Image analysis by topological gradient approach is a technique based upon the historic application of the topological asymptotic expansion to crack localization problem from boundary measurements. This paper aims at reviewing this methodology through various applications in image processing; in particular image restoration with edge detection, classification and segmentation problems for both grey level and color images is presented in this work. The numerical experiments show the efficiency of the topological gradient approach for modelling and solving different image analysis problems. However, the topological gradient approach presents a major drawback: the identified edges are not connected and then the results obtained particularly for the segmentation problem can be degraded. To overcome this inconvenience, we propose an alternative solution by combining the topological gradient approach with the watershed technique. The numerical results obtained using the coupled method are very interesting.

The goal of topological optimization is to find the optimal decomposition of a given domain in two parts: the optimal design and its complementary. Similarly in image processing, the goal is to split an image in several parts, in particular, in image restoration the detection of edges makes this operation straightforward.

The goal of this work is to show how it is possible to solve some image analysis problems using topological optimization tools. Let us recall that the basic and first idea is based on the work of Amstutz et al. [

Let

Let

We recall that a classical way to restore an image

The goal of this section is to use the topological gradient approach as a tool for detecting edges for image restoration [

Edge detection is equivalent to look for a subdomain of

Initialization:

Calculation of

Computation of the

Set

Calculation of

For numerical experiments, we considered, in a first step, grey level images (see Figure

(a) Initial

Initialization:

Calculation of

Computation of the

Set

Calculation of

In Figure

(a) zoom of the noisy Lena image (

We consider now color images such that the RGB model is used. Then, the image

Some other coupling techniques have been defined in order to process multichannel images. We can cite for instance the Beltrami diffusion-type process, in which images are considered as manifolds. A nonlinear structure tensor is defined and locally adjusted according to a gradient-type measure and allows one to control the diffusion process [

For the numerical tests, we first consider in Figure

(a) original image, (b) noisy image (

Then to improve the restoration process, we propose in Figure

(a) the initial

We must note here that, for the case corresponding to the classical topological gradient method, the solution

On the other hand, for the case corresponding to the topological gradient method using the Di Zenzo gradient, a multispectral tensor associated to the image vector field is considered and the largest eigenvalue of the tensor corresponds to the norm of the gradient, called the Di Zenzo gradient [

For

This section is concerned with the problem of classifying an original image using

Initialization:

Calculation of

Computation of the

Set

Calculation of

Application of the k-mean classification algorithm to

As in the restoration process, the classification algorithm is applied to color images by decomposing the image in the RGB color space and then we deal separately with the three component images. The topological gradient of the sum can be considered as the sum of the three topological gradients with respect to each channel. Figure

From (a) The unregularized classified image and and (b)the regularized classified image.

On the other hand, the unsupervised classification corresponds to a classification problem in which the classes are not given. In this case, it is possible to determine them in an optimal way, still by using the topological gradient method. The idea is to study the impact of changing the value of a class

Initialization with

Determination of the optimal value.

While

set

Stop if the value of the cost function at the optimum (with respect to

This algorithm has been applied in Figure

In this section, we use the topological gradient as a tool for the reconstruction problem in tomography. We recall that a standard approach for regularizing the ill-posed problem of tomographic imaging consists in the following optimization problem:

Let

Comparison between several reconstruction methods, quantified by several ratios and similarity indexes, for a noisy sinogram with

Method | PSNR | SNR | SSIM | MSE |
---|---|---|---|---|

FBP | 14.59 | 33.0 | 0.41 | 0.0342 |

FBP + Hamming | 15.91 | 31.71 | 0.52 | 0.0273 |

Topological gradient (TV) | 22.43 | 33.75 | 0.82 | 0.0042 |

Topological gradient ( | 26.18 | 34.01 | 0.94 | 0.0023 |

Evolution of the PSNR (in dB) of the reconstructed image by the topological gradient method, as a function of the level of the regularization coefficient

Reconstruction of the Shepp-Logan head phantom using the topological gradient method.

Original image

Noised sinogram (

Reconstructed image (

We propose in this part a new algorithm for the segmentation problem based on the topological gradient approach. Through the different and various applications which were presented, we can easily conclude that the topological gradient approach is a very interesting method in image analysis. First, the topological gradient method can easily be applied to many problems in image analysis. Second, we should mention that we obtain excellent numerical results and the computing time is very interesting and confirms the theoretical complexity

Segmentation using a watershed transformation combined with the topological gradient: (a) is a

Finally, due to a large number of approaches used for the segmentation problem, it clearly appears that it is important to compare our experimental results with methods already proposed in the literature. Particularly, we propose to compare our method with an active contour model based on the level set approach, which is well known to be an efficient method, extensively used in many applications over the last decade. The basic idea in active contour models is to evolve a curve subject to constraints from a given image, in order to detect different objects in that image. To achieve this goal, we start with a curve around the object to be detected, the curve moves toward its interior normal and has to stop on the boundary of the object. This was the first idea of classical snakes and active contour models proposed by [

Segmentation results of a synthetic image using an active contour model: different iterations are displayed from (b) to (e), (f) is the segmented image using our new approach.

We presented in this paper many applications in image processing of the topological gradient approach with an illustration of both advantages and inconveniences of this technique. We proved that the topological gradient approach provides an excellent frame for solving different image processing problems, same if an alternative way to overcome its major drawback to give not connected contours is to combine it with other approaches. It has been applied to image restoration, edge detection, image classification, and image segmentation for both grey level and color images. It has also been applied to tomography problem, and our main goal consists in applying it to other real life problems mainly that in all the applications proposed, we obtained good results and the computing time is very short. We propose in a forthcoming paper to perform the application of the topological gradient approach to image segmentation, eventually by combining it with other approaches and particularly using marker criteria. In fact, as the edges are detected in regions where the topological gradient is the most negative, then it suffices to extract some points which belong to the edge set and then use these points as a selected marker set. We also intend to extend this work to color image segmentation and three-dimensional segmentation.