Optimising adjacent membrane segmentation and parameterisation in multicellular aggregates by piecewise active contours

In fluorescence microscopy imaging, the segmentation of adjacent cell membranes within cell aggregates, multicellular samples, tissue, organs, or whole organisms remains a challenging task. The lipid bilayer is a very thin membrane when compared to the wavelength of photons in the visual spectra. Fluorescent molecules or proteins used for labelling membranes provide a limited signal intensity, and light scattering in combination with sample dynamics during in vivo imaging lead to poor or ambivalent signal patterns that hinder precise localisation of the membrane sheets. In the proximity of cells, membranes approach and distance each other. Here, the presence of membrane protrusions such as blebs; filopodia and lamellipodia; microvilli; or membrane vesicle trafficking, lead to a plurality of signal patterns, and the accurate localisation of two adjacent membranes becomes difficult. Several computational methods for membrane segmentation have been introduced. However, few of them specifically consider the accurate detection of adjacent membranes. In this article we present ALPACA (ALgorithm for Piecewise Adjacent Contour Adjustment), a novel method based on 2D piecewise parametric active contours that allows: (i) a definition of proximity for adjacent contours, (ii) a precise detection of adjacent, nonadjacent, and overlapping contour sections, (iii) the definition of a polyline for an optimised shared contour within adjacent sections and (iv) a solution for connecting adjacent and nonadjacent sections under the constraint of preserving the inherent cell morphology. We show that ALPACA leads to a precise quantification of adjacent and nonadjacent membrane zones in regular hexagons and live image sequences of cells of the parapineal organ during zebrafish embryo development. The algorithm detects and corrects adjacent, nonadjacent, and overlapping contour sections within a selected adjacency distance d, calculates shared contour sections for neighbouring cells with minimum alterations of the contour characteristics, and presents piecewise active contour solutions, preserving the contour shape and the overall cell morphology. ALPACA quantifies adjacent contours and can improve the meshing of 3D surfaces, the determination of forces, or tracking of contours in combination with previously published algorithms. We discuss pitfalls, strengths, and limits of our approach, and present a guideline to take the best decision for varying experimental conditions for in vivo microscopy.

An Image Segmentation Method Based on Improved Regularized Level Set Model

When the level set algorithm is used to segment an image, the level set function must be initialized periodically to ensure that it remains a signed distance function (SDF). To avoid this defect, an improved regularized level set method-based image segmentation approach is presented. First, a new potential function is defined and introduced to reconstruct a new distance regularization term to solve this issue of periodically initializing the level set function. Second, by combining the distance regularization term with the internal and external energy terms, a new energy functional is developed. Then, the process of the new energy functional evolution is derived by using the calculus of variations and the steepest descent approach, and a partial differential equation is designed. Finally, an improved regularized level set-based image segmentation (IRLS-IS) method is proposed. Numerical experimental results demonstrate that the IRLS-IS method is not only effective and robust to segment noise and intensity-inhomogeneous images but can also analyze complex medical images well.

A border-ownership model based on computational electromagnetism

The mathematical relation between a vector electric field and its corresponding scalar potential field is useful to formulate computational problems of lower/middle-order visual processing, specifically related to the assignment of borders to the side of the object: so-called border ownership (BO). BO coding is a key process for extracting the objects from the background, allowing one to organize a cluttered scene. We propose that the problem is solvable simultaneously by application of a theorem of electromagnetism, i.e., "conservative vector fields have zero rotation, or "curl." We hypothesize that (i) the BO signal is definable as a vector electric field with arrowheads pointing to the inner side of perceived objects, and (ii) its corresponding scalar field carries information related to perceived order in depth of occluding/occluded objects. A simple model was developed based on this computational theory. Model results qualitatively agree with object-side selectivity of BO-coding neurons, and with perceptions of object order. The model update rule can be reproduced as a plausible neural network that presents new interpretations of existing physiological results. Results of this study also suggest that T-junction detectors are unnecessary to calculate depth order.

Object-oriented Persistent Homology

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.

Harmonic active contours

We propose a segmentation method based on the geometric representation of images as 2-D manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and results in a segmentation criterion that exploits the directional information of image gradients to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: 1) the natural coupling of image channels proposed for anisotropic diffusion and 2) the ability of subjective surfaces to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours, the active contours without edges, and the robust edge integrator) to higher dimensional spaces, non-flat images, and feature spaces. Our experiments show that the proposed technique improves the segmentation of multi-channel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.

A Tangent Bundle Theory for Visual Curve Completion

Visual curve completion is a fundamental perceptual mechanism that completes the missing parts (e.g., due to occlusion) between observed contour fragments. Previous research into the shape of completed curves has generally followed an "axiomatic" approach, where desired perceptual/geometrical properties are first defined as axioms, followed by mathematical investigation into curves that satisfy them. However, determining psychophysically such desired properties is difficult and researchers still debate what they should be in the first place. Instead, here we exploit the observation that curve completion is an early visual process to formalize the problem in the unit tangent bundle R(2) × S(1), which abstracts the primary visual cortex (V1) and facilitates exploration of basic principles from which perceptual properties are later derived rather than imposed. Exploring here the elementary principle of least action in V1, we show how the problem becomes one of finding minimum-length admissible curves in R(2) × S(1). We formalize the problem in variational terms, we analyze it theoretically, and we formulate practical algorithms for the reconstruction of these completed curves. We then explore their induced visual properties vis-à-vis popular perceptual axioms and show how our theory predicts many perceptual properties reported in the corresponding perceptual literature. Finally, we demonstrate a variety of curve completions and report comparisons to psychophysical data and other completion models.

The collaboration of grouping laws in vision

Gestalt theory gives a list of geometric grouping laws that could in principle give a complete account of human image perception. Based on an extensive thesaurus of clever graphical images, this theory discusses how grouping laws collaborate, and conflict toward a global image understanding. Unfortunately, as shown in the bibliographical analysis herewith, the attempts to formalize the grouping laws in computer vision and psychophysics have at best succeeded to compute individual partial structures (or partial gestalts), such as alignments or symmetries. Nevertheless, we show here that a never formalized clever Gestalt experimental procedure, the Nachzeichnung suggests a numerical set up to implement and test the collaboration of partial gestalts. The new computational procedure proposed here analyzes a digital image, and performs a numerical simulation that we call Nachtanz or Gestaltic dance. In this dance, the analyzed digital image is gradually deformed in a random way, but maintaining the detected partial gestalts. The resulting dancing images should be perceptually indistinguishable if and only if the grouping process was complete. Like the Nachzeichnung, the Nachtanz permits a visual exploration of the degrees of freedom still available to a figure after all partial groups (or gestalts) have been detected. In the new proposed procedure, instead of drawing themselves, subjects will be shown samples of the automatic Gestalt dances and required to evaluate if the figures are similar. Several numerical preliminary results with this new Gestaltic experimental setup are thoroughly discussed.

Differential geometry based solvation model II: Lagrangian formulation

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent-solute interaction potential. The nonpolar solvation model is completed with a Poisson-Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent-solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace-Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent-solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein-protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein-protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.

2D/3D fetal cardiac dataset segmentation using a deformable model

PURPOSE

To segment the fetal heart in order to facilitate the 3D assessment of the cardiac function and structure.

METHODS

Ultrasound acquisition typically results in drop-out artifacts of the chamber walls. The authors outline a level set deformable model to automatically delineate the small fetal cardiac chambers. The level set is penalized from growing into an adjacent cardiac compartment using a novel collision detection term. The region based model allows simultaneous segmentation of all four cardiac chambers from a user defined seed point placed in each chamber.

RESULTS

The segmented boundaries are automatically penalized from intersecting at walls with signal dropout. Root mean square errors of the perpendicular distances between the algorithm's delineation and manual tracings are within 2 mm which is less than 10% of the length of a typical fetal heart. The ejection fractions were determined from the 3D datasets. We validate the algorithm using a physical phantom and obtain volumes that are comparable to those from physically determined means. The algorithm segments volumes with an error of within 13% as determined using a physical phantom.

CONCLUSIONS

Our original work in fetal cardiac segmentation compares automatic and manual tracings to a physical phantom and also measures inter observer variation.

Differential geometry based solvation model I: Eulerian formulation

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

Differential geometry based multiscale models

Large chemical and biological systems such as fuel cells, ion channels, molecular motors, and viruses are of great importance to the scientific community and public health. Typically, these complex systems in conjunction with their aquatic environment pose a fabulous challenge to theoretical description, simulation, and prediction. In this work, we propose a differential geometry based multiscale paradigm to model complex macromolecular systems, and to put macroscopic and microscopic descriptions on an equal footing. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum mechanical description of the aquatic environment with the microscopic discrete atomistic description of the macromolecule. Multiscale free energy functionals, or multiscale action functionals are constructed as a unified framework to derive the governing equations for the dynamics of different scales and different descriptions. Two types of aqueous macromolecular complexes, ones that are near equilibrium and others that are far from equilibrium, are considered in our formulations. We show that generalized Navier-Stokes equations for the fluid dynamics, generalized Poisson equations or generalized Poisson-Boltzmann equations for electrostatic interactions, and Newton's equation for the molecular dynamics can be derived by the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows. Comparison is given to classical descriptions of the fluid and electrostatic interactions without geometric flow based micro-macro interfaces. The detailed balance of forces is emphasized in the present work. We further extend the proposed multiscale paradigm to micro-macro analysis of electrohydrodynamics, electrophoresis, fuel cells, and ion channels. We derive generalized Poisson-Nernst-Planck equations that are coupled to generalized Navier-Stokes equations for fluid dynamics, Newton's equation for molecular dynamics, and potential and surface driving geometric flows for the micro-macro interface. For excessively large aqueous macromolecular complexes in chemistry and biology, we further develop differential geometry based multiscale fluid-electro-elastic models to replace the expensive molecular dynamics description with an alternative elasticity formulation.

Methods toward in vivo measurement of zebrafish epithelial and deep cell proliferation

We present a strategy for automatic classification and density estimation of epithelial enveloping layer (EVL) and deep layer (DEL) cells, throughout zebrafish early embryonic stages. Automatic cells classification provides the bases to measure the variability of relevant parameters, such as cells density, in different classes of cells and is finalized to the estimation of effectiveness and selectivity of anticancer drug in vivo. We aim at approaching these measurements through epithelial/deep cells classification, epithelial area and thickness measurement, and density estimation from scattered points. Our procedure is based on Minimal Surfaces, Otsu clustering, Delaunay Triangulation, and Within-R cloud of points density estimation approaches. In this paper, we investigated whether the distance between nuclei and epithelial surface is sufficient to discriminate epithelial cells from deep cells. Comparisons of different density estimators, experimental results, and extensively accuracy measurements are included.

Cells segmentation from 3-D confocal images of early zebrafish embryogenesis

We designed a strategy for extracting the shapes of cell membranes and nuclei from time lapse confocal images taken throughout early zebrafish embryogenesis using a partial-differential-equation-based segmentation. This segmentation step is a prerequisite for an accurate quantitative analysis of cell morphodynamics during embryogenesis and it is the basis for an integrated understanding of biological processes. The segmentation of embryonic cells requires live zebrafish embryos fluorescently labeled to highlight sub-cellular structures and designing specific algorithms by adapting classical methods to image features. Our strategy includes the following steps: the signal-to-noise ratio is first improved by an edge-preserving filtering, then the cell shape is reconstructed applying a fully automated algorithm based on a generalized version of the Subjective Surfaces technique. Finally we present a procedure for the algorithm validation either from the accuracy and the robustness perspective.

Online 3-D reconstruction of the right atrium from echocardiography data via a topographic cellular contour extraction algorithm

A computational method providing online, automated 3-D reconstruction of the right atrium of the human heart is presented in this paper. Endocardial boundaries were extracted from transesophageal ultrasound data by a topographic cellular contour extraction (TCCE) algorithm. The TCCE method was implemented on a continuous-time, analog, massively parallel processor, and on a digital serial processor. Processing speeds were 500 or 40 frames per second, depending on the hardware used. Extracted boundary point sets were rendered into a 3-D mesh and the volume of the cavity was quantified. Accuracy of volume quantification was validated on 60 in vitro static phantoms and 12 clinical subjects. For the clinical recordings, reference volumes were estimated from manually delineated endocardial boundaries. The average error of volume quantification by the TCCE method was 8% +/-5% ( n = 12), the average of the interobserver variability between two independent human experts was 5% +/-2% ( n = 6). Interactive planning of atrial septal defect closure in pediatric cardiology is presented as an example that demonstrates a potential clinical application of the method.

Incorporating temporal information into level set functional for robust ventricular boundary detection from echocardiographic image sequence

Echocardiographic images often suffer from dropouts that lead to loss of signals on the ventricular boundary and cause the level set curve used to detect the boundary leaking out from the gaps on the boundary. In this paper, a novel method that incorporates temporal information into the level set functional is proposed to solve the leakage problem encountered when detecting the heart wall boundary from the echocardiographic image sequence. The ventricular boundary is quantitatively partitioned and classified into strong and weak segments. The weak segments are considered to be weakened by dropouts and there is low confidence on the presence of boundary. Temporal information from neighboring frames is exploited as a regularizer into the level set equation. Hence, the original boundary information in the weak segments can be reconstructed and the curve leakage problem can be remedied. Experimental results demonstrate the advantages of the proposed method for the intended task.

Geometric and potential driving formation and evolution of biomolecular surfaces

This paper presents new geometrical flow equations for the theoretical modeling of biomolecular surfaces in the context of multiscale implicit solvent models. To account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we propose potential driven geometric flows, which balance the intrinsic geometric forces that would occur for a surface separating two homogeneous materials with the potential forces induced by the atomic interactions. Stochastic geometric flows are introduced to account for the random fluctuation and dissipation in density and pressure near the solvent-solute interface. Physical properties, such as free energy minimization (area decreasing) and incompressibility (volume preserving), are realized by some of our geometric flow equations. The proposed approach for geometric and potential forces driving the formation and evolution of biological surfaces is illustrated by extensive numerical experiments and compared with established minimal molecular surfaces and molecular surfaces. Local modification of biomolecular surfaces is demonstrated with potential driven geometric flows. High order geometric flows are also considered and tested in the present work for surface generation. Biomolecular surfaces generated by these approaches are typically free of geometric singularities. As the speed of surface generation is crucial to implicit solvent model based molecular dynamics, four numerical algorithms, a semi-implicit scheme, a Crank-Nicolson scheme, and two alternating direction implicit (ADI) schemes, are constructed and tested. Being either stable or conditionally stable but admitting a large critical time step size, these schemes overcome the stability constraint of the earlier forward Euler scheme. Aided with the Thomas algorithm, one of the ADI schemes is found to be very efficient as it balances the speed and accuracy.

Minimal molecular surfaces and their applications

This article presents a novel concept, the minimal molecular surface (MMS), for the theoretical modeling of biomolecules. The MMS can be viewed as a result of the surface free energy minimization when an apolar molecule, such as protein, DNA or RNA is immersed in a polar solvent. Based on the theory of differential geometry, the MMS is created via the mean curvature minimization of molecular hypersurface functions. A detailed numerical algorithm is presented for the practical generation of MMSs. Extensive numerical experiments, including those with internal and open cavities, are carried out to demonstrated the proposed concept and algorithms. The proposed MMS is typically free of geometric singularities. Application of the MMS to the electrostatic analysis is considered for a set of twenty six proteins.

Segmentation of cells from 3-D confocal images of live zebrafish embryo

In this paper, we use partial-differential-equation-based segmentation to accurately extract the shapes of membranes and nuclei from time lapse confocal microscopy images, taken throughout early Zebrafish embryogenesis. This strategy is a prerequisite for an accurate quantitative analysis of cell shape and morphodynamics during organogenesis and is the basis for an integrated understanding of biological processes. This data will also serve for the measurement of the variability between individuals in a population. The segmentation of cellular structures is achieved by first using an edge-preserving image filtering method for noise reduction and then applying an algorithm for cell shape reconstruction based on the Subjective Surfaces technique.

Ultrasound image segmentation: a survey

This paper reviews ultrasound segmentation paper methods, in a broad sense, focusing on techniques developed for medical B-mode ultrasound images. First, we present a review of articles by clinical application to highlight the approaches that have been investigated and degree of validation that has been done in different clinical domains. Then, we present a classification of methodology in terms of use of prior information. We conclude by selecting ten papers which have presented original ideas that have demonstrated particular clinical usefulness or potential specific to the ultrasound segmentation problem.

Segmentation of real-time three-dimensional ultrasound for quantification of ventricular function: a clinical study on right and left ventricles

Among screening modalities, echocardiography is the fastest, least expensive and least invasive method for imaging the heart. A new generation of three-dimensional (3-D) ultrasound (US) technology has been developed with real-time 3-D (RT3-D) matrix phased-array transducers. These transducers allow interactive 3-D visualization of cardiac anatomy and fast ventricular volume estimation without tomographic interpolation as required with earlier 3-D US acquisition systems. However, real-time acquisition speed is performed at the cost of decreasing spatial resolution, leading to echocardiographic data with poor definition of anatomical structures and high levels of speckle noise. The poor quality of the US signal has limited the acceptance of RT3-D US technology in clinical practice, despite the wealth of information acquired by this system, far greater than with any other existing echocardiography screening modality. We present, in this work, a clinical study for segmentation of right and left ventricular volumes using RT3-D US. A preprocessing of the volumetric data sets was performed using spatiotemporal brushlet denoising, as presented in previous articles Two deformable-model segmentation methods were implemented in 2-D using a parametric formulation and in 3-D using an implicit formulation with a level set implementation for extraction of endocardial surfaces on denoised RT3-D US data. A complete and rigorous validation of the segmentation methods was carried out for quantification of left and right ventricular volumes and ejection fraction, including comparison of measurements with cardiac magnetic resonance imaging as the reference. Results for volume and ejection fraction measurements report good performance of quantification of cardiac function on RT3-D data compared with magnetic resonance imaging with better performance of semiautomatic segmentation methods than with manual tracing on the US data.

Fast evolution of image manifolds and application to filtering and segmentation in 3D medical images

In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme that is fashioned after additive operator split (AOS) [1], [2] for Beltrami and the subjective surface computation. The Beltrami flow [3], [4], [5] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface [6] deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. In this paper, we first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition" the viewpoint surface using the Fast-Marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved.

Evolutionary partial differential equations for biomedical image processing

We are presenting here a model for processing space-time image sequences and applying them to 3D echo-cardiography. The non-linear evolutionary equations filter the sequence with keeping space-time coherent structures. They have been developed using ideas of regularized Perona-Malik an-isotropic diffusion and geometrical diffusion of mean curvature flow type (Malladi-Sethian), combined with Galilean invariant movie multi-scale analysis of Alvarez et al. A discretization of space-time filtering equations by means of finite volume method is discussed in detail. Computational results in processing of 3D echo-cardiographic sequences obtained by rotational acquisition technique and by real-time 3D echo volumetrics acquisition technique are presented. Quantitative error estimation is also provided.