Local Approximations in Signal and Image Processing
Local Approximations in Signal and Image Processing (LASIP) is a project dedicated to investigations in a wide class of novel efficient adaptive signal processing
techniques. Statistical methods for restoration from noisy and blurred observations of one-dimensional signals,
images, 3D microscopy, and video were recently developed.
We propose effective adaptive solutions for signal reconstruction problems based mainly on combining two
independent nonparametric estimation ideas: the local polynomial approximation (LPA) and
the intersection of confidence intervals (ICI) rule.
The LPA is a technique which is applied for nonparametric estimation using a polynomial data fit in a sliding window.
The ICI rule is a criterion used for the adaptive selection of the size (scale) of this window.
The resulting LPA-ICI estimators are nonlinear filters which are adaptive to the unknown smoothness of the signal.
The local polynomial approximation is originated from an old idea known under different names: moving (sliding, windowed) least-square, Savitzky-Golay filter, moment filters, reproducing kernels, singular convolution kernels, etc.
However, combined with the new adaptation technique it becomes a novel powerful tool.
The window size, interpreted also as scale, is the key parameter of this technique. The terms “window size”, “bandwidth”, and “scale” are interchangeable here.
The idea of the ICI scale-adaptation is as follows. The algorithm searches for a largest local vicinity of the point of estimation where the local polynomial approximation assumptions fit well to the data. The estimates are calculated for a number of different scales and compared. The adaptive scale is defined as the largest for which the estimate does not differ significantly from the estimates corresponding to the smaller scales.
The ICI rule defines the adaptive scale for each point (pixel, voxel) of the signal. In this way, we arrive to a pointwise-adaptive signal and image processing.
The resulting adaptive estimator is always nonlinear even for the linear local polynomial approximation as the nonlinearity of the method is incorporated in the ICI rule itself.
Asymptotically, these adaptive estimators allow to get a near-optimal quality of the signal recovery.
These new methods can be exploited as independent tools as well as jointly with conventional techniques, such as maximum likelihood and quasi-likelihood.
The anisotropic implementation of the LPA-ICI, based on the use of multi-directional kernels, gives further improvement to the adaptivity of the method, providing an efficient tool especially for image denoising, differentiation and inverse-imaging problems.
The new approach and new algorithms are mainly illustrated for image processing applications. However, they are quite general in nature and can be applied to multidimensional data.
Experiments demonstrate the state-of-art performance of the new algorithms which on many occasions visually and quantitatively outperform the best existing methods.
Historically, the nonparametric regression is a predecessor of wavelets.
In its modern development in the area of adaptive estimation this technique demonstrates tremendous new methods almost unknown to signal processing community dominated by the wavelet paradigm.
A web-presentation introducing the basic ideas of the LPA-ICI technique and showing its application to various kinds of image restoration problems is available here.
LASIP is also a set of MATLAB routines for signal and image processing.
They implement a recent new development in the area of statistical scale-adaptive local approximation techniques.
LASIP provides flexible tools for the design of filters equipped with scale (window size) parameters. Directional filters can also be designed.
The adaptivity of these filters is enabled by special statistical rules for a pointwise-adaptive selection of the scale values.
The multidirectional versions these filters are especially efficient for anisotropic data.
The main algorithms are prepared as demos, so that they can be executed in a straightforward manner. These demos reproduce figures and results from the publications by the authors of the LASIP project and their collaborators.
All the provided demos are open-source, and may be modified and tuned to be exploited with other data.
In this sense, we completely support the principle of reproducible research:
“An article about computational science in a scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures”[Buckheit & Donoho, 1995].
The LASIP routines are available free-of-charge for educational and non-profit scientific research, enabling others researchers to understand and reproduce our work.
Any unauthorized use of the LASIP routines for industrial or profit-oriented activities is expressively prohibited. Please read the LASIP limited license before you proceed with downloading the files.
The LASIP routines can be downloaded as four self-contained sets:
We propose a novel image denoising strategy based on an enhanced sparse representation in transform-domain. The enhancement of the sparsity is achieved by grouping similar 2D image fragments (e.g. blocks) into 3D data arrays which we call "groups". Collaborative filtering is a special procedure developed to deal with these 3D groups. We realize it using the three successive steps: 3D transformation of 3D group, shrinkage of transform spectrum, and inverse 3D transformation. The result is a 3D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions.
The Block-Matching and 3D Filtering (BM3D) algorithm is a computationally scalable algorithm based on this novel denoising strategy.
It achieves state-of-the-art denoising performance in terms of both peak signal-to-noise ratio and subjective visual quality.
We introduce a new approach to image reconstruction from highly incomplete data. The available data are assumed to be a small collection of spectral coefficients of an arbitrary linear transform. This reconstruction problem is the subject of intensive study in the recent field of “compressed sensing” (also known as “compressive sampling”). Our approach is based on a quite specific recursive filtering procedure. At every iteration the algorithm is excited by injection of random noise in the unobserved portion of the spectrum and a spatially adaptive image denoising filter, working in the image domain, is exploited to attenuate the noise and reveal new features and details out of the incomplete and degraded observations. This recursive algorithm can be interpreted as a special type of the Robbins-Monro stochastic approximation procedure with regularization enabled by a spatially adaptive filter. Overall, we replace the conventional parametric modeling used in compressed sensing by a nonparametric one.
We apply the local polynomial approximation (LPA) in order to estimate the absolute phase as the argument of cos and sin.
The LPA is a nonparametric regression technique with pointwise estimation in a sliding window. Using the intersection of
confidence interval (ICI) algorithm the window size is selected as adaptive pointwise varying. This adaptation gives the
phase estimate with the accuracy close to optimal in mean squared sense. For calculation we use a Gauss-Newton recursive
procedure initiated by the phase estimates obtained for the neighboring points. This initialization enables tracking
properties of the algorithm and its ability to go beyond the principal interval [0,2pi ) and to reconstruct the absolute
phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values. The algorithm
demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the
state-of-the-art algorithms developed for noisy phase unwrap. The theoretical analysis produced for the accuracy of the
pointwise estimates is used for justification of the ICI adaptation rule.
We consider a complex-valued wavefield reconstruction in an object plane from data in a sensor plane. A digital modeling for the forward propagation is presented in the algebraic matrix form M-DDT. The inverse propagation is formalized as an inverse problem.
The main algorithm of the complex-valued wavefield reconstruction with the proposed Matrix Discrete Diffraction Transform (M-DDT) is prepared as a Matlab demo.