New stereoscopic reconstruction protocol for scanning electron microscope images and its application to in vivo replicas of the shoot apical meristem
Anne-Lise Routier-Kierzkowska A C and Dorota Kwiatkowska BA Institute of Plant Biology, Wroclaw University, Kanonia 6/8, 50-328 Wroclaw, Poland.
B Department of Biophysics and Morphogenesis of Plants, University of Silesia, Jagiellonska 28, 40-032 Katowice, Poland.
C Corresponding author. Email: annelise.routier@gmail.com
This paper originates from a presentation at the 5th International Workshop on Functional–Structural Plant Models, Napier, New Zealand, November 2007.
Functional Plant Biology 35(10) 1034-1046 https://doi.org/10.1071/FP08047
Submitted: 5 March 2008 Accepted: 23 July 2008 Published: 11 November 2008
Abstract
The shoot apical meristem is a small and delicate organ, usually hidden between the young leaves and flowers that it produces. One approach to study meristem geometry and growth consists of taking consecutive replicas from the living meristem surface. In this paper, we present a new stereoscopic reconstruction method for this non-invasive replica protocol, which is applicable to study of growth and geometry of individual cells. This method had been used by the authors to study shoot apical meristem of two species: Arabidopsis thaliana (L.) Heynh. and Anagallis arvensis L., and can be extended to other species and organs. Scanning electron micrographs of the same replica are made at two different angles of view. The obtained stereopairs are used for the dense, three dimensional reconstruction of the replica surface. At the same time, some of the microscope parameters are refined based on the differences between the two micrographs. Three dimensional cell outlines are next extracted from the dense continuous reconstruction, and provide a basis for the quantification of meristem geometry and growth. The new reconstruction protocol can be used with different types of scanning electron microscopes, single- or multi-staged, does not require the identical working distance for the two micrographs of the stereopair, and can be used within a large range of magnifications, corresponding to the cases of either orthogonal or central projection model. It is based largely on recently published algorithms for stereoscopic vision. The reconstruction protocol can be used also for other stereoscopic applications based on scanning electron microscopy. The codes are written in Matlab and are freely available on request to the authors.
Additional keyword: replica technique.
Acknowledgements
The authors are grateful to Michal Huflejt (University of Hamburg), for his help with literature and valuable suggestions and ideas. We also thank Christophe Godin and members of the Virtual Plants Team (INRIA Montpellier) for discussions. This work received the financial support from the Marie Curie RTN grant SY-STEM.
Arun KS,
Huang TS, Blostein SD
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in terms of pLeft and lLeft:
Reciprocally, (FT * pLeft) defines lRight, the epipolar line associated to the point pLeft in the first view and that passes through the point pRight.
In theory, each point in one image should lie exactly on the corresponding epipolar line. In practice however, because of inevitable noise on the point coordinates, each point lies very close to its epipolar line, and the product of Eqn 2 is close to zero.
Appendix 2
Modelling of the image formation and of the rigid transformation between two view points
The pinhole camera model is described by the position vector of the principal point, the pixel aspect ratio, the angle between the projected vertical and horizontal axis (skew) and the focal lengths fx and fy (for vertical and horizontal axes). In order to reduce the model complexity, we made some reasonable assumptions on the imaging process of the scanning electron microscope, namely that the principal point is on the image centre, the pixels are square, fx and fy are nearly equal, the skew parameter is null. These assumptions constitute a good approximation of a scanning electron microscope (Cornille 2005) and allow us to describe the camera by its focal length only. The projection matrix of one view can be written as:
where the focal length fRight,Left is equal to the product of the working distance and the actual magnification of the view.
In the case of a rotation around the y-axis, the translation vector is given by:
where WDRight = fRight/magRight is the working distance for the right view, θ is the angle of rotation around the y-axis, and tx, ty are the translations along the x and y-axis, respectively.
The rotation matrix we use is:
The final form of the fundamental matrix is obtained by the multiplication (described in Eqn 1) of the matrices given by Eqns 4, 5 and 6. It is a 3 × 3 matrix which depends on seven parameters: WDRight, WDLeft, magLeft, magRight, θ, tx, and ty.
Appendix 3
Sampson cost
In theory, given at least seven pairs of matched points (pRight, pLeft) on the two stereoscopic images, one can compute the fundamental matrix obeying Eqn 2. In practice, however, because of inevitable noise on the point coordinates, each point lies very close to its corresponding epipolar line rather than on the line exactly, and the product of Eqn 2 is close, but not equal to zero. To get rid of the effect of noise, one should take a great number of points for the computation of the F matrix, and consider the ideal projection 
of each point pRight,Left, which obeys Eqn 2 exactly:
The sum of the geometric distances between each point pRight,Left and its ideal projection 
is called a reprojection error. A first order approximation of the reprojection error is given by the Sampson cost, which is the sum of the so-called Sampson distances d(pRight, pLeft) for each pair of points.
For a pair of matched points (pRight, pLeft), the Sampson distance is given by:
Given a fundamental matrix F and pairs of points (pRight, pLeft), the Sampson cost (S) is obtained by computing the sum of Sampson distances for all the pairs:
The optimal fundamental matrix, which is the closest to the true fundamental matrix, is the one giving the smallest value of the Sampson cost. Since, in our case, the fundamental matrix can be written as a function of seven parameters, the Sampson distances, and finally the Sampson cost, can also be parameterised with the same parameters. We can then, by changing iteratively some of the fundamental matrix parameters, find their optimal values, corresponding to the minimal value of the Sampson cost, at the same time refining the fundamental matrix.
