Affine Registration of label maps in Label Space

Two key aspects of coupled multi-object shape\ud analysis and atlas generation are the choice of representation\ud and subsequent registration methods used to align the sample\ud set. For example, a typical brain image can be labeled into\ud three structures: grey matter, white matter and cerebrospinal\ud fluid. Many manipulations such as interpolation, transformation,\ud smoothing, or registration need to be performed on these images\ud before they can be used in further analysis. Current techniques\ud for such analysis tend to trade off performance between the two\ud tasks, performing well for one task but developing problems when\ud used for the other.\ud This article proposes to use a representation that is both\ud flexible and well suited for both tasks. We propose to map object\ud labels to vertices of a regular simplex, e.g. the unit interval for\ud two labels, a triangle for three labels, a tetrahedron for four\ud labels, etc. This representation, which is routinely used in fuzzy\ud classification, is ideally suited for representing and registering\ud multiple shapes. On closer examination, this representation\ud reveals several desirable properties: algebraic operations may\ud be done directly, label uncertainty is expressed as a weighted\ud mixture of labels (probabilistic interpretation), interpolation is\ud unbiased toward any label or the background, and registration\ud may be performed directly.\ud We demonstrate these properties by using label space in a gradient\ud descent based registration scheme to obtain a probabilistic\ud atlas. While straightforward, this iterative method is very slow,\ud could get stuck in local minima, and depends heavily on the initial\ud conditions. To address these issues, two fast methods are proposed\ud which serve as coarse registration schemes following which the\ud iterative descent method can be used to refine the results. Further,\ud we derive an analytical formulation for direct computation of the\ud "group mean" from the parameters of pairwise registration of all\ud the images in the sample set. We show results on richly labeled\ud 2D and 3D data sets

A comprehensive analysis of the geometry of TDOA maps in localisation problems

In this manuscript we consider the well-established problem of TDOA-based source localization and propose a comprehensive analysis of its solutions for arbitrary sensor measurements and placements. More specifically, we define the TDOA map from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space. We then study the identifiability of the model, giving a complete analytical characterization of the image of this map and its invertibility. This analysis has been conducted in a completely mathematical fashion, using many different tools which make it valid for every sensor configuration. These results are the first step towards the solution of more general problems involving, for example, a larger number of sensors, uncertainty in their placement, or lack of synchronization.Comment: 51 pages (3 appendices of 12 pages), 12 figure

Towards Reliable and Accurate Global Structure-from-Motion

Reconstruction of objects or scenes from sparse point detections across multiple views is one of the most tackled problems in computer vision. Given the coordinates of 2D points tracked in multiple images, the problem consists of estimating the corresponding 3D points and cameras\u27 calibrations (intrinsic and pose), and can be solved by minimizing reprojection errors using bundle adjustment. However, given bundle adjustment\u27s nonlinear objective function and iterative nature, a good starting guess is required to converge to global minima. Global and Incremental Structure-from-Motion methods appear as ways to provide good initializations to bundle adjustment, each with different properties. While Global Structure-from-Motion has been shown to result in more accurate reconstructions compared to Incremental Structure-from-Motion, the latter has better scalability by starting with a small subset of images and sequentially adding new views, allowing reconstruction of sequences with millions of images. Additionally, both Global and Incremental Structure-from-Motion methods rely on accurate models of the scene or object, and under noisy conditions or high model uncertainty might result in poor initializations for bundle adjustment. Recently pOSE, a class of matrix factorization methods, has been proposed as an alternative to conventional Global SfM methods. These methods use VarPro - a second-order optimization method - to minimize a linear combination of an approximation of reprojection errors and a regularization term based on an affine camera model, and have been shown to converge to global minima with a high rate even when starting from random camera calibration estimations.This thesis aims at improving the reliability and accuracy of global SfM through different approaches. First, by studying conditions for global optimality of point set registration, a point cloud averaging method that can be used when (incomplete) 3D point clouds of the same scene in different coordinate systems are available. Second, by extending pOSE methods to different Structure-from-Motion problem instances, such as Non-Rigid SfM or radial distortion invariant SfM. Third and finally, by replacing the regularization term of pOSE methods with an exponential regularization on the projective depth of the 3D point estimations, resulting in a loss that achieves reconstructions with accuracy close to bundle adjustment

Segmentation of Fault Networks Determined from Spatial Clustering of Earthquakes

We present a new method of data clustering applied to earthquake catalogs, with the goal of reconstructing the seismically active part of fault networks. We first use an original method to separate clustered events from uncorrelated seismicity using the distribution of volumes of tetrahedra defined by closest neighbor events in the original and randomized seismic catalogs. The spatial disorder of the complex geometry of fault networks is then taken into account by defining faults as probabilistic anisotropic kernels, whose structures are motivated by properties of discontinuous tectonic deformation and previous empirical observations of the geometry of faults and of earthquake clusters at many spatial and temporal scales. Combining this a priori knowledge with information theoretical arguments, we propose the Gaussian mixture approach implemented in an Expectation-Maximization (EM) procedure. A cross-validation scheme is then used and allows the determination of the number of kernels that should be used to provide an optimal data clustering of the catalog. This three-steps approach is applied to a high quality relocated catalog of the seismicity following the 1986 Mount Lewis ($M_l=5.7$) event in California and reveals that events cluster along planar patches of about 2 km$^2$, i.e. comparable to the size of the main event. The finite thickness of those clusters (about 290 m) suggests that events do not occur on well-defined euclidean fault core surfaces, but rather that the damage zone surrounding faults may be seismically active at depth. Finally, we propose a connection between our methodology and multi-scale spatial analysis, based on the derivation of spatial fractal dimension of about 1.8 for the set of hypocenters in the Mnt Lewis area, consistent with recent observations on relocated catalogs

Image Based View Synthesis

This dissertation deals with the image-based approach to synthesize a virtual scene using sparse images or a video sequence without the use of 3D models. In our scenario, a real dynamic or static scene is captured by a set of un-calibrated images from different viewpoints. After automatically recovering the geometric transformations between these images, a series of photo-realistic virtual views can be rendered and a virtual environment covered by these several static cameras can be synthesized. This image-based approach has applications in object recognition, object transfer, video synthesis and video compression. In this dissertation, I have contributed to several sub-problems related to image based view synthesis. Before image-based view synthesis can be performed, images need to be segmented into individual objects. Assuming that a scene can approximately be described by multiple planar regions, I have developed a robust and novel approach to automatically extract a set of affine or projective transformations induced by these regions, correctly detect the occlusion pixels over multiple consecutive frames, and accurately segment the scene into several motion layers. First, a number of seed regions using correspondences in two frames are determined, and the seed regions are expanded and outliers are rejected employing the graph cuts method integrated with level set representation. Next, these initial regions are merged into several initial layers according to the motion similarity. Third, the occlusion order constraints on multiple frames are explored, which guarantee that the occlusion area increases with the temporal order in a short period and effectively maintains segmentation consistency over multiple consecutive frames. Then the correct layer segmentation is obtained by using a graph cuts algorithm, and the occlusions between the overlapping layers are explicitly determined. Several experimental results are demonstrated to show that our approach is effective and robust. Recovering the geometrical transformations among images of a scene is a prerequisite step for image-based view synthesis. I have developed a wide baseline matching algorithm to identify the correspondences between two un-calibrated images, and to further determine the geometric relationship between images, such as epipolar geometry or projective transformation. In our approach, a set of salient features, edge-corners, are detected to provide robust and consistent matching primitives. Then, based on the Singular Value Decomposition (SVD) of an affine matrix, we effectively quantize the search space into two independent subspaces for rotation angle and scaling factor, and then we use a two-stage affine matching algorithm to obtain robust matches between these two frames. The experimental results on a number of wide baseline images strongly demonstrate that our matching method outperforms the state-of-art algorithms even under the significant camera motion, illumination variation, occlusion, and self-similarity. Given the wide baseline matches among images I have developed a novel method for Dynamic view morphing. Dynamic view morphing deals with the scenes containing moving objects in presence of camera motion. The objects can be rigid or non-rigid, each of them can move in any orientation or direction. The proposed method can generate a series of continuous and physically accurate intermediate views from only two reference images without any knowledge about 3D. The procedure consists of three steps: segmentation, morphing and post-warping. Given a boundary connection constraint, the source and target scenes are segmented into several layers for morphing. Based on the decomposition of affine transformation between corresponding points, we uniquely determine a physically correct path for post-warping by the least distortion method. I have successfully generalized the dynamic scene synthesis problem from the simple scene with only rotation to the dynamic scene containing non-rigid objects. My method can handle dynamic rigid or non-rigid objects, including complicated objects such as humans. Finally, I have also developed a novel algorithm for tri-view morphing. This is an efficient image-based method to navigate a scene based on only three wide-baseline un-calibrated images without the explicit use of a 3D model. After automatically recovering corresponding points between each pair of images using our wide baseline matching method, an accurate trifocal plane is extracted from the trifocal tensor implied in these three images. Next, employing a trinocular-stereo algorithm and barycentric blending technique, we generate an arbitrary novel view to navigate the scene in a 2D space. Furthermore, after self-calibration of the cameras, a 3D model can also be correctly augmented into this virtual environment synthesized by the tri-view morphing algorithm. We have applied our view morphing framework to several interesting applications: 4D video synthesis, automatic target recognition, multi-view morphing

Ordinal depth from SFM and its application in robust scene recognition

Ph.DDOCTOR OF PHILOSOPH