ABSTRACT The minimal data necessary for projective reconstruction from point features is well-kno... more ABSTRACT The minimal data necessary for projective reconstruction from point features is well-known when each object point is visible in all images. In this paper, we formulate and propose solutions to a new family of reconstruction problems for multiple images from minimal data, where there are missing points in some of the images. The ability to handle the minimal cases with missing data is of great theoretical and practical importance. It is unavoidable to use them to bootstrap robust estimation such as RANSAC and LMS algorithms and optimal estimation such as bundle adjustment.
In this paper it is shown how to perform hand-eye calibration using only the normal flow field an... more In this paper it is shown how to perform hand-eye calibration using only the normal flow field and knowledge about the motion of the hand. The proposed method comprise a simple way to calculate the handeye calibration when a camera is mounted on a robot. Firstly, it is shown how the orientation of the optical axis can be estimated from at least two different translational motions of the robot. Secondly, it is shown how the other parameters can be obtained using at least two different motions containing also a rotational part. In both stages, only image gradients are used, i.e. no point matches are needed. As a by-product, both the motion field and the depth of the scene can be obtained. The proposed method is illustrated in experiments using both simulated and real data.
ABSTRACT In this paper we present an integrated approach that solves the structure and motion pro... more ABSTRACT In this paper we present an integrated approach that solves the structure and motion problem for affine cameras. Given images of corresponding points, lines and conics in any number of views, a reconstruction of the scene structure and the camera motion is calculated, up to an affine transformation. Starting with three views, two novel concepts are introduced. The first one is a quasi-tensor consisting of 20 components and the second one is another quasitensor consisting of 12 components. These tensors describe the viewing geometry for three views taken by an affine camera. It is shown how correspondences of points, lines and conics can be used to constrain the tensor components. A set of affine camera matrices compatible with the quasi-tensors can easily be calculated from the tensor components. The resulting camera matrices serve as an initial guess in a factorisation method, using points, lines and conics concurrently, generalizing the well-known factorisation method by Tomasi-Kanade. Finally, examples are given that illustrate the developed methods on both simulated and real data.
Water science and technology : a journal of the International Association on Water Pollution Research, 2007
This paper presents a complete mathematical model of a Moving Bed biofilm process for waste-water... more This paper presents a complete mathematical model of a Moving Bed biofilm process for waste-water treatment, in particular for the nitrification process. The model is based on a bottom up approach adopting a multidimensional model for the biofilm growth and metabolism and a global mass balance model for the whole reactor. The model shows that oxygen is limiting the amount of biomass involved in the nitrification process. Furthermore, it suggests the existence of an optimal amount biomass for an optimal reactor turnover rate. Studies of two specific new suspended carriers show that the model output is dependable on the geometry of the carrier, and to a satisfactory extent agreeable with measurements.
This paper deals with the problem of reconstructing the locations of n points in space from mdioe... more This paper deals with the problem of reconstructing the locations of n points in space from mdioeerent images without camera calibration. It is assumed that the correspondences between thepoints in the dioeerent images are known. It shows how these problems can be put into a similartheoretical framework and solves all minimal cases. When n = 7 and m = 2
In the high-level operations of computer vision it is taken for granted that image features have ... more In the high-level operations of computer vision it is taken for granted that image features have been reliably detected. This paper addresses the problem of feature extraction by scale-space methods. There has been a strong development in scale-space theory and its applications to low-level vision in the last couple of years. Scale-space theory for continuous signals is on a firm theoretical basis. However, discrete scale-space theory is known to be quite tricky, particularly for low levels of scale-space smoothing. The paper is based on two key ideas: to investigate the stochastic properties of scale-space representations and to investigate the interplay between discrete and continuous images. These investigations are then used to predict the stochastic properties of sub-pixel feature detectors. The modeling of image acquisition, image interpolation and scale-space smoothing is discussed, with particular emphasis on the influence of random errors and the interplay between the discrete and continuous representations. In doing so, new results are given on the stochastic properties of discrete and continuous random fields. A new discrete scale-space theory is also developed. In practice this approach differs little from the traditional approach at coarser scales, but the new formulation is better suited for the stochastic analysis of sub-pixel feature detectors. The interpolated images can then be analyzed independently of the position and spacing of the underlying discretisation grid. This leads to simpler analysis of sub-pixel feature detectors. The analysis is illustrated for edge detection and correlation. The stochastic model is validated both by simulations and by the analysis of real images.
ABSTRACT We present a novel calibration method for multi-camera platforms, based on multi-linear ... more ABSTRACT We present a novel calibration method for multi-camera platforms, based on multi-linear constraints. The calibration method can recover the relative orientation between the different cameras on the platform, even when there are no corresponding feature points between the cameras, i.e. there are no overlaps between the cameras. It is shown that two translational motions in different directions are sufficient to linearly recover the rotational part of the relative orientation. Then two general motions, including both translation and rotation, are sufficient to linearly recover the translational part of the relative orientation. However, as a consequence of the speed-scale ambiguity the absolute scale of the translational part can not be determined if no prior information about the motions are known, e.g. from dead reckoning. It is shown that in case of planar motion, the vertical component of the translational part can not be determined. However, if at least one feature point can be seen in two different cameras, this vertical component can also be estimated. Finally, the performance of the proposed method is shown in simulated experiments.
Three-dimensional structure information can be estimated from two-dimensional images using recurs... more Three-dimensional structure information can be estimated from two-dimensional images using recursive estimation methods. This paper investigates possibilities to improve structure filter performance for a certain class of stochastic perspective systems by utilizing mutual information, in particular when each observed point on a rigid object is affected by the same process noise. After presenting the dynamic system of interest, the method is applied, using an extended Kalman filter for the estimation, to a simulated time-varying multiple point vision system. The performance of a connected filter is compared, using Monte Carlo methods, to that of a set of independent filters. The idea is then further illustrated and analyzed by means of a simple linear system. Finally more formal stochastic differential equation aspects, especially the impact of transformations in the Itô sense, are discussed and related to physically realistic noise models in vision systems.
... Range identification for per-spective vision systems. IEEE Transactions on Automatic Control ... more ... Range identification for per-spective vision systems. IEEE Transactions on Automatic Control 48(12), 2232–2238. Jankovic, Mrdjan and Bijoy K. Ghosh (1995). ... Int. Journal of Control 18(6), 1201–1210. Ma, Yi, Stefano Soatto, Jana Košecká and S. Shankar Sastry (2004). ...
This paper deals with the problem of reconstructing the locations of five points in space from tw... more This paper deals with the problem of reconstructing the locations of five points in space from two different images taken by calibrated cameras. Equivalently, the problem can be formulated as finding the possible relative locations and orientations, in three-dimensional Euclidean space, of two labeled stars, of five lines each, such that corresponding lines intersect. The problem was first treated by Kruppa more than 50 years ago. He found that there were at most eleven solutions. Later Demazure and also Maybank showed that there were actually ten solutions. In this article will be given another proof of this theorem based on a different parameterisation of the problem neither using the epipoles nor the essential matrix. This is within the same point of view as direct structure recovery in the uncalibrated case. Instead of the essential matrix we use the kinetic depth vectors, which has shown to be were useful in the uncalibrated case. We will also present an algorithm that in most cases calculates the ten different solutions, although some may be complex and some may not be physically realisable. The algorithm is based on a homotopy method and tracks solutions on the so called Chasles' manifold. One of the major contributions of this paper is to bridge the gap between reconstruction methods for calibrated and uncalibrated cameras. Furthermore, we show that the twisted pair solutions are natural in this context because the kinetic depths are the same for both components.
2.2. 1 Definitions (i) The index rule: When an index appears once in a formula, the formula is va... more 2.2. 1 Definitions (i) The index rule: When an index appears once in a formula, the formula is valid for every value of that index.(ii) The summation rule: When an index appears twice in a term, it is assumed that a summation is taken place over all possible values of that index.(iii) The maximim rule: An index can not be used more than twice in the same term.
There exist intricate geometric relations between multiple views of a 3D scene. These relations a... more There exist intricate geometric relations between multiple views of a 3D scene. These relations are related to the camera motion and calibration as well as to the scene structure. In this chapter, we introduce these concepts and discuss how they can be applied to recover 3D models from images. In Section 3.2, a rather thorough description of projective geometry is given. Section 3.3 gives a short introduction to tensor calculus, and Section 3.4 describes in detail the camera model used. In Section 3.5, a modern approach to multiple view ...
ABSTRACT The minimal data necessary for projective reconstruction from point features is well-kno... more ABSTRACT The minimal data necessary for projective reconstruction from point features is well-known when each object point is visible in all images. In this paper, we formulate and propose solutions to a new family of reconstruction problems for multiple images from minimal data, where there are missing points in some of the images. The ability to handle the minimal cases with missing data is of great theoretical and practical importance. It is unavoidable to use them to bootstrap robust estimation such as RANSAC and LMS algorithms and optimal estimation such as bundle adjustment.
In this paper it is shown how to perform hand-eye calibration using only the normal flow field an... more In this paper it is shown how to perform hand-eye calibration using only the normal flow field and knowledge about the motion of the hand. The proposed method comprise a simple way to calculate the handeye calibration when a camera is mounted on a robot. Firstly, it is shown how the orientation of the optical axis can be estimated from at least two different translational motions of the robot. Secondly, it is shown how the other parameters can be obtained using at least two different motions containing also a rotational part. In both stages, only image gradients are used, i.e. no point matches are needed. As a by-product, both the motion field and the depth of the scene can be obtained. The proposed method is illustrated in experiments using both simulated and real data.
ABSTRACT In this paper we present an integrated approach that solves the structure and motion pro... more ABSTRACT In this paper we present an integrated approach that solves the structure and motion problem for affine cameras. Given images of corresponding points, lines and conics in any number of views, a reconstruction of the scene structure and the camera motion is calculated, up to an affine transformation. Starting with three views, two novel concepts are introduced. The first one is a quasi-tensor consisting of 20 components and the second one is another quasitensor consisting of 12 components. These tensors describe the viewing geometry for three views taken by an affine camera. It is shown how correspondences of points, lines and conics can be used to constrain the tensor components. A set of affine camera matrices compatible with the quasi-tensors can easily be calculated from the tensor components. The resulting camera matrices serve as an initial guess in a factorisation method, using points, lines and conics concurrently, generalizing the well-known factorisation method by Tomasi-Kanade. Finally, examples are given that illustrate the developed methods on both simulated and real data.
Water science and technology : a journal of the International Association on Water Pollution Research, 2007
This paper presents a complete mathematical model of a Moving Bed biofilm process for waste-water... more This paper presents a complete mathematical model of a Moving Bed biofilm process for waste-water treatment, in particular for the nitrification process. The model is based on a bottom up approach adopting a multidimensional model for the biofilm growth and metabolism and a global mass balance model for the whole reactor. The model shows that oxygen is limiting the amount of biomass involved in the nitrification process. Furthermore, it suggests the existence of an optimal amount biomass for an optimal reactor turnover rate. Studies of two specific new suspended carriers show that the model output is dependable on the geometry of the carrier, and to a satisfactory extent agreeable with measurements.
This paper deals with the problem of reconstructing the locations of n points in space from mdioe... more This paper deals with the problem of reconstructing the locations of n points in space from mdioeerent images without camera calibration. It is assumed that the correspondences between thepoints in the dioeerent images are known. It shows how these problems can be put into a similartheoretical framework and solves all minimal cases. When n = 7 and m = 2
In the high-level operations of computer vision it is taken for granted that image features have ... more In the high-level operations of computer vision it is taken for granted that image features have been reliably detected. This paper addresses the problem of feature extraction by scale-space methods. There has been a strong development in scale-space theory and its applications to low-level vision in the last couple of years. Scale-space theory for continuous signals is on a firm theoretical basis. However, discrete scale-space theory is known to be quite tricky, particularly for low levels of scale-space smoothing. The paper is based on two key ideas: to investigate the stochastic properties of scale-space representations and to investigate the interplay between discrete and continuous images. These investigations are then used to predict the stochastic properties of sub-pixel feature detectors. The modeling of image acquisition, image interpolation and scale-space smoothing is discussed, with particular emphasis on the influence of random errors and the interplay between the discrete and continuous representations. In doing so, new results are given on the stochastic properties of discrete and continuous random fields. A new discrete scale-space theory is also developed. In practice this approach differs little from the traditional approach at coarser scales, but the new formulation is better suited for the stochastic analysis of sub-pixel feature detectors. The interpolated images can then be analyzed independently of the position and spacing of the underlying discretisation grid. This leads to simpler analysis of sub-pixel feature detectors. The analysis is illustrated for edge detection and correlation. The stochastic model is validated both by simulations and by the analysis of real images.
ABSTRACT We present a novel calibration method for multi-camera platforms, based on multi-linear ... more ABSTRACT We present a novel calibration method for multi-camera platforms, based on multi-linear constraints. The calibration method can recover the relative orientation between the different cameras on the platform, even when there are no corresponding feature points between the cameras, i.e. there are no overlaps between the cameras. It is shown that two translational motions in different directions are sufficient to linearly recover the rotational part of the relative orientation. Then two general motions, including both translation and rotation, are sufficient to linearly recover the translational part of the relative orientation. However, as a consequence of the speed-scale ambiguity the absolute scale of the translational part can not be determined if no prior information about the motions are known, e.g. from dead reckoning. It is shown that in case of planar motion, the vertical component of the translational part can not be determined. However, if at least one feature point can be seen in two different cameras, this vertical component can also be estimated. Finally, the performance of the proposed method is shown in simulated experiments.
Three-dimensional structure information can be estimated from two-dimensional images using recurs... more Three-dimensional structure information can be estimated from two-dimensional images using recursive estimation methods. This paper investigates possibilities to improve structure filter performance for a certain class of stochastic perspective systems by utilizing mutual information, in particular when each observed point on a rigid object is affected by the same process noise. After presenting the dynamic system of interest, the method is applied, using an extended Kalman filter for the estimation, to a simulated time-varying multiple point vision system. The performance of a connected filter is compared, using Monte Carlo methods, to that of a set of independent filters. The idea is then further illustrated and analyzed by means of a simple linear system. Finally more formal stochastic differential equation aspects, especially the impact of transformations in the Itô sense, are discussed and related to physically realistic noise models in vision systems.
... Range identification for per-spective vision systems. IEEE Transactions on Automatic Control ... more ... Range identification for per-spective vision systems. IEEE Transactions on Automatic Control 48(12), 2232–2238. Jankovic, Mrdjan and Bijoy K. Ghosh (1995). ... Int. Journal of Control 18(6), 1201–1210. Ma, Yi, Stefano Soatto, Jana Košecká and S. Shankar Sastry (2004). ...
This paper deals with the problem of reconstructing the locations of five points in space from tw... more This paper deals with the problem of reconstructing the locations of five points in space from two different images taken by calibrated cameras. Equivalently, the problem can be formulated as finding the possible relative locations and orientations, in three-dimensional Euclidean space, of two labeled stars, of five lines each, such that corresponding lines intersect. The problem was first treated by Kruppa more than 50 years ago. He found that there were at most eleven solutions. Later Demazure and also Maybank showed that there were actually ten solutions. In this article will be given another proof of this theorem based on a different parameterisation of the problem neither using the epipoles nor the essential matrix. This is within the same point of view as direct structure recovery in the uncalibrated case. Instead of the essential matrix we use the kinetic depth vectors, which has shown to be were useful in the uncalibrated case. We will also present an algorithm that in most cases calculates the ten different solutions, although some may be complex and some may not be physically realisable. The algorithm is based on a homotopy method and tracks solutions on the so called Chasles' manifold. One of the major contributions of this paper is to bridge the gap between reconstruction methods for calibrated and uncalibrated cameras. Furthermore, we show that the twisted pair solutions are natural in this context because the kinetic depths are the same for both components.
2.2. 1 Definitions (i) The index rule: When an index appears once in a formula, the formula is va... more 2.2. 1 Definitions (i) The index rule: When an index appears once in a formula, the formula is valid for every value of that index.(ii) The summation rule: When an index appears twice in a term, it is assumed that a summation is taken place over all possible values of that index.(iii) The maximim rule: An index can not be used more than twice in the same term.
There exist intricate geometric relations between multiple views of a 3D scene. These relations a... more There exist intricate geometric relations between multiple views of a 3D scene. These relations are related to the camera motion and calibration as well as to the scene structure. In this chapter, we introduce these concepts and discuss how they can be applied to recover 3D models from images. In Section 3.2, a rather thorough description of projective geometry is given. Section 3.3 gives a short introduction to tensor calculus, and Section 3.4 describes in detail the camera model used. In Section 3.5, a modern approach to multiple view ...
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