We study an inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion. By incorporating edges found at multiple scales via an adaptive edge detector-based indicator function, the proposed scheme removes noise while respecting salient boundaries. We create a smooth transition region around probable edges found and reduce the diffusion rate near it by a gradient-based diffusion coefficient. In contrast to the previous anisotropic diffusion schemes, we prove the well-posedness of our scheme in the space of bounded variation. The proposed scheme is general in the sense that it can be used with any of the existing diffusion equations. Numerical simulations on noisy images show the advantages of our scheme when compared to other related schemes.

Anisotropic diffusion-based image denoising [

There are efforts to remedy the ill-posedness and to use better edge indicators in the diffusion process. Strong [

Apart from staircasing artifacts in flat regions and noise amplification along edges, capturing multiscale edges are difficult in previous schemes. To circumvent the drawbacks a more stable and robust edge indicator function using the multiscale edge detectors as discussed in [

The rest of the paper is organized as follows. In Section

The diffusion function

Image gradients based edge indicators. (a) Original image. (b) Noisy image. (c)

Instead of using only the edge indicator function

at edge pixels to reduce the diffusion:

in flat regions to allow smoothing:

near edge pixels

We choose a term of the form

Various edge detectors [

convolution of the given image

estimating the second derivative

again using convolution for

thresholding the gradient of Step

zero-crossings of Step

Canny edge map

Canny edge detector-based indicator function. (a) Canny binary output

One can use any of the mentioned edge detectors into the spatially adaptive term

Let

There exist

An important example in this class is the minimal surface function

(a) Diffusion functions

Results of (a) Perona and Malik scheme [

PM [

CL [

RO [

ST [

CD [

KH [

YW [

BB [

Our scheme

Let

The total variation is in fact a Radon measure; see [

The function space

Let the regularization function

Let

Note that

Let

Consider the approximation PDE

Since

To get the

The weak solution of the approximation PDE

The proof of the inequality follows from the identity

From Lemma

Let

There exists a subsequence

By Lemma

Since

Now, we state and prove our main result.

If

From Lemma

As a consequence of Theorem

We use finite differences to discretize the proposed edge adaptive PDE (

We use the Canny edge detector with its default settings for (

We use the peak signal-to-noise ratio (PSNR) for comparing our scheme with other schemes. The parameters are tuned to get the best possible PSNR value for each of the scheme compared. Figure

(a) Noise level

A class of well-posed inhomogeneous diffusion schemes for image denoising is studied in this paper. By integrating multiscale edge detectors into the divergence term of anisotropic diffusion PDE we obtain edge preserving restoration of noisy images. Unlike the classical anisotropic diffusion schemes, which use only gradients to find edge pixels, we have proposed to include an adaptive parameter computed from Canny edge detector. Using an approximation scheme and the theory of monotone operators, well-posedness of the proposed inhomogeneous PDE is proved in the space of functions of bounded variation. Numerical examples illustrate that the scheme performs well under noise. Comparison with related schemes is undertaken to show that the expected improvement can really be achieved. Similar analysis for higher-order diffusion schemes and comparing various edge detectors for performance evaluation provide future directions.

The authors thank the anonymous reviewers for their comments which improved the content and presentation of the paper and the handling editor Professor Malgorzata Peszynska for her patience during the review period.