20,923 research outputs found
Variational and Partial Differential Equation Models for Color Image Denoising and Their Numerical Approximations using Finite Element Methods
Image processing has been a traditional engineering field, which has a broad range of applications in science, engineering and industry. Not long ago, statistical and ad hoc methods had been main tools for studying and analyzing image processing problems. In the past decade, a new approach based on variational and partial differential equation (PDE) methods has emerged as a more powerful approach. Compared with old approaches, variational and PDE methods have remarkable advantages in both theory and computation. It allows to directly handle and process visually important geometric features such as gradients, tangents and curvatures, and to model visually meaningful dynamic process such as linear and nonlinear diffusions. Computationally, it can greatly benefit from the existing wealthy numerical methods for PDEs.
Mathematically, a (digital) greyscale image is often described by a matrix and each entry of the matrix represents a pixel value of the image and the size of the matrix indicates the resolution of the image. A (digital) color image is a digital image that includes color information for each pixel. For visually acceptable results, it is necessary (and almost sufficient) to provide three color channels for each pixel, which are interpreted as coordinates in some color space. The RGB (Red, Green, Blue) color space is commonly used in computer displays. Mathematically, a RGB color image is described by a stack of three matrices so that each color pixel value of the RGB color image is represented by a three-dimensional vector consisting values from the RGB channels. The brightness and chromaticity (or polar) decomposition of a color image means to write the three-dimensional color vector as the product of its length, which is called the brightness, and its direction, which is defined as the chromaticity. As a result, the chromaticity must lie on the unit sphere S2 in R3.
The primary objectives of this thesis are to present and to implement a class of variational and PDE models and methods for color image denoising based on the brightness and chromaticity decomposition. For a given noisy digital image, we propose to use the well-known Total Variation (TV) model to denoise its brightness and to use a generalized p-harmonic map model to denoise its chromaticity. We derive the Euler-Lagrange equations for these models and formulate the gradient descent method (in the name of gradient flows) for computing the solutions of these equations. We then formulate finite element schemes for approximating the gradient flows and implement these schemes on computers using Matlab® and Comsol Multiphysics® software packages. Finally, we propose some generalizations of the p-harmonic map model, and numerically compare these models with the well-known channel-by-channel model
Spectral proper orthogonal decomposition
The identification of coherent structures from experimental or numerical data
is an essential task when conducting research in fluid dynamics. This typically
involves the construction of an empirical mode base that appropriately captures
the dominant flow structures. The most prominent candidates are the
energy-ranked proper orthogonal decomposition (POD) and the frequency ranked
Fourier decomposition and dynamic mode decomposition (DMD). However, these
methods fail when the relevant coherent structures occur at low energies or at
multiple frequencies, which is often the case. To overcome the deficit of these
"rigid" approaches, we propose a new method termed Spectral Proper Orthogonal
Decomposition (SPOD). It is based on classical POD and it can be applied to
spatially and temporally resolved data. The new method involves an additional
temporal constraint that enables a clear separation of phenomena that occur at
multiple frequencies and energies. SPOD allows for a continuous shifting from
the energetically optimal POD to the spectrally pure Fourier decomposition by
changing a single parameter. In this article, SPOD is motivated from
phenomenological considerations of the POD autocorrelation matrix and justified
from dynamical system theory. The new method is further applied to three sets
of PIV measurements of flows from very different engineering problems. We
consider the flow of a swirl-stabilized combustor, the wake of an airfoil with
a Gurney flap, and the flow field of the sweeping jet behind a fluidic
oscillator. For these examples, the commonly used methods fail to assign the
relevant coherent structures to single modes. The SPOD, however, achieves a
proper separation of spatially and temporally coherent structures, which are
either hidden in stochastic turbulent fluctuations or spread over a wide
frequency range
A new nonlocal nonlinear diffusion equation for image denoising and data analysis
In this paper we introduce and study a new feature-preserving nonlinear
anisotropic diffusion for denoising signals. The proposed partial differential
equation is based on a novel diffusivity coefficient that uses a nonlocal
automatically detected parameter related to the local bounded variation and the
local oscillating pattern of the noisy input signal. We provide a mathematical
analysis of the existence of the solution of our nonlinear and nonlocal
diffusion equation in the two dimensional case (images processing). Finally, we
propose a numerical scheme with some numerical experiments which demonstrate
the effectiveness of the new method
Discrete spherical means of directional derivatives and Veronese maps
We describe and study geometric properties of discrete circular and spherical
means of directional derivatives of functions, as well as discrete
approximations of higher order differential operators. For an arbitrary
dimension we present a general construction for obtaining discrete spherical
means of directional derivatives. The construction is based on using the
Minkowski's existence theorem and Veronese maps. Approximating the directional
derivatives by appropriate finite differences allows one to obtain finite
difference operators with good rotation invariance properties. In particular,
we use discrete circular and spherical means to derive discrete approximations
of various linear and nonlinear first- and second-order differential operators,
including discrete Laplacians. A practical potential of our approach is
demonstrated by considering applications to nonlinear filtering of digital
images and surface curvature estimation
An Image Morphing Technique Based on Optimal Mass Preserving Mapping
©2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TIP.2007.896637Image morphing, or image interpolation in the time domain, deals with the metamorphosis of one image into another. In this paper, a new class of image morphing algorithms is proposed based on the theory of optimal mass transport. The 2 mass moving energy functional is modified by adding an intensity penalizing term, in order to reduce the undesired double exposure effect. It is an intensity-based approach and, thus, is parameter free. The optimal warping function is computed using an iterative gradient descent approach. This proposed morphing method is also extended to doubly connected domains using a harmonic parameterization technique, along with finite-element methods
- …