Continuous-time liquid neural networks constitute a novel class of machine learning models that emulate the dynamics of biological neurons and synapses using ordinary differential equations. Despite their promising applications in predicting spatiotemporal dynamics, the adoption of these models has been constrained by their reliance on computationally expensive numerical differential equation solvers or approximate solutions. In this work, we propose a redefinition of the network's core neuron that accommodates for multiple presynaptic connections. We then derive a closed-form solution and present an implementation through a computationally efficient recursive algorithm. Our solution is validated both at the level of individual neurons and within a deep neural network architecture. Experimental results on sequential modeling tasks with image, sensor or medical data, demonstrate improved performance compared to state-of-the-art numerical and approximate methods.

We provide a generic framework to learn shape dictionaries of landmark-based curves that are defined in the continuous domain. We first present an unbiased alignment method that involves the construction of a mean shape as well as training sets whose elements are subspaces that contain all affine transformations of the training samples. The alignment relies on orthogonal projection operators that have a closed form. We then present algorithms to learn shape dictionaries according to the structure of the data that needs to be encoded: 1) projection-based functional principal-component analysis for homogeneous data and 2) continuous-domain sparse shape encoding to learn dictionaries that contain imbalanced data, outliers, or different types of shape structures. Through parametric spline curves, we provide a detailed and exact implementation of our method. We demonstrate that it requires fewer parameters than purely discrete methods and that it is computationally more efficient and accurate. We illustrate the use of our framework for dictionary learning of structures in biomedical images as well as for shape analysis in bioimaging.

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

We present a generic method to construct orthogonal projectors for two-dimensional landmark-based parametric spline curves. We construct vector spaces that define a geometric transformation (e.g., affine, similarity, and scaling) that is applied to a reference curve. These vector spaces contain all parametric curves up to the chosen transformation. We define the vector spaces implicitly through an orthogonal projection operator and present a theorem that characterizes the projector for landmark-based spline curves, which are popular for the user-interactive analysis of biomedical images. Finally, we show how shape priors are constructed with the spline projector and provide an example of application for the segmentation of microscopy images in biology.

We present a new family of snakes that satisfy the property of multiresolution by exploiting subdivision schemes. We show in a generic way how to construct such snakes based on an admissible subdivision mask. We derive the necessary energy formulations and provide the formulas for their efficient computation. Depending on the choice of the mask, such models have the ability to reproduce trigonometric or polynomial curves. They can also be designed to be interpolating, a property that is useful in user-interactive applications. We provide explicit examples of subdivision snakes and illustrate their use for the segmentation of bioimages. We show that they are robust in the presence of noise and provide a multiresolution algorithm to enlarge their basin of attraction, which decreases their dependence on initialization compared to singleresolution snakes. We show the advantages of the proposed model in terms of computation and segmentation of structures with different sizes.

Biomedical image segmentation is an active field of research where deformable models have proved to be efficient. The geometric representation of such models determines their ability to approximate the shape of interest as well as the speed of convergence of related optimization algorithms. We present a new tensor-product parameterization of surfaces that offers the possibility of local refinement. The goal is to allocate additional degrees of freedom to the surface only where an increase in local detail is required. We introduce the possibility of locally increasing the number of control points by inserting basis functions at specific locations. Our approach is generic and relies on refinable functions which satisfy the refinement relation. We show that the proposed method improves brain segmentation in 3D MRI images.

Our motivation is the design of efficient algorithms to process closed curves represented by basis functions or wavelets. To that end, we introduce an inner-product calculus to evaluate correlations and L2distances between such curves. In particular, we present formulas for the direct and exact evaluation of correlation matrices in the case of closed (i.e., periodic) parametric curves and periodic signals. We give simplifications for practical cases that involve B-splines. To illustrate this approach, we also propose a least-squares approximation scheme that is able to resample curves while minimizing aliasing artifacts. Another application is the exact calculation of the enclosed area.

Shape segmentation is an active field of research in biomedical imaging. In this context, we present a new parameterization of a snake that is locally refinable. We introduce the possibility of locally increasing the approximation power of the parametric model by inserting basis functions at a specific location. This is controlled by a user-interface that permits the refinement of an initial segmentation around an anchor position selected by a user. Our approach relies on scaling functions that satisfy the refinement relation and are related to wavelets. We also derive explicit formulas for the energy functions associated to our new parameterization. We demonstrate the accuracy of our snake and its robustness under noisy conditions on phantom data. We also present segmentation results on real cell images, which are our main target. The algorithm is made freely available as a plugin for the open source platform Icy.

We propose a new parametric 3D snake with cylindrical topology. Its construction is based on interpolatory basis functions which facilitates user-interaction because the control points of the snake directly lie on the surface of the deformable cylinder. We prove that the basis functions exactly reproduce a cylinder and propose a new parametrization as a tensor-product spline surface. We provide explicit formulas for the energy function based on Green's theorem that speed up the computation of the optimization algorithm. We have implemented the proposed framework as a freely available open-source plugin for the bioimaging platform Icy. Its utility has been tested on phantom data as well as on real 3D data to segment the spinal cord and the descending aorta.

We present a new trigonometric basis function that is capable of perfectly reproducing circles, spheres and ellipsoids while at the same time being interpolatory. Such basis functions have the advantage that they allow to construct shapes through a sequence of control points that lie on their contour (2-D) or surface (3-D) which facilitates user-interaction, especially in 3-D. Our piecewise exponential basis function has finite support, which enables local control for shape modification. We derive and prove all the necessary properties of the kernel to represent shapes that can be smoothly deformed and show how idealized shapes such as ellipses and spheres can be constructed.

Parametric active contours are an attractive approach for image segmentation, thanks to their computational efficiency. They are driven by application-dependent energies that reflect the prior knowledge on the object to be segmented. We propose an energy involving shape priors acting in a regularization-like manner. Thereby, the shape of the snake is orthogonally projected onto the space that spans the affine transformations of a given shape prior. The formulation of the curves is continuous, which provides computational benefits when compared with landmark-based (discrete) methods. We show that this approach improves the robustness and quality of spline-based segmentation algorithms, while its computational overhead is negligible. An interactive and ready-to-use implementation of the proposed algorithm is available and was successfully tested on real data in order to segment Drosophila flies and yeast cells in microscopic images.