configuration to
a meaningful one. This obviates the need for an involved initialization procedure germane to most of the
traditional methods used to segment fluorescence microscope images. While we provide the mathematical
details of the functions used to segment fluorescence microscope images, this is only an instantiation of the
active mask framework. We suggest some other instantiations of the framework to segment different types
In recent years, the focus in biological science has shifted to understanding complex systems at the cellular and molecular levels, a task greatly facilitated by fluorescence microscopy. Segmentation, a fundamental yet difficult problem, is often the first processing step following acquisition. We have previously demonstrated that a stochastic active contour based algorithm together with the concept of topology preservation (TPSTACS) successfully segments single cells from multicell images. In this paper we demonstrate that TPSTACS successfully segments images from other imaging modalities such as DIC microscopy, MRI and fMRI. While this method is a viable alternative to hand segmentation, it is not yet ready to be used for high-throughput applications due to its large run time. Thus, we highlight some of the benefits of combining TPSTACS with the multiresolution approach for the segmentation of fluorescence microscope images. Here we propose a multiscale active contour (MSAC)
transformation framework for developing a family of modular algorithms for the segmentation of fluorescence microscope images in particular, and biomedical images in general. While this framework retains the flexibility and the high quality of the segmentation provided by active contour-based algorithms, it offers a boost in the
efficiency as well as a framework to compute new features that further enhance the segmentation.
We survey our work on adaptive multiresolution (MR) approaches to the classification of biological and fingerprint
images. The system adds MR decomposition in front of a generic classifier consisting of feature computation
and classification in each MR subspace, yielding local decisions, which are then combined into a global decision
using a weighting algorithm. The system is tested on four different datasets, subcellular protein location images,
drosophila embryo images, histological images and fingerprint images. Given the very high accuracies obtained for
all four datasets, we demonstrate that the space-frequency localized information in the multiresolution subspaces
adds significantly to the discriminative power of the system. Moreover, we show that a vastly reduced set of
features is sufficient. Finally, we prove that frames are the class of MR techniques that performs the best in this
context. This leads us to consider the construction of a new family of frames for classification, which we term
lapped tight frame transforms.
We give a physical interpretation for finite tight frames along the lines of Columb's Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an N-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a fundamental inequality that all finite tight frames must satisfy.
It is a challenging task to design orthogonal filter banks, especially multidimensional (MD) ones. In the one-dimensional (1D) two-channel finite impulse response (FIR) filter bank case, several design methods exist. Among them, designs based on spectral factorizations (by Smith and Barnwell) and designs based on lattice
factorizations (by Vaidynanathan and Hoang) are the most effective and widely used. The 1D two-channel infinite impulse response (IIR) filter banks and associated wavelets were considered by Herley and Vetterli. All of these design methods are based on spectral factorization. Since in multiple dimensions, there is no factorization
theorem, traditional 1D design methods fail to generalize. Tensor products can be used to construct MD orthogonal filter banks from 1D orthogonal filter banks, yielding separable filter banks. In contrast to separable filter banks, nonseparable filter banks are designed directly, and result in more freedom and better frequency selectivity. In the FIR case, Kovacevic and Vetterli designed specific two-dimensional and three-dimensional nonseparable FIR orthogonal filter banks. In the IIR case, there are few design results (if any) for MD orthogonal IIR filter banks. To design orthogonal filter banks, we must design paraunitary matrices,
which leads to solving sets of nonlinear equations. The Cayley transform establishes a one-to-one mapping between paraunitary
matrices and para-skew-Hermitian matrices. In contrast to nonlinear equations, the para-skew-Hermitian condition amounts to linear constraints on the matrix entries which are much easier to
solve. We present the complete characterization of both paraunitary FIR matrices and paraunitary IIR matrices in the Cayley domain. We also propose efficient design methods for MD orthogonal filter banks and corresponding methods to impose the vanishing-moment condition.
While it is recognized that images are described through color, texture and shapes of objects in the scene, the general image understanding is still very difficult. Thus, to perform an image retrieval in a human-like manner one has to choose a specific domain, understand how users achieve similarity within that domain and then build a system that duplicates human performance. Since color and texture are fundamental aspects of human perception we propose a set of techniques for retrieval of color patterns. To determine how humans judge similarity of color patterns we performed a subjective study. Based on the result of the study five most relevant visual categories for the perception of pattern similarity were identified. We also determined the hierarchy of rules governing the use of these categories. Based on these results we designed a system which accepts one or more texture images as input, and depending on the query, produces a set of choices that follow human behavior in pattern matching. Processing steps in our model follow those of the human visual system, resulting in perceptually based features and distance measures. As expected, search results closely correlate wit human choices.
26 August 2007 | San Diego, California, United States
31 July 2005 | San Diego, California, United States
4 August 2003 | San Diego, California, United States
SC203: Wavelets and Applications: State-of-the-Art
In this course, a comprehensive presentation of discrete and continuous wavelets, filter banks and subband coding, and multiresolution signal processing, is given. Techniques recently developed in different fields have converged to a unified theory. Wavelets provide an interesting alternative to Fourier transform methods. This course explains the successive approximation or multiresolution essential to wavelets and subhand coding: a signal can be seen as a coarse version plus added details.