Catherine Mennessier

Plane-based camera self-calibration by metric rectification of images

Plane-based self-calibration aims at the computation of camera intrinsic parameters from homographies relating multiple views of the same unknown planar scene. This paper proposes a straightforward geometric statement of plane-based self-calibration, through the concept of metric rectification of images. A set of constraints is derived from a decomposition of metric rectification in terms of intrinsic parameters and planar scene orientation. These constraints are then solved using an optimization framework based on the minimization of a geometrically motivated cost function. The link with previous approaches is demonstrated and our method appears to be theoretically equivalent but conceptually simpler. Moreover, a solution dealing with radial distortion is introduced. Experimentally, the method is compared with plane-based calibration and very satisfactory results are obtained. Markerless self-calibration is demonstrated using an intensity-based estimation of the inter-image homographies.

Centers and centroids of the cone-beam projection of a ball

In geometric calibration of cone-beam (CB) scanners, point-like marker objects such as small balls are imaged to obtain positioning information from which the unknown geometric parameters are extracted. The procedure is sensitive to errors in the positioning information, and one source of error is a small bias which can occur in estimating the detector locations of the CB projections of the centers of the balls. We call these detector locations the center projections. In general, the CB projection of a ball of uniform density onto a flat detector forms an ellipse. Inside the ellipse lie the center projection M, the ellipse center C and the centroid G of the intensity values inside the ellipse. The center projection is invariably estimated from C or G which are much easier to extract directly from the data. In this work, we quantify the errors incurred in using C or G to estimate M. We prove mathematically that the points C, G, M and O are always distinct and lie on the major axis of the ellipse, where O is the detector origin, defined as the orthogonal projection of the cone vertex onto the detector. (The ellipse can only degenerate to a circle if the ball is along the direct line of sight to O, and in this case all four points coincide.) The points always lie in the same order: O, M, G, C which establishes that the centroid has less geometric bias than the ellipse center for estimating M. However, our numerical studies indicate that the centroid bias is only 20% less than the ellipse center bias so the benefit in using centroid estimates is not substantial. For the purposes of quantifying the bias in practice, we show that the ellipse center bias ||CM|| can be conveniently estimated by eA/(π ƒ(≈) where A is the area of the elliptical projection, e is the eccentricity of the ellipse and ƒ(≈) is an estimate of the focal length of the system. Finally, we discuss how these results are affected by physical factors such as beam hardening, and indicate extensions to balls of non-uniform density.

A new representation and projection model for tomography, based on separable B-splines

Data modelization in tomography is a key point for iterative reconstruction. The design of the projector, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a discrete basis of functions.

3D Interventional Imaging with 2D X-Ray Detectors

Pre-operative images, such as CT or MRI, are often necessary for CAMI. However, they could be replaced by interventional 3D reconstruction from 2D x-ray sensors. 3D reconstruction from classical image amplifiers needs the correction of geometric distortions due to the magnetic fields. We investigate new calibration marker schemes exploiting spectral properties of the x-ray transform. According to Shannon theory, no information is lost with these new schemes, even if the markers can be seen in each image. Numerical experiments from both phantom and real data are provided.

Cone-beam calibration using small balls: Centroids or ellipse centers?

When small dense balls are used in geometric calibration of x-ray cone-beam (CB) scanners, it is vital to accurately identify the CB projections of the centers of the balls, called the “center projections.” The detector location of the center projection is usually estimated either from the center of the elliptical projection, or from the centroid of the intensity values inside the ellipse. The ellipse center is easily seen to differ from the center projection, and probably for this reason the centroid has been the preferred estimate.

Algebraic and analytic reconstruction methods for dynamic tomography

In this work, we discuss algebraic and analytic approaches for dynamic tomography. We present a framework of dynamic tomography for both algebraic and analytic approaches. We finally present numerical experiments.

A simple analytic method for cone beam calibration

In cone-beam tomography, for accurate image reconstruction to be achieved, the system first has to be calibrated. Few methods exist for direct cone-beam scanner calibration. The advantages of direct methods is that they do not need human intervention and can be very general with respect to the source/detector trajectory. With the assumption of a vertical scanner, Noo et al. obtained analytic equations for the calibration parameters with respect to the projected images of a calibration object. Without assumptions on the scanner geometry, a simple and direct method is described here, using a calibration object made of three orthogonal line segments with a common center. First, the projected center is identified. Then, three intermediate vectors are calculated from the data. Then, intrinsic (focal length and principal point) and extrinsic (rotation angles and translation vector from the calibration object to the scanner) parameters are obtained analytically from the three vectors

Two-dimensional region-of-interest reconstruction: Analyzing the difference between virtual fanbeam and DBP-Hilbert reconstructions

This work addresses theoretical advances classical (2D) tomographic image reconstruction. During the past several years, inversion formulas have been established that allow ROI reconstruction from incomplete (yet sufficient) data. Such reconstructions have important consequences in certain practical situations, such as truncated projections. The precise relationship between the largest ROI that can be reconstructed and the incompleteness of the sinogram is a complex question which has still not been completely answered in the 2D case. These relationships are inherent to the system and have consequences for iterative/statistical reconstruction methods, because they describe which part of the reconstructed image is determined completely by the data; the other parts of the image will have been more heavily influenced by the regularization method or by the nature of the objective function. Our understanding of the nature of reconstruction from incomplete yet sufficient data relies mainly on formulas obtained from the virtual fanbeam (VFB) method and from the DBP-Hilbert method. The purpose of this work is to provide a structure in which to examine the inherent differences in these two approaches. Using a common reconstruction problem, we reformulate VFB and DBP-Hilbert reconstruction formulas into weight functions that are applied in the sense of an inner product to the sinogram. A common regularization is used for the Hilbert transform in both methods. Unlike the usual Fourier windows used in analytic methods, the regularization we used is applied locally to the singularity to avoid the regularization obscuring the nature of the reconstruction. The weight functions clearly show how truncated projections are being correctly handled. The dissimilarity in the weight functions of the two methods illustrates fundamental differences in managing incomplete data, and suggests that many other such methods exist.

Efficient Sampling of the Rotation Invariant Radon Transform

In this chapter we extend the efficient multidimensional sampling results well known in tomography to the rotation invariant Radon transform with a polynomial weight. Numerical experiments are given. Application of this result to Doppler Imaging, for the particular stellar inclination angle 7r/2 is presented.