Traveling Wave Solutions of Fourth Order PDEs for Image Processing

Greer, John B.; Bertozzi, Andrea L.

doi:10.1137/s0036141003427373

Cited by 82 publications

(56 citation statements)

References 43 publications

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“…Unfortunately, the Perona-Malik equation is ill-posed in continuous space, making the model somewhat theoretically unsatisfying. On the other hand, LCIS preserves the anisotropic diffusion properties of the PeronaMalik model while being globally well-posed [4,16] and making more realistic curvature connections.…”

Section: Fourth-order Inpainting Methodsmentioning

confidence: 99%

Enhancement and recovery in atomic force microscopy images

Chen¹

2012

J Material Sci Eng

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Summary. Atomic force microscopy (AFM) images have become increasingly useful in the study of biological, chemical and physical processes at the atomic level. The acquisition of AFM images takes more time than the acquisition of most optical images, so that the avoidance of unnecessary scanning becomes important. Details that are unclear from a scan may be enhanced using various image processing techniques. This chapter reviews various interpolation and inpainting methods and considers them in the specific application of AFM images. Lower-resolution AFM data is simulated by subsampling the number of scan lines in an image, and reconstruction methods are used to recreate an image on the original domain.The methods considered are classified in the categories of linear interpolation, nonlinear interpolation, and inpainting. These techniques are evaluated based on qualitative and quantitative measures, showing the extent to which scans times can be reduced while preserving the essence of the original features. A further application is in the removal of streaks, which can occur due to scanning errors and post-processing corrections. Identified streaks are removed, and the resulting unknown region is filled using inpainting. IntroductionThe atomic force microscope (AFM) is an extremely high magnification microscope [5]. It achieves its high resolution by moving an atomically sharp probe over surfaces and recording the highly localized interaction force. Isolating specific interaction forces such as electrostatic, magnetic, specific chemical interactions, van der Waals attraction, and Pauli repulsion enable the AFM to measure many surface properties in addition to topography [20,12,22,2,29,17]. The AFM is also able to measure surfaces in any environment from liquids, to corrosive gases and vacuum. The high resolution, versatility, and broad information content make AFM a frequent choice for nanoscience imaging.The current standard method of AFM data collection is the raster scan. The probe starts by traveling along the "fast scan direction," or in the +x direction. As it reaches the end of the scan region, it takes a small step in the "slow can direction," or +y direction, and scans in the −x direction until it retraces the x displacement. Another small step in the +y direction is taken and the scanner moves in the +x direction to initiate another scan line. Continuing in this manner, an image is formed. The backward (retrace) scans are often displayed independently from the forward (trace) scans due to errors in position from scanner nonlinearities and hysteresis. A feedback mechanism maintains the probe-sample interaction at constant force to ensure that the probe is not damaged by contact with the sample.Because the AFM is a local probe it must collect data serially to construct an image over time which can be a significant disadvantage. The sample and probe are massive objects that are difficult to accelerate requiring relatively slow scan velocities otherwise the feedback mechanism that holds the interaction force...

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Section: Fourth-order Inpainting Methodsmentioning

confidence: 99%

Enhancement and recovery in atomic force microscopy images

Chen¹

2012

J Material Sci Eng

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“…As a result, one can bypass the image preprocessing, i.e., the convolution of the noise image with a smooth mask in the application of the PDE operator to noisy images, which is a very tricky process in the application of geometric flows to noisy iamges. High order geometric PDEs have been widely applied to image and surface analysis [4, 13, 14, 28, 29, 34, 53, 59, 66]. Recently, arbitrarily high-order geometric PDEs have been modified for molecular surface formation and evolution [4]

\frac{\partial S}{\partial t} = {false(- 1 false)}^{q} \sqrt{g (| \nabla \nabla^{2 q} S |)} \nabla \cdot true(\frac{\nabla false(\nabla^{2 q} S false)}{\sqrt{g false(false| \nabla \nabla^{2 q} S false| false)}} true) + P false(S, false| \nabla S false| false),

where S is the hypersurface function, g (|∇∇ 2 q S |) = 1 + |∇∇ 2 q S | 2 is the generalized Gram determinant and P is a generalized potential term, including microscopic interactions in biomolecular surface construction.…”

Section: Introductionmentioning

confidence: 99%

“…Over the past decade, the mathematical analysis of high-order geometric PDEs has attracted much attention. For example, Bertozzi and Greer have analyzed fourth order nonlinear PDEs in the Sobolev space and proved the existence and uniqueness of the solution to a case with H 1 initial data and a regularized operator [7, 28, 29]. Xu and Zhou [65] showed the well-posedness of the solution of fourth order nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

High-order fractional partial differential equation transform for molecular surface construction

Chen

Wei

2012

Computational and Mathematical Biophysics

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Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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“…As another remedy to the undesired dissipation, fourth-order PDE models have been emerged [19,26,40]. In particular, the Laplacian mean-curvature (LMC) model has been paid a particular attention due to its potential capability to preserve edges of linear curvatures.…”

Section: Introductionmentioning

confidence: 99%