Compressed Detection via Manifold Learning

In many imaging applications such as Computed Tomography (CT) in medical imaging and Synthetic Aperture Radar (SAR) imaging, the collected raw data, residing in R , of the receiver or detector can be modeled as data lying in the Fourier space of the target reflectivity function. The magnitude of the reflectivity is considered as the image of the target, and we are often interested in detecting specific features in the image, e.g. tumor in medical imaging and military weapons in SAR imaging. A natural way to achieve this goal is to form an image in R , where N > M , and detect the feature in the spatial domain. Recent development of a theory in [1] states that under certain conditions, random linear projections, Φ : R 7→ R, guarantees, with high probability, that all pairwise Euclidean and geodesic distances between points on M ⊂ R are well preserved under the mapping Φ. This made us wonder if the geodesic distances of the original image, R , and the corresponding raw data, R , in SAR imaging were also reasonably preserved. Motivated by satisfactory results of simple tests to check this, which will be discussed in more detail later, we tried to detect features directly from the lower dimensional, or compressed, SAR raw data, without involving any image reconstruction step. In this report, manifold learning techniques are applied to reduce the dimension of the raw data, and is followed by a detection step to achieve our goal. The theoretical framework will be discussed later in more detail. To our best knowledge, there has not yet been a successful image detection or classification algorithm that takes advantage of this framework and works directly on the domain where the raw data resides in. Since existing algorithms work in the spatial domain, the algorithms first have to transform the raw data to the spatial domain. This transformation step could be time consuming, and in general results in loss of information. Also, working with M dimensional data will be computationally less demanding. Thus, it is interesting to look at the problem of working directly on the raw data. The rest of the report is structured as follows. First, technical background such as the theoretical framework mentioned above, the Laplacian Eigenmaps as a method of dimensionality reduction, and semi-supervised learning algorithm will be discussed. Carrying on, experimental results will be presented. Finally, we will conclude this report by giving discussion on the challenges and future works.