@article{yi_krim_2014, title={Subspace Learning of Dynamics on a Shape Manifold: A Generative Modeling Approach}, volume={23}, ISSN={["1941-0042"]}, DOI={10.1109/tip.2014.2358200}, abstractNote={In this paper, we propose a novel subspace learning algorithm of shape dynamics. Compared to the previous works, our method is invertible and better characterizes the nonlinear geometry of a shape manifold while retaining a good computational efficiency. In this paper, using a parallel moving frame on a shape manifold, each path of shape dynamics is uniquely represented in a subspace spanned by the moving frame, given an initial condition (the starting point and starting frame). Mathematically, such a representation may be formulated as solving a manifold-valued differential equation, which provides a generative modeling of high-dimensional shape dynamics in a lower dimensional subspace. Given the parallelism and a path on a shape manifold, the parallel moving frame along the path is uniquely determined up to the choice of the starting frame. With an initial frame, we minimize the reconstruction error from the subspace to shape manifold. Such an optimization characterizes well the Riemannian geometry of the manifold by imposing parallelism (equivalent as a Riemannian metric) constraints on the moving frame. The parallelism in this paper is defined by a Levi-Civita connection, which is consistent with the Riemannian metric of the shape manifold. In the experiments, the performance of the subspace learning is extensively evaluated using two scenarios: 1) how the high dimensional geometry is characterized in the subspace and 2) how the reconstruction compares with the original shape dynamics. The results demonstrate and validate the theoretical advantages of the proposed approach.}, number={11}, journal={IEEE TRANSACTIONS ON IMAGE PROCESSING}, author={Yi, Sheng and Krim, Hamid}, year={2014}, month={Nov}, pages={4907–4919} }
 @inproceedings{yi_krim_norris_2012, title={Human activity modeling as Brownian motion on shape manifold}, volume={6667}, DOI={10.1007/978-3-642-24785-9_53}, abstractNote={In this paper we propose a stochastic modeling of human activity on a shape manifold. From a video sequence, human activity is extracted as a sequence of shape. Such a sequence is considered as one realization of a random process on shape manifold. Then Different activities are modeled by manifold valued random processes with different distributions. To solve the problem of stochastic modeling on a manifold, we first regress a manifold values process to a Euclidean process. The resulted process then could be modeled by linear models such as a stationary incremental process and a piecewise stationary incremental process. The mapping from manifold to Euclidean space is known as a stochastic development. The idea is to parallelly transport the tangent along curve on manifold to a single tangent space. The advantage of such technique is the one to one correspondence between the process in Euclidean space and the one on manifold. The proposed algorithm is tested on database [5] and compared with the related work in [5]. The result demonstrate the high accuracy of our modeling in characterizing different activities.}, booktitle={Scale space and variational methods in computer vision}, author={Yi, S. and Krim, H. and Norris, L. K.}, year={2012}, pages={628–639} }
 @inproceedings{yi_krim_norris_2011, title={A invertible dimension reduction of curves on a manifold}, DOI={10.1109/iccvw.2011.6130412}, abstractNote={In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential geometry tools such as moving frame and parallel transportation are successfully adapted into the dimension reduction problem of high dimensional curves. Intuitively, instead of searching for a global flat subspace for curve embedding, we deployed a sequence of local flat subspaces adaptive to the geometry of both of the curve and the manifold it lies on. In practice, the experimental results of the dimension reduction and reconstruction algorithms well illustrate the advantages of the proposed theoretical innovation.}, booktitle={2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops)}, author={Yi, S. and Krim, H. and Norris, L. K.}, year={2011} }
 @article{yi_labate_easley_krim_2009, title={A Shearlet Approach to Edge Analysis and Detection}, volume={18}, ISSN={["1941-0042"]}, DOI={10.1109/TIP.2009.2013082}, abstractNote={It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions.}, number={5}, journal={IEEE TRANSACTIONS ON IMAGE PROCESSING}, author={Yi, Sheng and Labate, Demetrio and Easley, Glenn R. and Krim, Hamid}, year={2009}, month={May}, pages={929–941} }