Image Anal Stereol 2005;24:127-134 Original Research Paper ESTIMATING THE SECTION ELEVATION ANGLE OF CUBES ON A CUBIC MESH. APPLICATION TO NICKEL MICROSTRUCTURE SIZE ESTIMATION Christian Germain, Rémy Blanc, Marc Donias, Olivier Lavialle, Jean-Pierre da Costa and Pierre Baylou Equipe Signal et Image, LAPS - UMR 5131 CNRS – ENSEIRB, ENITAB, Bordeaux1 University, 351 cours de la Libération - 33405 Talence cedex - France e-mail: christian.germain@laps.u-bordeaux1.fr, remy.blanc@laps.u-bordeaux1.fr , donias@enseirb.fr, jean-pierre.dacosta@laps.u-bordeaux1.fr, olivier.lavialle@laps.u-bordeaux1.fr (Accepted September 27, 2005) ABSTRACT This paper discusses two new image analysis methods for estimating the elevation angle of the section plane of a material. These methods are applicable to materials such as nickel base superalloys, the microstructure of which shows cubes arranged on a cubic regular grid. 3-D models were proposed that help interpret the section images and validate our approach. Our first method operates in the Fourier domain, and is based on the estimation of the spatial frequencies of the network of lines observed on the section. The second method is based on the average distance measured between hazy areas. Both methods are independent. Applied to synthetic images or to real material samples, they produce comparable estimations. The values of the elevation angle allow us to cancel the bias associated with the estimation of the material pattern dimensions. Keywords: 3-D simulation, cubic mesh, elevation angle, nickel superalloy, size estimation. INTRODUCTION The microstructure of monocrystalline nickel base superalloys consists of two phases called ? and ?'. The ?' phase appears as a rectangular grid of quasi-parallelepipedal particles. The mean size of a particle side is about 400 nanometres. The ? phase constitutes walls and provides a partition between the ?' nodules. The thickness of the ? walls is roughly ten nanometres. Appropriate specimen etching causes some altitude differences between the two phases. The scanning electron microscope transforms the altitude differences into grey scale variations. In Fig. 1a, ?' sections appear in dark grey and they most frequently have the shape of parallelograms. The intersections of ? walls appear in light grey. Some irregular and intermediate grey shapes also appear. These are called hazy areas. They are usually considered phases with intermediate chemical properties. We propose an interpretation for these hazy areas in section ’Material and method.’ Estimating the size of ?' particles is necessary in order to control the physical and mechanical properties of the nickel base superalloy. One approach to such estimation consists of characterising the material using a morphological analysis of y' particles (Georget et al, 1990). More specifically, this approach is based on statistical measurement of the sizes of the parallelepiped sides and on the estimation of the average side size, believed to be identical along each of the three main crystal axes. Unfortunately, these measurements are affected by the elevation angle between the sectioning plane and the main directions of the crystal. Indeed, a nonzero elevation angle leads to the overestimation of the parallelepiped sizes, whereas an oblique section plane has no effect on the relationship between surface and volume ratios (Russ et al, 2000; Blanc et al, 2004). In this paper, we address the correction of the influence of the section plane elevation angle on the stereological size estimation of a set of cubes arranged on a cubic mesh. For this purpose, we propose two methods, each allowing the estimation of the azimuth q> and the elevation angle Öof the section plane. Both angles correspond to the 3-D reference system associated with the main directions of the crystal as described in Fig. 1b. 127 Germain C et al: Estimating the section angle of cubes on a cubic mesh for nickel superalloy characterisation In a first section, we propose a 3-D model of the material and provide 3-D simulations of section images. Within the context of these simulations, we present the effect of the inclination of the section plane on the image aspect. This allows us to propose an interpretation for the hazy areas observed in real images. Then, in a second section, we propose an initial method for estimating the elevation angle ? from the trans-formation of the section image in the Fourier domain. Indeed, in this space, the mean orientations of the sides of ?' modules are detectable. In a third section, we introduce a competing approach for elevation angle ? estimation. This approach is based on the analysis of the hazy areas. We provide some results obtained with SEM images of nickel base superalloy using both estimation approaches. We also specify the correction to apply to the morphological measurements of ?' particles and discuss the results obtained. The last section is devoted to the conclusion. MATERIAL AND METHOD THREE-DIMENSIONAL MODEL AND SIMULATIONS In this section we introduce a 3-D model that describes the morphology of the y and y' phases that compose a mono-crystal nickel base superalloy. This model allows the simulation of the y/y' intersection with a sloped plane. In a first stage, our model uses identical cubes arranged on a regular cubic mesh. The length of the cube side is represented by the constant value c and the width of the separation wall is the constant value e (Fig. 2a). For such a material, a horizontal section parallel to one of the principal directions of the cube could produce either an image showing squares arranged on a regular grid (Fig. 2b, section plane A) or a uniformly white picture (section plane B) . When the section plane parameters are q> = 0° and 0 * 0° (Fig. 3), the intersection shows rectangles. a) b) Fig. 1. a) Section of a nickel base superalloy. b) Section plane azimuth ? and elevation ?. a) b) Fig. 2. a) Cubic grid model with ? = 0° and ? = 0°. b) Image of section plane A. 128 Image Anal Stereol 2005;24:127-134 Fig. 3. Cubic grid model for (p = 0° and 6 ?0° and the related image of the section plane. a) b) Fig. 4. a) Vertical view for ? = 0° and ? = 20°. b) Image of the oblique section plane. Some rectangle sides are related to the size of the cubes (c and c/cos?), and some other rectangles have one side related to the size of the cube c and the other side related to the vertical position of the section plane. Only the rectangles of the first kind allow the reconstruction of the morphological parameters of the material. Fig. 4a shows the various intersections observed in a vertical view. Fig. 4b shows the image of the corresponding section plane. In order to qualify our model, we have simulated some images of section planes with various azimuth and elevation angles. Fig. 5a shows such a simulation with ? = 0° and ? = 20°. We observe that this image is in accordance with the prediction given in Fig. 4b. In Fig. 5b we have carried out another simulation with ? = 45° and ? = 20°. Using this azimuth, we see that the complete intersections of cubic particles are diamond-shaped, whereas incomplete intersections show triangular and pentagonal shapes. In order to obtain more realistic simulations, we have also introduced fluctuations in the previous model. We have replaced the cubes with parallelepipeds and slightly shifted their locations. The model becomes irregular and thus provides a better description of the actual material layout. Fig. 6a shows the simulation of a section drawn according to this new model. The section plane crosses several y layers, thus forming several large y areas surrounded by y' squares. Theses areas give an interpretation for the hazy areas observed on the real sections. Indeed, both kinds of areas have consistent shapes. Moreover, the grey scale variations inside the hazy areas can be explained by the chemical etching of the y phase. The results in section ‘Results’ confirm this interpretation for various azimuth and elevation angles. Fig. 6b confirms that our model is realistic when q>± 0° and 0^0°. Let us note that the implementation of our models was programmed in C language using a Microsoft Windows station. 129 Germain C et al: Estimating the section angle of cubes on a cubic mesh for nickel superalloy characterisation a) Fig. 5. Section simulations a) ? = 0°, ? = 20°, b) ? = 45°, ? = 20°. b) a) b) Fig. 6. Real section and simulations using the realistic model a) ? = 0°, ? = 0°, b) ? ? 0°, ? ? 0°. ESTIMATION OF THE ELEVATION ANGLE IN THE FOURIER DOMAIN The complete intersection between a cube and an oblique section plane is a parallelogram. In our case, the set of cubes forms two networks of equidistant parallel lines on the section plane. Let d1 be the period of one network, d2 the period of the second one, and ? the angle between the two networks. d1 and d2 are related to the parallelogram shape of the ?' particle (Fig. 7). Fig. 7. Parallelogram-shaped ?’ particle. 130 Image Anal Stereol 2005;24:127-134 We can deduce the azimuth q> and the elevation angle 0 from d1, d2 and». (p = Arc tan 1-r2 +JÄ -1+4Ä and 6 = Arc tan 24Ä 1 + r2 i~A with A = r4+(4cos20-2)r2+1 and r = d/d (1) (2) The estimation of d1 and d2 is much easier in the Fourier domain, considering the periodicity of networks. Actually, the 2-D Fourier transform F(i,j) computed from the grey levels f(x,y) of an image represent the luminance changes with respect to spatial distances (Jain, 1989). Thus, when a peak appears in the Fourier spectrum, the coordinates (ij) of this peak make it possible to directly obtain the frequency and the orientation of the corresponding periodic pattern appearing in the image. Fig. 8b shows the central area of the Fourier spectrum of the image in Fig. 8a. The contrast has been enhanced in order to improve the visibility. Indeed, two significant pairs of peaks appear on this spectrum around the central blob. They correspond to the frequencies f 1 = 1/d1 and f2 = 1/d2. Knowing f 1 and f2 allows us to compute d1 and d2 and then 0 and