A Grassmann integral equation

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{\bar\Psi},{\Psi}] = G_0[{\lambda\bar\Psi}, {\lambda\Psi}] + const., 0<\lambda\in R (\bar\Psi_k, \Psi_k, k=1,...,n, are the generators of the Grassmann algebra G_2n), between the finite-dimensional analogues G_0 and G of the (``classical'') action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If \lambda \not= 1, solutions to this Grassmann integral equation exist for n=2 (and consequently, also for any even value of n, specifically, for n=4) but not for n=3. If \lambda=1, the considered Grassmann integral equation has always a solution which corresponds to a Gaussian integral, but remarkably in the case n=4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G_2n with arbitrarily chosen n.Comment: 58 pages LaTeX (v2: mainly, minor updates and corrections to the reference section; v3: references [4], [17]-[21], [39], [46], [49]-[54], [61], [64], [139] added

Spatially Adaptive Wiener Filtering For Image Denoising Using Undecimated Wavelet Transform

Recently wavelet thresholding has been a popular approach to the 1-D and 2-D signal (image) denoising. In this work, instead of thresholding the wavelet coefficients, estimation approaches are proposed in the wavelet domain to reduce the noise. The fundamental philosophy is to consider the wavelet coefficients as a stationary random signal. Therefore, an optimal linear mean squared error estimate can be obtained from the corrupted observations. It turns out this is the Wiener filtering. In this paper we propose an FIR approximation approach to the IIR Wiener filter and use it in image denoising. Key words Undecimated wavelet transform, autocorrelation function, linear minimum mean squared error estimation 1 Contents 1 Introduction 2 2 Adaptive Wiener filtering 2 2.1 Wiener filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Adaptive Wiener filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Wavelet domain adaptive..