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In the first part of this dissertation, we address the problem of representing 2D and 3D shapes. In particular, we introduce a novel implicit shape representation based on Support Vector Machine (SVM) theory. Each shape is represented by an analytic decision function obtained by training an SVM, with a Radial Basis Function (RBF) kernel, so that the interior shape points are given higher values. This empowers support vector shape (SVS) with multifold advantages. First, the representation uses a sparse subset of feature points determined by the support vectors, which significantly improves the discriminative power against noise, fragmentation and other artifacts that often come with the data. Second, the use of the RBF kernel provides scale, rotation, and translation invariant features, and allows a shape to be represented accurately regardless of its complexity. Finally, the decision function can be used to select reliable feature points. These features are described using gradients computed from highly consistent decision functions instead of conventional edges. Our experiments on 2D and 3D shapes demonstrate promising results.

The availability of inexpensive 3D sensors like Kinect necessitates the design of new representation for this type of data. We present a 3D feature descriptor that represents local topologies within a set of folded concentric rings by distances from local points to a projection plane. This feature, called as Concentric Ring Signature (CORS), possesses similar computational advantages to point signatures yet provides more accurate matches. CORS produces compact and discriminative descriptors, which makes it more robust to noise and occlusions.

It is also well-known to computer vision researchers that there is no universal representation that is optimal for all types of data or tasks. Sparsity has proved to be a good criterion for working with natural images. This motivates us to develop efficient sparse and non-linear learning techniques for automatically extracting useful information from visual data. Specifically, we present dictionary learning methods for sparse and redundant representations in a high-dimensional feature space. Using the kernel method, we describe how the well-known dictionary learning approaches such as the method of optimal directions and KSVD can be made non-linear. We analyse their kernel constructions and demonstrate their effectiveness through several experiments on classification problems. It is shown that non-linear dictionary learning approaches can provide significantly better discrimination compared to their linear counterparts and kernel PCA, especially when the data is corrupted by different types of degradations.

Visual descriptors are often high dimensional. This results in high computational complexity for sparse learning algorithms.

Motivated by this observation, we introduce a novel framework, called sparse embedding (SE), for simultaneous dimensionality reduction and dictionary learning. We formulate an optimization problem for learning a transformation from the original signal domain to a lower-dimensional one in a way that preserves the sparse structure of data. We propose an efficient optimization algorithm and present its non-linear extension based on the kernel methods.

One of the key features of our method is that it is computationally efficient as the learning is done in the lower-dimensional space and it discards the irrelevant part of the signal that derails the dictionary learning process. Various experiments show that our method is able to capture the meaningful structure of data and can perform significantly better than many competitive algorithms on signal recovery and object classification tasks.

In many practical applications, we are often confronted with the situation where the data that we use to train our models are different from that presented during the testing. In the final part of this dissertation, we present a novel framework for domain adaptation using a sparse and hierarchical network (DASH-N), which makes use of the old data to improve the performance of a system operating on a new domain. Our network jointly learns a hierarchy of features together with transformations that rectify the mismatch between different domains. The building block of DASH-N is the latent sparse representation. It employs a dimensionality reduction step that can prevent the data dimension from increasing too fast as traversing deeper into the hierarchy. Experimental results show that our method consistently outperforms the current state-of-the-art by a significant margin. Moreover, we found that a multi-layer {DASH-N} has an edge over the single-layer DASH-N.