Steerable Discrete Fourier Transform

Directional transforms have recently raised a lot of interest thanks to their numerous applications in signal compression and analysis. In this letter, we introduce a generalization of the discrete Fourier transform, called steerable DFT (SDFT). Since the DFT is used in numerous fields, it may be of interest in a wide range of applications. Moreover, we also show that the SDFT is highly related to other well-known transforms, such as the Fourier sine and cosine transforms and the Hilbert transforms

Finding a Hamiltonian Path in a Cube with Specified Turns is Hard

We prove the NP-completeness of finding a Hamiltonian path in an N × N × N cube graph with turns exactly at specified lengths along the path. This result establishes NP-completeness of Snake Cube puzzles: folding a chain of N3 unit cubes, joined at face centers (usually by a cord passing through all the cubes), into an N × N × N cube. Along the way, we prove a universality result that zig-zag chains (which must turn every unit) can fold into any polycube after 4 × 4 × 4 refinement, or into any Hamiltonian polycube after 2 × 2 × 2 refinement

Design and Optimization of Graph Transform for Image and Video Compression

The main contribution of this thesis is the introduction of new methods for designing adaptive transforms for image and video compression. Exploiting graph signal processing techniques, we develop new graph construction methods targeted for image and video compression applications. In this way, we obtain a graph that is, at the same time, a good representation of the image and easy to transmit to the decoder. To do so, we investigate different research directions. First, we propose a new method for graph construction that employs innovative edge metrics, quantization and edge prediction techniques. Then, we propose to use a graph learning approach and we introduce a new graph learning algorithm targeted for image compression that defines the connectivities between pixels by taking into consideration the coding of the image signal and the graph topology in rate-distortion term. Moreover, we also present a new superpixel-driven graph transform that uses clusters of superpixel as coding blocks and then computes the graph transform inside each region. In the second part of this work, we exploit graphs to design directional transforms. In fact, an efficient representation of the image directional information is extremely important in order to obtain high performance image and video coding. In this thesis, we present a new directional transform, called Steerable Discrete Cosine Transform (SDCT). This new transform can be obtained by steering the 2D-DCT basis in any chosen direction. Moreover, we can also use more complex steering patterns than a single pure rotation. In order to show the advantages of the SDCT, we present a few image and video compression methods based on this new directional transform. The obtained results show that the SDCT can be efficiently applied to image and video compression and it outperforms the classical DCT and other directional transforms. Along the same lines, we present also a new generalization of the DFT, called Steerable DFT (SDFT). Differently from the SDCT, the SDFT can be defined in one or two dimensions. The 1D-SDFT represents a rotation in the complex plane, instead the 2D-SDFT performs a rotation in the 2D Euclidean space

Bent Hamilton cycles in d-dimensional grid graphs, Electron

1 Introduction The figure below shows two incarnations of a popular "snake " puzzle. The figure represents a flattened view of a series of 27 unit cubes that are held together by a shock cord running from one end to the other. The cubes can rotate at those faces that are held together by the cord. The object of the puzzle is to arrange the snake into a 3 \Theta 3 \Theta 3 cube

Snake cube puzzles : Hamilton paths in grid graphs

This thesis is devoted to the investigation of the mathematics behind a popular puzzle known as the snake cube. The puzzle traditionally consists of a string of 27 small cubes that, when folded correctly, form a larger 3 x 3 x 3 cube. About six different versions are sold commercially, but many more are possible. Each cube that is not a beginning or ending cube is between two other cubes; either these two cubes are adjacent to directly opposite faces, or they are not. If they are, we say that the cube in question is a straight , and if not we say that the cube is a bend . All forms of the puzzle must correspond to at least one Hamilton path through the 3 x 3 x 3 grid graph. A large portion of the work and research in this paper was completed in Professor Atela\u27s MTH 227 class in the Spring semester of 2008. The only previous work done on snake cubes has been concerning bent Hamilton paths and cycles, described by Rusky and Sawada in their paper Bent Hamilton cycles in d-dimensional grid graphs . A bent path consists of entirely bend cubes, and when unfolded will have a zig-zag appearance. This paper describes for which d-dimensional grid graphs a Hamilton cycle or Hamilton path is possible.The Smith Math Department posesses a 5 x 5 x 5 cube which unfolds into a bent Hamilton path, designed and built by previous students of Professor Atela. Of particular interest were cubes that would unfold into various knot configurations. This twist on the traditional snake cube puzzle is, to my knowledge, an original one. A large number of \Hamilton Knots were found during my work in MTH 227. The existance of these cycles through three-dimensional grid graphs suggests further questions. Of any knot configuration, we can ask, What is the smallest cube that can contain such a knot? Can the knot span the entire cube? Section II deals with the material in previously published work, while Sections III and IV are composed entirely of original work