ON SURFACE TOPOGRAPHY RECONSTRUCTION FROM GRADIENT FIELDS Toni Kuparinen 1 , Ville Kyrki 1 , Jarno Mielikainen 2 , and Pekka Toivanen 2 1 Department of Information Technology, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland 2 Department of Computer Science, University of Kuopio, P.O.Box 1627, 70211 Kuopio, Finland {tkuparin, kyrki}@lut.fi, {mielikai, pekka.toivanen}@uku.fi ABSTRACT In this paper, we propose and study surface reconstruction techniques for surfaces with high frequency height variation, which are common for example, in paper and textile man- ufacturing. Traditionally, photometric stereo methods have been developed and evaluated on objects with additive Gaus- sian noise. The minimization based methods may perform well on large objects, but they smooth the inherent high fre- quency variation of machined surfaces in the reconstruction. We extend a Fourier integration method with Wiener filter to reconstruct surfaces from two gradient fields. The experimen- tal results validate that the proposed method performs well on surfaces with high frequency height variation. Index Terms— Topography, surface reconstruction, gra- dient fields, Fourier transform, Wiener filter 1. INTRODUCTION Surface topography is an important quality parameter in many industrial applications, such as paper and textile manufac- turing. Undesired surface topography variations can reflect imperfections in manufacturing process, product operational efficiency, and life expectancy. Depth recovery techniques, such as shape from shading (SfS) [1], and photometric stereo (PS) [2], provide surface gradients in a fast and non-contact manner. In order to obtain surface topography, the relative height values of the surface, the surface gradients have to be integrated. However, in practice the surface gradients contain noise, which can be derived from imaging and other measure- ment errors. Several solutions have been proposed to inte- grate of the calculated gradient fields. The traditional method for integrating the surface height from gradient information is the Frankot-Chellappa algorithm [3]. The recent develop- ments, such as α surfaces, M-estimators, Regularization and Diffusion, in surface reconstruction from gradient fields have been compared to minimization based Frankot-Chellappa and Poisson [4] methods in [5]. Traditionally, the performance of the surface reconstruction methods have been evaluated on surfaces with objects, such as flower pots, faces, peaks, The author gratefully appreciate the provided funding from European Regional Development Fund (ERDF), National Technology Agency of Fin- land (TEKES), Stora Enso, UPM, Metso, and Future Printing Center (FPC). Stora Enso is acknowledged also for profilometer measurements. and ramps with rather strong additive Gaussian noise. How- ever, the monitoring of surface roughness and texture, that is smaller scale variations with limited noise levels, are fre- quently of interest in manufacturing processes. Recently Hans- son [6, 7] has studied two- and three-light photometric stereo in paper surface reconstruction. In his methods, the paper surface topography is calculated from one and three gradient fields in two- and three-light methods, respectively. Unfortu- nately, he does not provide comparison to alternative surface reconstruction methods. The contributions of this paper are an extension and com- parison of surface reconstruction methods in surface topogra- phy reconstruction. A four-light photometric stereo method is introduced, which is developed from Hansson’s two-light method. In the experiments, surface reconstruction techniques are evaluated with gradient fields calculated from surfaces containing high frequency variation. The proposed method is shown to preserve the original small scale variation in re- constructed surfaces. 2. REVIEW OF PHOTOMETRIC STEREO In photometric stereo, the viewing direction is held constant while the direction of the illumination between successive images is varied. Thus, the correspondence between image points is known a priori. The use of the radiance values at a single image location, in successive views, makes the tech- nique photometric. The technique can be used to determine the surface orientation at each image point [2]. For Lambertian surfaces the reflected intensity is indepen- dent of the viewing direction. However, the intensity depends on the direction of the light source. Lambert’s Law [8] repre- sents the image intensity i at the point (x, y) i = ρλ(l T · n) , (1) where ρ is the surface albedo, λ is the intensity of the light source, n =[n 1 ,n 2 ,n 3 ] T = [p,q,1] T √ p 2 +q 2 +1 is the unit normal to the surface and l =[cos(τ )sin(σ), sin(τ )sin(σ), cos(τ )] T is the unit vector toward the light source. Elements p and q are surface partial derivatives measured along the x and y axes, respectively. τ is the tilt angle of illumination; the angle that the projection of the illuminant vector incident onto the test surface plane makes with an axis in that plane. σ is the slant II - 545 1-4244-1437-7/07/$20.00 ©2007 IEEE ICIP 2007