In this paper we discuss a novel patch-based framework for image denoising through local geometric representations
of an image. We learn local data adaptive bases that best capture the underlying geometric information
from noisy image patches. To do so we first identify regions of similar structure in the given image and group
them together. This is done by the use of meaningful features in the form of local kernels that capture similarities
between pixels in a neighborhood. We then learn an informative basis (called a dictionary) for each
cluster that best describes the patches in the cluster. Such a data representation can be achieved by performing
a simple principal component analysis (PCA) on the member patches of each cluster. The number of principal
components to consider in a particular cluster is dictated by the underlying geometry captured by the cluster
and the strength of the corrupting noise. Once a dictionary is defined for a cluster, each patch in the cluster is
denoised by expressing it as a linear combination of the dictionary elements. The coefficients of such a linear
combination for any particular patch is determined in a regression framework using the local dictionary for the
cluster. Each step of our method is well motivated and is shown to minimize some cost function. We then
present an iterative extension of our algorithm that results in further performance gain. We validate our method
through experiments with simulated as well as real noisy images. These indicate that our method is able to
produce results that are quantitatively and qualitatively comparable to those obtained by some of the recently
proposed state of the art denoising techniques.
The Non-Local Means (NLM) method of denoising has received considerable attention in the image processing
community due to its performance, despite its simplicity. In this paper, we show that NLM is a zero-th order
kernel regression method, with a very specific choice of kernel. As such, it can be generalized. The original
method of NLM, we show, implicitly assumes local constancy of the underlying image data. Once put in the
context of kernel regression, we extend the existing Non-Local Means algorithm to higher orders of regression
which allows us to approximate the image data locally by a polynomial or other localized basis of a given order.
These extra degrees of freedom allow us to perform better denoising in texture regions. Overall the higher order
method displays consistently better denoising capabilities compared to the zero-th order method. The power of
the higher order method is amply illustrated with the help of various denoising experiments.
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