Karthik S. Gurumoorthy

A Method for Compact Image Representation Using Sparse Matrix and Tensor Projections Onto Exemplar Orthonormal Bases

We present a new method for compact representation of large image datasets. Our method is based on treating small patches from a 2-D image as matrices as opposed to the conventional vectorial representation, and encoding these patches as sparse projections onto a set of exemplar orthonormal bases, which are learned a priori from a training set. The end result is a low-error, highly compact image/patch representation that has significant theoretical merits and compares favorably with existing techniques (including JPEG) on experiments involving the compression of ORL and Yale face databases, as well as a database of miscellaneous natural images. In the context of learning multiple orthonormal bases, we show the easy tunability of our method to efficiently represent patches of different complexities. Furthermore, we show that our method is extensible in a theoretically sound manner to higher-order matrices (¿tensors¿). We demonstrate applications of this theory to compression of well-known color image datasets such as the GaTech and CMU-PIE face databases and show performance competitive with JPEG. Lastly, we also analyze the effect of image noise on the performance of our compression schemes.

A Schrödinger Equation for the Fast Computation of Approximate Euclidean Distance Functions

Computational techniques adapted from classical mechanics and used in image analysis run the gamut from Lagrangian action principles to Hamilton-Jacobi field equations: witness the popularity of the fast marching and fast sweeping methods which are essentially fast Hamilton-Jacobi solvers. In sharp contrast, there are very few applications of quantum mechanics inspired computational methods. Given the fact that most of classical mechanics can be obtained as a limiting case of quantum mechanics (as Plancks constant h tends to zero), this paucity of quantum mechanics inspired methods is surprising. In this work, we derive relationships between nonlinear Hamilton-Jacobi and linear Schrodinger equations for the Euclidean distance function problem (in 1D , 2D and 3D ). We then solve the Schrodinger wave equation instead of the corresponding Hamilton-Jacobi equation. We show that the Schrodinger equation has a closed form solution and that this solution can be efficiently computed in O (N logN ), N being the number of grid points. The Euclidean distance can then be recovered from the wave function. Since the wave function is computed for a small but non-zero h , the obtained Euclidean distance function is an approximation. We derive analytic bounds for the error of the approximation and experimentally compare the results of our approach with the exact Euclidean distance function on real and synthetic data.

The Schrödinger distance transform (SDT) for point-sets and curves

Despite the ubiquitous use of distance transforms in the shape analysis literature and the popularity of fast marching and fast sweeping methods - essentially Hamilton-Jacobi solvers, there is very little recent work leveraging the Hamilton-Jacobi to Schrödinger connection for representational and computational purposes. In this work, we exploit the linearity of the Schrödinger equation to (i) design fast discrete convolution methods using the FFT to compute the distance transform, (ii) derive the histogram of oriented gradients (HOG) via the squared magnitude of the Fourier transform of the wave function, (iii) extend the Schrödinger formalism to cover the case of curves parametrized as line segments as opposed to point-sets, (iv) demonstrate that the Schrödinger formalism permits the addition of wave functions - an operation that is not allowed for distance transforms, and finally (v) construct a fundamentally new Schrödinger equation and show that it can represent both the distance transform and its gradient density - not possible in earlier efforts.

Directly Measuring Material Proportions Using Hyperspectral Compressive Sensing

A compressive sensing framework is described for hyperspectral imaging. It is based on the widely used linear mixing model, LMM, which represents hyperspectral pixels as convex combinations of small numbers of endmember (material) spectra. The coefficients of the endmembers for each pixel are called proportions. The endmembers and proportions are often the sought-after quantities; the full image is an intermediate representation used to calculate them. Here, a method for estimating proportions and endmembers directly from compressively sensed hyperspectral data based on LMM is shown. Consequently, proportions and endmembers can be calculated directly from compressively sensed data with no need to reconstruct full hyperspectral images. If spectral information is required, endmembers can be reconstructed using compressive sensing reconstruction algorithms. Furthermore, given known endmembers, the proportions of the associated materials can be measured directly using a compressive sensing imaging device. This device would produce a multiband image; the bands would directly represent the material proportions.

A Schrödinger Wave Equation Approach to the Eikonal Equation: Application to Image Analysis

As Plancks constant

Beyond SVD: Sparse projections onto exemplar orthonormal bases for compact image representation

\hbar

Distance Transform Gradient Density Estimation Using the Stationary Phase Approximation

(treated as a free parameter) tends to zero, the solution to the eikonal equation

The complex wave representation of distance transforms

|\nabla S(X)|=f(X)

Dynamics of 2-worker bucket brigade assembly line with blocking and instantaneous walk-back

can be increasingly closely approximated by the solution to the corresponding Schrodinger equation. When the forcing function f (X ) is set to one, we get the Euclidean distance function problem. We show that the corresponding Schrodinger equation has a closed form solution which can be expressed as a discrete convolution and efficiently computed using a Fast Fourier Transform (FFT). The eikonal equation has several applications in image analysis, viz. signed distance functions for shape silhouettes, surface reconstruction from point clouds and image segmentation being a few. We show that the sign of the distance function, its gradients and curvature can all be written in closed form, expressed as discrete convolutions and efficiently computed using FFTs. Of note here is that the sign of the distance function in 2D is expressed as a winding number computation. For the general eikonal problem, we present a perturbation series approach which results in a sequence of discrete convolutions once again efficiently computed using FFTs. We compare the results of our approach with those obtained using the fast sweeping method, closed-form solutions (when available) and Dijkstras shortest path algorithm.

Color Image Compression Using a Learned Dictionary of Pairs of Orthonormal Bases

We present a new method for compact representation of large image datasets. Our method is based on treating small patches from an image as matrices as opposed to the conventional vectorial representation, and encoding those patches as sparse projections onto a set of exemplar orthonormal bases, which are learned a priori from a training set. The end result is a low-error, highly compact image/patch representation that has significant theoretical merits and compares favorably with existing techniques on experiments involving the compression of ORL and Yale face databases.