Many physical phenomena are described by first-order differential equations whose solution is an exponential decay. Determining the time constants and amplitudes of exponential decays from the experimental data is a common task in semiconductor physics (deep level transient spectroscopy), biophysics (fluorescence decay analysis), nuclear physics and chemistry (radioactive decays, nuclear magnetic resonance), chemistry and electrochemistry (reaction kinetics) and medical imaging. This review article discusses the fundamental mathematical limitations of exponential analysis, outlines the critical aspects of acquisition of exponential transients for subsequent analysis, and gives a comprehensive overview of numerical algorithms used in exponential analysis. In the first part of the article the resolution of exponential analysis as a function of noise in input decays is discussed. It is shown that two exponential decays can be resolved in a transient only if the ratio of their time constants is greater than t...

Exponential analysis in physical phenomena

Speckle noise is a ubiquitous artifact that limits the interpretation of optical coherence tomography images. Here we apply various speckle-reduction digital filters to optical coherence tomography images and compare their performance. Our results indicate that shift-invariant, nonorthogonal wavelet-transform-based filters together with enhanced Lee and adaptive Wiener filters can significantly reduce speckle and increase the signal-to-noise ratio, while preserving strong edges. The speckle reduction capabilities of these filters are also compared with speckle reduction from incoherent angular compounding. Our results suggest that by using these digital filters, the number of individual angles required to attain a certain level of speckle reduction can be decreased.

Speckle reduction in optical coherence tomography images using digital filtering.

Structural transitions in the tertiary structure of plasmid DNA have been investigated using atomic force microscopy. Changes in superhelical stress were induced by ethidium bromide intercalation, and conformational effects monitored by recording topographic images from DNA complexes of various ethidium bromide:base pair stoichiometry. Significant changes in the tertiary structure of individual DNA molecules were observed with increasing ethidium bromide concentration. The first distinct conformational transition was from a predominantly relaxed structure to one consisting solely of toroidal supercoils. A further increase in ethidium bromide concentration resulted in the formation of regions of plectonemic supercoiling. The ratio of plectonemic:toroidal supercoiling gradually increased until an extremely tightly interwound structure of solely plectonemic supercoiling was finally adopted. The toroidal form of supercoiling observed in this study is unusual as both atomic force microscopy and electron microscopy techniques have previously shown that plectonemic supercoiling is the predominant form adopted by plasmid DNA.

Atomic force microscopy studies of intercalation-induced changes in plasmid DNA tertiary structure.

We present a method for removing noise from scanning tunneling microscopy images based on least-squares fitting of spatial data. Modeling the known structure of the surface, including isolated features and surface steps, allows for effective discrimination of signal from noise and produces minimal processing artifacts, even for very noisy images. This approach is effective for removing external noise due to vibrational or acoustic interference, and can also be applied to correct for tip-related height jumps as well as to flatten images warped by thermal effects or nonlinearity of the microscope scanner.

Minimizing image-processing artifacts in scanning tunneling microscopy using linear-regression fitting

Tikhonov's regularization approach applied to image restoration, stated in terms of ill-posed problems, has proved to be a powerful tool to solve noisy and incomplete data. This work proposes a variable norm discrepancy function as the regularization term of a Tikhonov expression, where the cross-entropy functional was derived. Our method was applied to true Atomic Force Microscopy (AFM) images obtained from biological samples, producing satisfactory results towards the most probable sample morphological aspect. These images represent a mapping of local interaction forces exerted between a reduced scaled AFM sensing tip and the biological sample surface, kept alive in aqueous or air environment.

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A generalized approach for atomic force microscopy image restoration with bregman distances as tikhonov regularization terms

During the acquisition of an image from any probe microscope instrument, various noise sources cause distortion in the observed image. It is often the case that impulsive disturbances cause bright groups of pixels to replace the actual image data in these locations. Furthermore, the images from a probe microscope show some amount of blurring caused both by the instrument function and the material properties. In almost all image-processing applications it is important to remove any impulsive distortion that may be present before deblurring can be attempted. We give a technique for detecting these impulses and reconstructing the image. This technique is superior to the standard global application of median filters for the case considered. The reconstruction is limited only to the affected regions and therefore results in a much sharper and more meaningful image. With the assumption of Gaussian blur it is then possible to propose several different deblurring methodologies. We present a novel Wiener-filter deblurring implementation and compare it to both maximum-entropy and Richardson–Lucy deblurring.

N. Persad

Papers

Restoration of images from the scanning-tunneling microscope.