Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Using Commodity Graphics Hardware for Real-Time Digital Hologram View-Reconstruction

View-reconstruction and display is an important part of many applications in digital holography such as computer vision and microscopy. Thus far, this has been an offline procedure for megapixel sized holograms. This paper introduces an implementation of real-time view-reconstruction using programmable graphics hardware. The theory of Fresnel-based view-reconstruction is introduced, after which an implementation using stream programming is presented. Two different fast Fourier transform (FFT)-based reconstruction methods are implemented, as well as two different FFT strategies. The efficiency of the methods is evaluated and compared to a CPU-based implementation, providing over 100 times speedup for a hologram size of 2048 x 2048

Using Commodity Graphics Hardware for Real-Time Digital Hologram View-Reconstruction

View-reconstruction and display is an important part of many applications in digital holography such as computer vision and microscopy. Thus far, this has been an offline procedure for megapixel sized holograms. This paper introduces an implementation of real-time view-reconstruction using programmable graphics hardware. The theory of Fresnel-based view-reconstruction is introduced, after which an implementation using stream programming is presented. Two different fast Fourier transform (FFT)-based reconstruction methods are implemented, as well as two different FFT strategies. The efficiency of the methods is evaluated and compared to a CPU-based implementation, providing over 100 times speedup for a hologram size of 2048 x 2048

Automated Three-Dimensional Microbial Sensing and Recognition Using Digital Holography and Statistical Sampling

We overview an approach to providing automated three-dimensional (3D) sensing and recognition of biological micro/nanoorganisms integrating Gabor digital holographic microscopy and statistical sampling methods. For 3D data acquisition of biological specimens, a coherent beam propagates through the specimen and its transversely and longitudinally magnified diffraction pattern observed by the microscope objective is optically recorded with an image sensor array interfaced with a computer. 3D visualization of the biological specimen from the magnified diffraction pattern is accomplished by using the computational Fresnel propagation algorithm. For 3D recognition of the biological specimen, a watershed image segmentation algorithm is applied to automatically remove the unnecessary background parts in the reconstructed holographic image. Statistical estimation and inference algorithms are developed to the automatically segmented holographic image. Overviews of preliminary experimental results illustrate how the holographic image reconstructed from the Gabor digital hologram of biological specimen contains important information for microbial recognition