Kedar Khare

Sampling theory approach to prolate spheroidal wavefunctions

We use the Whittaker–Shannon sampling theorem to show that the eigenvalue problem for the sinc-kernel is equivalent to a discrete eigenvalue problem. The well-known eigenfunctions, namely, the prolate spheroidal wavefunctions, their corresponding eigenvalues and the orthogonality and completeness properties are determined without invoking the prolate spheroidal differential equation. This analysis based on the sampling theorem may be used for calculating the eigenvalues and eigenfunctions of bandlimited kernels in general as we illustrate with an additional example of the sinc2-kernel.

Direct coarse sampling of electronic holograms

We describe a series of experiments in order to test a new sampling criterion for wavefront reconstruction from a carrier-frequency signal as obtained in electronic holography. Both a Fourier-transform configuration and a Fresnel-zone form of a digital holographic microscope are described. Direct coarse sampling of holograms with no phase-shifting elements can be used to obtain wavefront reconstruction, with the number of samples required reduced by the ratio f0/(2Bs), where f0 is the carrier frequency and 2Bs is the signal bandwidth.

Direct sampling and demodulation of carrier-frequency signals

We show that a generalized carrier-frequency signal can be directly and efficiently coarse-sampled based upon the notion of the space-bandwidth product. Using an envelope function form, we treat both amplitude-modulation and frequency-modulation cases simultaneously. We derive exact formulas for the sampled carrier-frequency signal and for the sampled envelope, showing that it is possible to recover the envelope from the coarsely sampled carrier without conventional mixing or heterodyning. This alternative to the conventional demodulation methods is illustrated with numerical examples including phase retrieval. Extension to the two-dimensional case is also included.

Sampling-theory approach to eigenwavefronts of imaging systems

We present a direct method based on the sampling theorem for computing eigenwavefronts associated with linear space-invariant imaging systems (including aberrated imaging systems). A potential application of the eigenwavefronts to inverse problems in imaging is discussed. A noise-dependent measure for the information-carrying capacity of an imaging system is also proposed.

Fractional finite Fourier transform

We show that a fractional version of the finite Fourier transform may be defined by using prolate spheroidal wave functions of order zero. The transform is linear and additive in its index and asymptotically goes over to Namiass definition of the fractional Fourier transform. As a special case of this definition, it is shown that the finite Fourier transform may be inverted by using information over a finite range of frequencies in Fourier space, the inversion being sensitive to noise. Numerical illustrations for both forward (fractional) and inverse finite transforms are provided.

Eigenwavefronts for Image Restoration

Eigenwavefronts are complex wavefronts that pass through a given imaging system unchanged apart from a constant multiplier. This paper is aimed at studying an application of eigenwavefronts to the problem of image restoration.