Variational Image Segmentation with Constraints

The research of Huizhu Pan addresses the problem of image segmentation with constraints though designing and solving various variational models. A novel constraint term is designed for the use of landmarks in image segmentation. Two region-based segmentation models were proposed where the segmentation contour passes through landmark points. A more stable and memory efficient solution to the self-repelling snakes model, a variational model with the topology preservation constraint, was also designed

Geodesic Models with Convexity Shape Prior

The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with geometric regularization terms, such as Euclidean curve length or curvature-penalized length, for computing geodesic curves. In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting, by which a planar curve can be mapped to an high-dimensional orientation-dependent space. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a particular curvature constraint. Then the geodesic distances and the corresponding closed geodesic paths in the orientation-lifted space can be efficiently computed through state-of-the-art Hamiltonian fast marching method. In addition, we apply the proposed geodesic models to the active contours, leading to efficient interactive image segmentation algorithms that preserve the advantages of convexity shape prior and curvature penalization.Comment: This paper has been accepted by TPAM

Shape/image registration for medical imaging : novel algorithms and applications.

This dissertation looks at two different categories of the registration approaches: Shape registration, and Image registration. It also considers the applications of these approaches into the medical imaging field. Shape registration is an important problem in computer vision, computer graphics and medical imaging. It has been handled in different manners in many applications like shapebased segmentation, shape recognition, and tracking. Image registration is the process of overlaying two or more images of the same scene taken at different times, from different viewpoints, and/or by different sensors. Many image processing applications like remote sensing, fusion of medical images, and computer-aided surgery need image registration. This study deals with two different applications in the field of medical image analysis. The first one is related to shape-based segmentation of the human vertebral bodies (VBs). The vertebra consists of the VB, spinous, and other anatomical regions. Spinous pedicles, and ribs should not be included in the bone mineral density (BMD) measurements. The VB segmentation is not an easy task since the ribs have similar gray level information. This dissertation investigates two different segmentation approaches. Both of them are obeying the variational shape-based segmentation frameworks. The first approach deals with two dimensional (2D) case. This segmentation approach starts with obtaining the initial segmentation using the intensity/spatial interaction models. Then, shape model is registered to the image domain. Finally, the optimal segmentation is obtained using the optimization of an energy functional which integrating the shape model with the intensity information. The second one is a 3D simultaneous segmentation and registration approach. The information of the intensity is handled by embedding a Willmore flow into the level set segmentation framework. Then the shape variations are estimated using a new distance probabilistic model. The experimental results show that the segmentation accuracy of the framework are much higher than other alternatives. Applications on BMD measurements of vertebral body are given to illustrate the accuracy of the proposed segmentation approach. The second application is related to the field of computer-aided surgery, specifically on ankle fusion surgery. The long-term goal of this work is to apply this technique to ankle fusion surgery to determine the proper size and orientation of the screws that are used for fusing the bones together. In addition, we try to localize the best bone region to fix these screws. To achieve these goals, the 2D-3D registration is introduced. The role of 2D-3D registration is to enhance the quality of the surgical procedure in terms of time and accuracy, and would greatly reduce the need for repeated surgeries; thus, saving the patients time, expense, and trauma

Arbitrary Order Total Variation for Deformable Image Registration

In this work, we investigate image registration in a variational framework and focus on regularization generality and solver efficiency. We first propose a variational model combining the state-of-the-art sum of absolute differences (SAD) and a new arbitrary order total variation regularization term. The main advantage is that this variational model preserves discontinuities in the resultant deformation while being robust to outlier noise. It is however non-trivial to optimize the model due to its non-convexity, non-differentiabilities, and generality in the derivative order. To tackle these, we propose to first apply linearization to the model to formulate a convex objective function and then break down the resultant convex optimization into several point-wise, closed-form subproblems using a fast, over-relaxed alternating direction method of multipliers (ADMM). With this proposed algorithm, we show that solving higher-order variational formulations is similar to solving their lower-order counterparts. Extensive experiments show that our ADMM is significantly more efficient than both the subgradient and primal-dual algorithms particularly when higher-order derivatives are used, and that our new models outperform state-of-the-art methods based on deep learning and free-form deformation. Our code implemented in both Matlab and Pytorch is publicly available at https://github.com/j-duan/AOTV

Proceedings of the First International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'06) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceNon-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. Contributions were solicited in the following areas: * Riemannian and group theoretical methods on non-linear transformation spaces * Advanced statistics on deformations and shapes * Metrics for computational anatomy * Geometry and statistics of surfaces 26 submissions of very high quality were recieved and were reviewed by two members of the programm committee. 12 papers were finally selected for oral presentations and 8 for poster presentations. 16 of these papers are published in these proceedings, and 4 papers are published in the proceedings of MICCAI'06 (for copyright reasons, only extended abstracts are provided here)

An evaluation of partial differential equations based digital inpainting algorithms

Partial Differential equations (PDEs) have been used to model various phenomena/tasks in different scientific and engineering endeavours. This thesis is devoted to modelling image inpainting by numerical implementations of certain PDEs. The main objectives of image inpainting include reconstructing damaged parts and filling-in regions in which data/colour information are missing. Different automatic and semi-automatic approaches to image inpainting have been developed including PDE-based, texture synthesis-based, exemplar-based, and hybrid approaches. Various challenges remain unresolved in reconstructing large size missing regions and/or missing areas with highly textured surroundings. Our main aim is to address such challenges by developing new advanced schemes with particular focus on using PDEs of different orders to preserve continuity of textural and geometric information in the surrounding of missing regions. We first investigated the problem of partial colour restoration in an image region whose greyscale channel is intact. A PDE-based solution is known that is modelled as minimising total variation of gradients in the different colour channels. We extend the applicability of this model to partial inpainting in other 3-channels colour spaces (such as RGB where information is missing in any of the two colours), simply by exploiting the known linear/affine relationships between different colouring models in the derivation of a modified PDE solution obtained by using the Euler-Lagrange minimisation of the corresponding gradient Total Variation (TV). We also developed two TV models on the relations between greyscale and colour channels using the Laplacian operator and the directional derivatives of gradients. The corresponding Euler-Lagrange minimisation yields two new PDEs of different orders for partial colourisation. We implemented these solutions in both spatial and frequency domains. We measure the success of these models by evaluating known image quality measures in inpainted regions for sufficiently large datasets and scenarios. The results reveal that our schemes compare well with existing algorithms, but inpainting large regions remains a challenge. Secondly, we investigate the Total Inpainting (TI) problem where all colour channels are missing in an image region. Reviewing and implementing existing PDE-based total inpainting methods reveal that high order PDEs, applied to each colour channel separately, perform well but are influenced by the size of the region and the quantity of texture surrounding it. Here we developed a TI scheme that benefits from our partial inpainting approach and apply two PDE methods to recover the missing regions in the image. First, we extract the (Y, Cb, Cr) of the image outside the missing region, apply the above PDE methods for reconstructing the missing regions in the luminance channel (Y), and then use the colourisation method to recover the missing (Cb, Cr) colours in the region. We shall demonstrate that compared to existing TI algorithms, our proposed method (using 2 PDE methods) performs well when tested on large datasets of natural and face images. Furthermore, this helps understanding of the impact of the texture in the surrounding areas on inpainting and opens new research directions. Thirdly, we investigate existing Exemplar-Based Inpainting (EBI) methods that do not use PDEs but simultaneously propagate the texture and structure into the missing region by finding similar patches within the rest of image and copying them into the boundary of the missing region. The order of patch propagation is determined by a priority function, and the similarity is determined by matching criteria. We shall exploit recently emerging Topological Data Analysis (TDA) tools to create innovative EBI schemes, referred to as TEBI. TDA studies shapes of data/objects to quantify image texture in terms of connectivity and closeness properties of certain data landmarks. Such quantifications help determine the appropriate size of patch propagation and will be used to modify the patch propagation priority function using the geometrical properties of curvature of isophotes, and to improve the matching criteria of patches by calculating the correlation coefficients from the spatial, gradient and Laplacian domains. The performance of this TEBI method will be tested by applying it to natural dataset images, resulting in improved inpainting when compared with other EBI methods. Fourthly, the recent hybrid-based inpainting techniques are reviewed and a number of highly performing innovative hybrid techniques that combine the use of high order PDE methods with the TEBI method for the simultaneous rebuilding of the missing texture and structure regions in an image are proposed. Such a hybrid scheme first decomposes the image into texture and structure components, and then the missing regions in these components are recovered by TEBI and PDE based methods respectively. The performance of our hybrid schemes will be compared with two existing hybrid algorithms. Fifthly, we turn our attention to inpainting large missing regions, and develop an innovative inpainting scheme that uses the concept of seam carving to reduce this problem to that of inpainting a smaller size missing region that can be dealt with efficiently using the inpainting schemes developed above. Seam carving resizes images based on content-awareness of the image for both reduction and expansion without affecting those image regions that have rich information. The missing region of the seam-carved version will be recovered by the TEBI method, original image size is restored by adding the removed seams and the missing parts of the added seams are then repaired using a high order PDE inpainting scheme. The benefits of this approach in dealing with large missing regions are demonstrated. The extensive performance testing of the developed inpainting methods shows that these methods significantly outperform existing inpainting methods for such a challenging task. However, the performance is still not acceptable in recovering large missing regions in high texture and structure images, and hence we shall identify remaining challenges to be investigated in the future. We shall also extend our work by investigating recently developed deep learning based image/video colourisation, with the aim of overcoming its limitations and shortcoming. Finally, we should also describe our on-going research into using TDA to detect recently growing serious “malicious” use of inpainting to create Fake images/videos