A simple methodology to segment X-ray tomographic images of a multiphasic building stone.

Assessment of the weathering of a particular limestone, the tuffeau, used in historical monuments requires an accurate description of its microstructure. An efﬁcient tools to obtain such a description is X-ray microtomography. However the segmentation of the images of this multiphasic material is not trivial. As the identiﬁcation of pertinent markers of the structural components to extract is difﬁcult, a two steps ﬁltering approach is chosen. Alternate sequential ﬁlters are shown to efﬁciently remove the noise but, as they destroy the structural components smaller than the structuring element used, they cannot be carried out far enough. Hence as a second step in the ﬁltering process, a mosaic operator, relying on a pragmatic yet efﬁcient marker determination, is implemented to simplify further the images.


INTRODUCTION
Exposed to their climatic environment, the building stones of heritage monuments are often altered and eventually destroyed. This phenomenon called weathering is visible throughout the world and many studies in this field, led with architects and restorers, aimed at finding processes to slow down, control or ultimately avoid this decay (Amoroso and Fassina, 1983;Tiano et al., 2006;Torraca, 1976). A way to achieve such a goal is to understand the weathering mechanisms of building stones, i.e. to relate the microscopic mechanisms occurring at the pore scale (dissolution of minerals, transport, precipitation, etc.) to their consequences at the macro-scale (desquamation, powdering, etc.) (Amoroso and Fassina, 1983;Camuffo, 1995;Török, 2002). Some studies comparing weathered stones with unweathered stones were performed. The chemical and mineralogical composition, as well as the porosity were analyzed (Rozenbaum et al., 2007;Maravelaki-Kalaitzaki et al., 2002;Galan et al., 1999). The main processes of weathering were thus qualitatively inferred from the identified differences. However, a more quantitative understanding of these weathering mechanisms and their consequences is to simulate them via a computerized model. This requires in turn a quantitative, realistic description of the three dimensional structure of the porous medium (Dullien, 1992;Adler, 1992;Anguy et al., 2001). An increasingly developing technique to achieve such a goal is Xray tomography which produces 3-d images related to the absorption coefficients of the various phases constituting the material (Kak and Slaney, 2001). In this contribution the images have been acquired on the ID-19 micro-tomographic beamline at the European Synchrotron Radiation Facility. These facilities have several advantages compared to desktop devices : smaller pixel size, X-ray beam quality (monochromaticity, stability, high flux) that lead to high quality and high resolution images Baruchel et al. (2006).
However the segmentation of a raw tomographic image is seldom a trivial process, highly dependent on the starting image and the objects to extract from it. Segmentation is the process of partitioning billions of gray-level voxels of a 3-d image into distinct objects or phases. The goal, in the context of building stones, is be to separate the void phase from 2/24 some distinct solid phases (two for the stone under study). Most of the segmentation complexity is related to the presence of noise (voxels with the same gray value can actually belong to two different phases) and blur (the borders between the phases are not well defined). Moreover, because the images are big and three-dimensional, the analysis cannot be done by hand (e.g. by marking the objects of interest) and must be as automated as possible. The main techniques found in the literature are:

-
Thresholding the gray levels histogram, with  or without (Appoloni et al., 2007) a former filtering, with automatically (Sezgin and Sankur, 2004) or manually (Appoloni et al., 2007) determined thresholds. The thresholded images sometimes have to undergo a binary post-treatment to adjust the results of these approaches. Most of the time it is a reconstruction of the connected components of interest (du Roscoat et al., 2005;Lambert et al., 2005;Kaestner et al., 2006;Erdogan et al., 2006;Ketcham, 2005).
-Active contours on the image considered as a level set (Ramlau and Ring, 2007;Chung and kin Ho, 2000;Maksimovic et al., 2000;Qatarneh et al., 2001). These techniques are mostly used in medical applications and usually require some a priori knowledge of the locations of the interfaces.
The subject of this contribution is to provide a segmentation methodology of the images of a limestone described hereafter. The technique used falls into the first category, i.e.
filtering followed by thresholding. Nevertheless one of the filters used is the mosaic operator, based on a watershed Beucher (1990); Beucher et al. (1990). This methodology 3/24 has been developed because it does not require the marking of each object to be retrieved which appears difficult in this context of multiphased noisy images. Following this introduction, section 2 presents the stone used in this study that serves as model stone, the sample preparation, and the images obtained.
Section 3 details the image analysis procedure, based on mathematical morphology tools, used to segment the images. Section 4 concludes this contribution.  (Dessandier, 1995;Brunet-Imbault, 1999;Rozenbaum et al., 2007) showed that minerals are essentially sparitic (large grains) or micritic (small grains) calcite (≈50%), silica (≈45%) in the form of opal cristobalite-tridymite spheres and quartz crystals, and some secondary minerals such as clays and micas in much smaller proportion (a few %). The scanning electron microscopic (SEM) image in figure 1 illustrates this structural variety, in shapes and sizes, of the main phases of tuffeau. Xray tomography (Kak and Slaney, 2001) is a choice technology to extract the structure of samples of various porous materials: rocks (e.g. Lindquist and Venkatarangan (1999); Cnudde and Jacobs (2003) µm in size) and opal spheres (10 to 20 µm in size), suggest the use of a high resolution tomograph, which leads toward synchrotron radiation facilities. However smaller structures, such as those related to the opal spheres roughness or phyllosilicates, will not be accessible as their size is far below the best resolution any X-ray tomographic facility can achieve nowadays. The microtomographic images presented in this study were collected at the ID-19 beamline of the ESRF (European Synchrotron Radiation Facility, Grenoble, France) (Salvo et al., 2003;Baruchel et al., 2006)  to the X-ray absorption of the sample at the voxel position (Baruchel et al., 2000). Thus, 5/24 the pores appear in dark gray, the silica compounds in medium gray and the calcite compounds in light gray in figure 2. The different phases are easily distinguishable to the naked-eye: calcite is present in the form of large irregular grains (sparitic calcite) or small grains that look like crumble (micritic calcite); silica has the form of large quartz crystals or small spheres of opal. Nevertheless a direct threshold of the raw images is not possible considering their histogram (figure 4, red curve); one cannot identify the expected three peaks. This is due to the presence of noise as illustrated on figure 3-a, which clearly shows the impossibility to select two gray levels that would distinguish the three phases from each other.

TWO STEPS IMAGE FILTERING
Segmentation can be based on the automatic marking of each structural component to be extracted followed by the identification of the zone of influence of each marker (via a watershed operator (Beucher and Lantuejoul, 1979;Vincent and Soille, 1991)).
This approach requires either the definition of markers based on a shape and/or size criterion Beucher et al. (1990) or markers based on a rough a priori knowledge of the gray level of the background and the foreground, possibly cleaned-up or merged by swamping (Beucher (1992)). From our point of view (i) the multiphasic nature, (ii) the variety of shapes and sizes of the structural components of each phase and (iii) the presence of noise, does not allow to propose a solid criterion for the identification of such markers. Hence segmentation can only rely on the gray level of each phase and thus our proposed method will consist of denoising the images, prior to a mere threshold.
Classical denoising tools like linear (e.g. mean) or non-linear (e.g. median) filters are usually efficient but introduce a blur, which in turn has to be dealt with via some edgeenhancing techniques (e.g. janaki and ebenezer (2006)). The (noisy) images under consideration have a good sharpness (as illustrated on figure 3-a) one would like to preserve.
Mathematical morphology (Matheron, 1975;Serra, 1982;1988) proposes an efficient denoising tool called alternate sequential filters (ASF) that does not smooth images. It The image is here considered cubic for the sake of simplicity. Each voxel can be located in the image with a unique triplet of numbers (coordinates).
On a gray level image, the erosion ε and the dilation δ by a structuring element B are defined at every point x by Serra (1982) δ where ∨ if the supremum (or maximum) operator and ∧ the infimum (minimum) operator and B(x) is the structuring element centered at point x. The opening γ and closing ϕ are defined by the adjunctions Serra (1982)

7/24
A particular family of digital balls B λ , with radius λ , have been used for the structuring where d(x, y) is the euclidean distance between the centers of the two voxels at coordinates x and y in voxel-size unit. This choice was made because these balls are a better approximation of the euclidean sphere than those based on the digital distance d 26 (which are euclidean cubes) and they do not impact the efficiency of the implementation. The sequential alternate filtering are brought up to λ = 3, yielding the filtered image h The The results of such a threshold are illustrated on figure 5. For the "simple" zone (left column) the result is acceptable but for a more "complicated" zone, containing micritic calcite (right column) it is not. A lot of inexistent silica is artificially identified between the calcite and the resin. Fig. 3

FILTERING WITH MOSAIC
In order to denoise the image further, it is possible to carry on ASF filtering with bigger structuring elements, but the loss of structural components of these sizes will not be acceptable. Another denoising operation is introduced, the mosaic. The mosaic operator was invented by Beucher (1990). The idea is to simplify the image in an assemblage zones of constant gray level. Beucher then originally performed a merging of neighboring 8/24 zones following a given criterion (hence performing a watershed operation on the graph of the connected zones, see Beucher (1990)), in order to simplify further the images.
The idea here is simpler, the mosaic operator is merely viewed as an additional filtering step in order to remove more noise and improve the separation between the phases in terms of gray levels. The marker determination is a key feature of the method. After the filtering by ASF the image is already essentially composed of flat zones (see figure 3- (d)) each of which being a minima of the gradient of the image. The use of these minima as markers to build a mosaic leads to the creation of too many zones and does not simplify enough the image. In fact the mosaic constructed upon such a marker is very similar to the filtered image and does bring any improvement. In order to build a more efficient marking criterion, a strong assumption is made about the images: each separated structural component inside a given phase (e.g. a block of sparitic calcite, a small grain of micritic calcite or an isolated opal sphere) contains at least one local extremum (minimum or maximum). Inside a "big" component (e.g. a block of sparitic calcite, an opal sphere) this assumes that there is still enough noise to create such local extrema, i.e. the zone is not completely flat. In which case, as it is likely to be surrounded by voxels of higher and lower gray value, the whole flat zone would not be an extremum. The "small" components of one phase (e.g. micritic calcite grains or resin between these grains) are likely to be isolated inside another phase and are de facto a local maxima (e.g. calcite inside resin) or a local minima (conversely).
The morphological gradient g of the filtered image is obtained in the following way (Serra (1982)) The min operator yields a binary image locating the minima of the function h. It is defined for every voxel by Portions of this study were conducted as part of the Région Centre/SOLEIL project funded by the Région Centre that granted one of us.