The need to reduce the dimensionality of a dataset whilst retaining its inherent manifold structure is key to many pattern recognition, machine learning, and computer vision problems. This process is often referred to as manifold learning since the structure is preserved during dimensionality reduc tion by learning the intrinsic low-dimensional manifold that the data lies upon. In this paper a heuristic approach is presented to tackle this problem by approximating the manifold as a set of piecewise linear models. By merging these linear models in an order defined by their global topology a globally stable and locally accurate model of the manifold can be obtained. A detailed analysis of the proposed approach is presented along with comparison with existing manifold learning techniques. Results obtained on both artificial and image based data show that in many cases this heuristic approach to manifold learning is able to out-perform traditional techniques.

In mammographic imaging, the presence of microcalcifications, small deposits of calcium in the breast, is a primary indicator of breast cancer. However, not all microcalcifications are malignant and their distribution within the breast can be used to indicate whether clusters of microcalcifications are benign or malignant. Computer-aided diagnosis (CAD) systems can be employed to help classify such microcalcification clusters. In this paper a novel method for classifying microcalcification clusters is presented by representing discrete mereotopological relations between the individual microcalcifications over a range of scales in the form of a mereotopological barcode. This barcode based representation is able to model complex relations between multiple regions and the results on mammographic microcalcification data shows the effectiveness of this approach. Classification accuracies of 95% and 80% are achieved on the MIAS and DDSM datasets, respectively. These results are comparable to existing state-of-the art methods. This work also demonstrates that mereotopological barcodes could be used to help trained clinicians in their diagnosis by providing a clinical interpretation of barcodes that represent both benign and malignant cases.

Group-wise image alignment or image congealing is an image processing technique which allows the joint alignment of a collection of images. Typically, information theoretic metrics have been employed as the objective function for the assessment of the process of alignment of the images for such methods. However, these objective functions rely on their probabilistic foundations and cannot model the underlying vagueness or uncertainty that is captured by approaches such as those based on fuzzy sets. In this paper a novel fuzzy-entropy based approach is presented for the task of image congealing. This approach allows for much flexibility in terms of employing different definitions for both similarity and fuzzy-entropy. Indeed, the existing approach for image congealing is subsumed as a special case of the approach proposed in this paper. The novel fuzzy-entropy congealing technique is applied to different benchmark problems and also to a medical imaging dataset with good results.

Manifold learning is a powerful tool for reducing the dimensionality of a dataset by finding a low-dimensional embedding that retains important geometric and topological features. In many applications it is desirable to add new samples to a previously learnt embedding, this process of adding new samples is known as the out-of-sample extension problem. Since many manifold learning algorithms do not naturally allow for new samples to be added we present an easy to implement generalized solution to the problem that can be used with any existing manifold learning algorithm. Our algorithm is based on simple geometric intuition about the local structure of a manifold and our results show that it can be effectively used to add new samples to a previously learnt embedding. We test our algorithm on both artificial and real world image data and show that our method significantly out performs existing out-of-sample extension strategies.

We present a framework for the reduction of dimensionality of a data set via manifold learning. Using the building blocks of local hyperplanes we show how a global manifold can be reconstructed by iteratively merging these hyperplanes. A Minimum Spanning Tree provides the skeleton needed to traverse the manifold so that the local hyperplanes can be used to build a global, locally stable, embedding. We show state of the art results when compared against existing manifold learning approaches using benchmark synthetic data. We also show how our technique can be used on real world image data.

Manifold learning is a widely used statistical tool which reduces the dimensionality of a data set while aiming to maintain both local and global properties of the data. We present a novel manifold learning technique which aligns local hyper planes to build a global representation of the data. A Minimum Spanning Tree provides the skeleton needed to traverse the manifold so that the local hyper planes can be merged using parallel projections to build a global hyper plane of the data. We show state of the art results when compared against existing manifold learning algorithm on both artificial and real world image data.