We propose a new method that is aimed at denoising images having textures. The method combines a balanced nonlinear partial differential equation driven by optimal parameters, mathematical morphology operators, weighting techniques, and some recent works in harmonic analysis. Furthermore, the new scheme decomposes the observed image into three components that are well defined as structure/cartoon, texture, and noise-background. Experimental results are provided to show the improved performance of our method for the texture-preserving denoising problem.

A major topic in the image processing community is concerning the extraction of features. Some of the current research has been aimed at decomposing an image into various auxiliary images, each representing a specific set of characteristics such as edges, contours, structure, noise, texture, among others. In this context, an important application is the denoising problem, where many models assume that a noisy image

In most cases, when an image has only few regions defined by texture, the term

In order to overcome these problems there is even a third strand of studies that considers a combination of the mentioned techniques allied with recent approaches of harmonic analysis, as in [

To solve this problem, in this paper we propose a methodology capable of restoring a noisy image having a high concentration of textures and fine details. The proposed model not only keeps the oscillatory well-placed characteristics in the image but also maintains more sensitive textures like intrinsic contours, and edges. Moreover, motivated by [

The proposed scheme seeks to combine the ideas described in [

The remainder of the paper is outlined as follows: in Section

Let

Furthermore, we assumed that the original image

The objective here is to minimize the noise level of the input image

Let

Figure

Illustrative diagram of the proposed algorithm.

Details of step 1. (a) Input image

Details of step 2. (a) Input image

In the first step of the proposed method, the idea is to decompose the initial image

In this step, the algorithm aims to smooth the observed image

An alternative to implement this step of the algorithm is to use any model that is capable of a good image smoothing, such as [

The justification to use the nonlinear PDE proposed in [

The numerical algorithm used to implement this step of our scheme follows from [

The purpose of this step is to remove noise from component

Generally, the diffusivity term

With the intent to automate the computation of (

The equation (

The first great advantage of using the nonlinear model (

The second great advantage is that there are only two parameters in the numerical solution of the model given in [

In [

Encouraged by the authors of [

To implement the numerical equation (

That is one of the most important steps of the proposed method, because it is in it that we extract the oriented texture and most of the oscillatory details of the image. To do this, we use a recent study of wavelet variants presented in [

Wave atoms are a variant obtained through a 2D wavelet packet obeying the important parabolic scaling relation

Compared to other transforms, wave atoms have two great advantages: the ability to arbitrarily adapt in localities defined by a certain pattern and the ability to sparsely represent anisotropic patterns aligned with the axes. Wave atoms composition elements have a high direction sensibility and anisotropy, which makes them ideal to apply wherever the intention is to identify regions characterized by oscillatory patterns such as texture, as is the case here presented.

In the following, based on [

Consider wave atoms given by

To construct wave atoms for our problem, we first considered the case of a family

Considering

Thus, we can write functions that make up the base as

According [

For the denoising problem, it is recommendable to use wave atom shrinkage, which is formulated in most cases by

In this work we use wave atom shrinkage to extract oriented texture from image component

The implementation of our wave-atoms shrinkage transform consists of the execution of three steps. First, we apply wave atom transform

The main advantage of methods based on wavelet variants is space-frequency localization and multiscale view of the features of surfaces. However, it is known that traditional wavelets are not good to analyze surfaces with “scratches” or textures, due to wavelets ignoring properties defined by geometric features of edges and textures, which leads to strong oscillation along these “scratches”.

In contrast, curvelet transforms, such as [

Curvelets are good for representing edges while wave atoms are good for representing oscillatory patterns and textures. Wave atom texture-shape elements not only capture the coherence along the oscillations like curvelets but also take into consideration patterns across the oscillations (see Figures

Elements of the curvelets and wave atoms. (a) A digital curvelet [

Details of step 3. (a) Input image

Details of step 4. (a) Input image

Details of step 5. (a) Input image

Details of final step 6. (a) Input image

In this step of the process, the method is to produce a fuzzy representation (in

For the treatment of the image

According to [

In this step of the algorithm, the objective is to emphasize the texture and simultaneously remove heterogeneous parts of the image. In such case, we opted to use a Top-hat transform. Precisely, we applied transformation (

To finish this step, we convert the preprocessed image to the interval of shades of gray

This step is aimed to synthesize final component

To do this, the idea is to combine auxiliary components generated in steps

Here, the proposed idea is very similar to that used by the diffusivity term

This highlighting among pixels does not contribute for a variation in the range of the input image

Because of having applied the closing top-hat transform (

The last step of the propose scheme consists of obtaining the recovered image

As component

The characterizing noise is done by calculating the residue between the restored image

Finally, besides generating reconstructed image

Now we present some experiments obtained by our scheme, where images in a grayscale defined in the standard interval

In order to validity our approach with respect to the tested methods, we used the statistical measure PSNR (peak signal-to-noise ratio), which is measured in dB.

In the first step of the algorithm, we use (

In Section

In the following, we show two experiments done on images having different levels of complexity: a highly-detailed real image and one of fingerprint.

Our first experiment mentions the real image of Barbara. Here the image contaminated with noise (

Decomposition into three components. (a) Observed image

In the second experiment we take a synthetic image of fingerprint having a considerable noise level (

Decomposition into three components. (a) Noisy image

Restoring the image

To attest to the good performance of the proposed method, we compared it to recent models in literature. Parameters adopted in each of the models tested were chosen according to the best visual quality obtained from each one of those models, in addition to computation of PSNR between the original image and the compared image. Classical models that remove noise but do not cover treating texture were not considered.

Figure

Comparison to models existing in literature. (a) Noisy image, (b) denoising by curvelets, (c) by wave atoms, (d) by adaptive fidelity term, (e) by nonlinear diffusion combined with curvelet, and (f) by proposed scheme.

Components removed using each of the methods. (a) Residual by curvelets, (b) by wave atoms, (c) by adaptive fidelity term, (d) by nonlinear diffusion combined with curvelet, and (e) by proposed scheme.

In this work we gather important mathematical techniques for image processing and we combine these techniques to generate an efficient algorithm for decomposition and noise removal in image processing. The new method has as its aim treating images contaminated with noise, having a high texture concentration, intrinsic contours, and irregular patterns. The scheme combines elementary techniques, such as classic morphological operators, with more sophisticated harmonic analysis models, such as wave atoms. Moreover, the scheme has, in some of its processing steps, a best parameter automatic selector. Based on the proposed method, we propose an efficient decomposition standard to separate the observed image into three well-defined components, as was shown previously. One of the advantages of this type of decomposition is that there is possible, from the degraded image, the individual treatment of each component, which allows a range of image processing applications such as image segmentation and digital inpainting. Experimental tests show the efficiency of the new method, even when compared to recent harmonic analysis techniques and to models based on nonlinear diffusion for the processing of images with texture.

The authors thank the São Paulo State Research Foundation (FAPESP) and the Brazilian Commission for Higher Education (CAPES) for financial support. They also thank Alagacone Sri Ranga and the unknown referees for suggestions on the improvement of the paper.