Biomedical subjects

K G Larkin

Publications and source records attributed to K G Larkin.

5 recordsLinked to original sources

PubMed2001

Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform.

It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.

Journal Article↗

PubMed2001

Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform.

Utilizing the asymptotic method of stationary phase, I derive expressions for the Fourier transform of a two-dimensional fringe pattern. The method assumes that both the amplitude and the phase of the fringe pattern are well-behaved differentiable functions. Applying the limits in two distinct ways, I show, first, that the spiral phase (or vortex) transform approaches the ideal quadrature transform asymptotically and, second, that the approximation errors increase with the relative curvature of the fringes. The results confirm the validity of the recently proposed spiral phase transform method for the direct demodulation of closed fringe patterns.

Journal Article↗

PubMed2001

Wigner function for nonparaxial wave fields.

The generalized optical transfer function and the spectral correlation function are investigated for nonparaxial two-dimensional wave fields. The angle-impact marginal of the four-dimensional Wigner function is derived directly. For focused wave fields of semiangle greater than 90 degrees, the spectral correlation function exhibits overlapping and interference. For focused wave fields for which the semiangle is known to be less than 180 degrees, the magnitude and phase can be recovered directly from knowledge of the intensity in the focal region.

Journal Article↗

PubMed2000

Using the Hilbert transform for 3D visualization of differential interference contrast microscope images.

Differential interference contrast (DIC) is frequently used in conventional 2D biological microscopy. Our recent investigations into producing a 3D DIC microscope (in both conventional and confocal modes) have uncovered a fundamental difficulty: namely that the phase gradient images of DIC microscopy cannot be visualized using standard digital image processing and reconstruction techniques, as commonly used elsewhere in microscopy. We discuss two approaches to the problem of preparing gradient images for 3D visualization: integration and the Hilbert transform. After applying the Hilbert transform, the dataset can then be visualized in 3D using standard techniques. We find that the Hilbert transform provides a rapid qualitative pre-processing technique for 3D visualization for a wide range of biological specimens in DIC microscopy, including chromosomes, which we use in this study.

Algorithms↗

PubMed2000

Focal shift, optical transfer function, and phase-space representations.

The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner distribution, and the ambiguity function are described, leading to a model for effects of Fresnel number in terms of a rotation in phase space. The uncertainty principle is discussed, including the significance of the beam propagation factor M2 and the width of optical fiber beam modes. Calculation of the moments in terms of the modulus and the phase of the illuminating wave is presented, and the use of the Kaiser-Teager energy operator is also described.

Fourier Analysis↗