Section: New Results

Variational models

Detection of filaments in 2D and 3D images

Keywords : filament, segmentation, variational method, Ginzburg-Landau.

Participants : Alexis Baudour, Laure Blanc-Féraud.

This Ph.D. is co-supervised by Gilles Aubert, professor of the J.-A. Dieudonné Mathematics Laboratory of the University of Nice Sophia Antipolis [http://math1.unice.fr/ ]. It is performed as part of the ANR Detecfine and the CNRS/Math STIC.

Our work currently deals with filament spotting in skeletons of cells. These filaments are detected by locally approximating the 2D or 3D image function by its second order Taylor polynomial. The direction of the filament is determined from the Hessian matrix of the Taylor polynomial. The results obtained are shown in figure 19 . This first method is not sufficiently robust in the presence of noise. Thus we have to complete the missing parts of the filaments.

We use the Ginzburg-Landau model (from physics) to find the filaments, inspired by its use in the inpainting of singularities with codimension greater than one in an image. The filaments are the zero-level set of a complex-valued function defined on 3D space that minimizes an energy derived from the Ginzburg-Landau model. We use variational methods to find this minimum.

Figure 19. Detection of filaments.

We also use the Ginzburg-Landau energy to complete triple junctions (see figure 20 ).

Figure 20. Completion in a triple junction case.

Some applications of the norm in image processing

Keywords : l ∞ -norm, total variation, bounded noise, duality, quantization noise, compression noise, projected subgradient descent.

Participants : Pierre Weiss, Laure Blanc-Féraud.

This Ph.D. is co-supervised by Gilles Aubert, professor of the J.-A. Dieudonné Mathematics Laboratory of the University of Nice Sophia Antipolis [http://math1.unice.fr/ ].

The goal of this work is to propose algorithms for minimizing generic regularizing functionals under an -constraint [28] .

We show that many models using Total Variation can be stated in this formalism, among others, the Rudin-Osher-Fatemi model, the BV-l1 model, the BV- model, and Y. Meyer's cartoon-plus-texture decomposition model.

Another class of applications which include constraints arises in the processing of bounded noise, for example, quantization noise, or the noise that appears in all compression algorithms (notably JPEG and JPEG2000). For these types of noise, we show that Total Variation is not the only prior which preserves edges, and that simpler priors lead to better results.

All the models we study are difficult to handle both theoretically and numerically, because of the non-differentiability of the functionals and of the domains.

We describe a general convergent algorithm to solve such problems. This technique is the projected subgradient descent algorithm. We also give its worst case rate of convergence. We apply it successfully to all the problems mentioned above. The computation times we obtain are good compared to state of the art methods, and the models prove to be effective. An example of the processing of bounded noise is shown in figure 21 .

Figure 21. Left: image quantized into ten levels; right: restored image using a model with constraints.

Variational models for road network updating in dense urban areas from very high resolution optical images

Keywords : GIS, dense urban area, road network extraction, multi-scale analysis, higher-order, active contour, shape prior.

Participants : Ting Peng, Ian Jermyn, Josiane Zerubia.

This Ph.D. is co-supervised by Baogang Hu, from LIAMA/CASA, Chinese Academy of Sciences [http://liama.ia.ac.cn ], and by Véronique Prinet from the same laboratory. The data (Quickbird images and GIS of Beijing urban areas) were respectively provided by DigitalGlobe [http://www.digitalglobe.com ] and Beijing Institute of Survey and Mapping [http://www.bism.cn ]. It is partially supported by Alcatel Alenia Space.

The aim of this work is the extraction of road networks from very high resolution (VHR) satellite images ( 0.6m/pixel). Rochery et al.  [41] developed a `phase field higher-order active contour (HOAC)' model for road network extraction from images of 10m/pixel, but when applied to VHR data, this model was not sufficient to solve the problem. The main difficulties lie in the appearance of detail invisible in lower resolution images, (e.g. cars, road markings, shadows, and other linear but non-road features), which can easily disrupt the recognition process, and in the greater diversity of road widths and behaviours. We aim to overcome these difficulties using a more sophisticated image model, in particular, at multiple scales and including inter-pixel interactions, and a multiscale prior model, still within the phase field HOAC framework. Our preliminary work uses the same prior as [41] , but introduces a multiscale image model. The energy is a sum of energies at three different scales. Within each scale we model the one-point statistics of the scaling coefficients using mixtures of two Gaussians, and the two-point statistics (variance histograms) using Gamma distributions. The models were learnt using GIS data to build masks for the main roads and the background.

The original Quickbird image, a GIS mask for the main roads and a preliminary result are shown in figure 22 . The result is not perfect, but is very promising considering the complexity of the image. There are still some problems with spurious noise in the background, and there is geometric noise along the boundaries of the road, sometimes resulting in a gap. The former indicates a lack in the image model, while the latter seems more likely to be due to a weakness in the prior model, which therefore needs improving in order to enforce the road geometry more effectively. Our current work is focused on further study of the two-point statistics of the wavelet and scaling coefficients to improve the image model, and also on the development of a coherent multiscale prior (as opposed to data) energy to reduce computational time and to overcome geometric noise.

Figure 22. Left: Quickbird ©image; Middle: GIS mask of main roads; Right: Preliminary result of the model.

Higher-order active contours for tree detection

Keywords : tree crown extraction, aerial image, gas of circles, higher-order, active contour, shape prior.

Participants : Peter Horvath, Ian Jermyn, Josiane Zerubia.

This Ph.D. is co-supervised by Zoltan Kato, University of Szeged, Hungary [http://www.inf.u-szeged.hu/~kato ]. It is performed as part of an Egide PAI Balaton. The data (aerial images of French forests) were provided by the French National Forest Inventory (IFN) [http://www.ifn.fr ].

Many image processing problems involve identifying the region in the image domain occupied by a given entity in the scene. Automatic solution of these problems requires models that incorporate significant prior knowledge about the shape of the region. Many methods for including such knowledge run into difficulties when the topology of the region is unknown a priori, for example when the entity is composed of an unknown number of similar objects. Higher-order active contours (HOACs) represent one method for the modelling of non-trivial prior knowledge about shape without necessarily constraining region topology, via the inclusion of non-local interactions between region boundary points in the energy defining the model.

The case of an unknown number of circular objects arises in a number of domains, e.g. medical, biological, nanotechnological, and remote sensing imagery. Regions composed of an a priori unknown number of circles may be referred to as a `gas of circles'. In the first part of this work, we developed a HOAC model of a `gas of circles' [19] . In order to guarantee stable circles, we conducted a stability analysis via a functional Taylor expansion of the HOAC energy around a circular shape. Demanding that circles of a given radius be local minima of the energy fixes one of the model parameters in terms of the others and constrains the rest. In conjunction with a suitable likelihood energy, the model was applied to the extraction of tree crowns from aerial imagery. Experimentally, it was found that making a circle of the given radius a local minimum created problems in conjunction with the gradient descent algorithm. Image forces were not always sufficient to overcome these local minima, resulting in the formation of `phantom circles'. We solved the problem of phantom circles by refining the stability analysis, making circles into energy inflection points rather than minima [20] . This constraint further reduces the number of free parameters, and severely constrains one of the two that remain, while improving the empirical success of the model. The results can be seen in figure 23 .

Figure 23. Left: an aerial image of a planted forest; right: the segmentation result