Local structure tensor for multidimensional signal processingApplications to medical image analysis

Feature extraction and, particularly, orientation estimation of multidimensional images is of paramount importance for the Image Processing and Computer Vision communities. This dissertation focuses on this topic; specifically, we deal with the problem of local structure tensor (LST) estimation, as a mean of characterizing the local behavior of a multidimensional signal. The LST can be seen as a measure of the uncertainty of a multidimensional signal with respect to a given orientation.

LST estimation can be achieved by estimating the local energy of a signal in different orientations. Then, the LST is computed as a linear combination of the local energy for each orientation with a tensor basis whose elements are calculated for each orientation. This kind of methods for the estimation of the LST are based on quadrature filters to obtain the local energy of the signal. While the LST based on quadrature filters is well defined for signals that vary locally only in one orientation (simple signals), the estimation method fails with complex signals, i.e. signals that consist of several differently-oriented simple signals. In this dissertation, an analytical study of the distortions of the tensor eigenvalues due to such complex neighborhoods is carried out. From this analytical study, two constructive methods are proposed for the estimation of the LST. The first method is based on a maximum likelihood estimation of the quadrature filter outputs. The second method uses a measure of phase consistency based on generalized quadrature filters which are formally derived from an extension of the analytic signal to multidimensional signals known as the monogenic signal.

The interpretation of a multidimensional image as a function graph, i.e. a Riemannian manifold, instead of just intensity variations on the Euclidean space, has important implications that are exploited in this dissertation. Image processing tasks can then be performed by solving partial differential equations on the Riemannian manifold. In this dissertation, Riemannian geometry is used to study the evolution of fronts under mean curvature flow on a Riemannian manifold using a level set framework. For our purposes, the Riemannian manifold is defined by the induced metric of the image that is related to the LST. The Riemannian mean curvature flow is the theoretical basis for the definition of a level set segmentation method.

The methods proposed in this dissertation are applied to two medical image applications. The first consists in a freehand 3D ultrasound reconstruction technique that uses the LST to perform an adaptive interpolation based on normalized convolution. Our results show that our method outperforms traditional technique for this interpolation problem. The second application uses the level set method based on Riemannian mean curvature flow to segment anatomical structures in dataset from magnetic resonance imaging (MRI), computed tomography (CT) and ultrasound (US). This novel method reveals as a feasible approach to medical image segmentation.

Feature extraction and, particularly, orientation estimation of multidimensional images is of paramount importance for the Image Processing and Computer Vision communities. This dissertation focuses on this topic; specifically, we deal with the problem of local structure tensor (LST) estimation, as a mean of characterizing the local behavior of a multidimensional signal. The LST can be seen as a measure of the uncertainty of a multidimensional signal with respect to a given orientation.

LST estimation can be achieved by estimating the local energy of a signal in different orientations. Then, the LST is computed as a linear combination of the local energy for each orientation with a tensor basis whose elements are calculated for each orientation. This kind of methods for the estimation of the LST are based on quadrature filters to obtain the local energy of the signal. While the LST based on quadrature filters is well defined for signals that vary locally only in one orientation (simple signals), the estimation method fails with complex signals, i.e. signals that consist of several differently-oriented simple signals. In this dissertation, an analytical study of the distortions of the tensor eigenvalues due to such complex neighborhoods is carried out. From this analytical study, two constructive methods are proposed for the estimation of the LST. The first method is based on a maximum likelihood estimation of the quadrature filter outputs. The second method uses a measure of phase consistency based on generalized quadrature filters which are formally derived from an extension of the analytic signal to multidimensional signals known as the monogenic signal.

The interpretation of a multidimensional image as a function graph, i.e. a Riemannian manifold, instead of just intensity variations on the Euclidean space, has important implications that are exploited in this dissertation. Image processing tasks can then be performed by solving partial differential equations on the Riemannian manifold. In this dissertation, Riemannian geometry is used to study the evolution of fronts under mean curvature flow on a Riemannian manifold using a level set framework. For our purposes, the Riemannian manifold is defined by the induced metric of the image that is related to the LST. The Riemannian mean curvature flow is the theoretical basis for the definition of a level set segmentation method.

The methods proposed in this dissertation are applied to two medical image applications. The first consists in a freehand 3D ultrasound reconstruction technique that uses the LST to perform an adaptive interpolation based on normalized convolution. Our results show that our method outperforms traditional technique for this interpolation problem. The second application uses the level set method based on Riemannian mean curvature flow to segment anatomical structures in dataset from magnetic resonance imaging (MRI), computed tomography (CT) and ultrasound (US). This novel method reveals as a feasible approach to medical image segmentation.