This licentiate thesis describes computational methods that solve problems occurring in industrial on-line shape quality inspection of produced items. These items are measured and compared with their corresponding CAD object. The meaning of on-line is that the inspection is done on-line in the production line, i.e. the items are not removed from the line. In practice this means that the inspection must be done very fast, both the measurement and the data analysis. The measurement is done using an optical non-contact method based on projection of fringes.The presented methods are mainly based on finding a transformation, a rotation and a translation, of the measurement values which consists of a point cloud representing the measured surface. This transformation is calculated using the iterative closest point (ICP) method such that the point cloud fits the corresponding surface of the CAD object properly. The method for finding this transformation is adapted for reiterated use, i.e. it makes use of the fact that the same CAD object is used several times for different measurements. A search tree making it possible to do this fast is proposed.When dealing with real measurements obtained from optical methods undesired measurement errors will occur, caused by reflections, dirt on lenses or other likely matters in the industrial environment. The iteratively re-weighted least squares (IRLS) method for different robust functions are used in combination with ICP for handling these errors, in order to do a correct surface matching. This result in much higher matching accuracy and almost no additional computations are needed.

Beräkningsmetoder för formverifiering av friformsytor utgör huvudinnehållet i denna doktorsavhandling. En gemensam egenskap för dessa metoder är att de möjliggör formverifiering online direkt i produktionslinan. Av den anledningen måste metoderna vara snabba och robusta. Ett av problemen som uppkommer i formverifieringen av friformsytor är registrering. Det är problemet med att matcha datapunkter i 3D-rymden, som representerar den uppmätta ytan, med ett CAD-objekt genom att ansätta en stelkropps transformation. En metod för att utföra registreringen snabbt och robust är utvecklad. Metoden är en utveckling av ”the iterative closest point method, ICP”. Vi förprocessar CAD-objektet genom att skapa en datastruktur för att möjliggöra snabb närmsta-punkt sökning. Initialt läggs mycket tid på att skapa datastrukturen för att de enskilda registreringarna skall gå snabbt. Den robusta registreringen baserar sig på teorier från robust statistik genom att tillämpa ”iteratively re-weighted least squares” i kombination med ICP metoden. Detta resulterar i en snabb registreringsmetod som är okänslig för avvikande data. Metoden med registreringen används i en tillämpning för att hitta avvikelser mellan formen för ett objekt och dess ideala form. Den ideala formen är känd och ges av ett CAD-objekt. En optisk formmätningsmetod, projicerade fransar med en enda mönsterdetektering, används för att skapa datapunkter av den uppmätta ytan. Denna metod är snabb och okänslig för vibrationer men datapunkterna kan innehålla fel i vissa regioner, vilket hanteras av registreringen. Ett inversproblem som uppkommer i många optiska formmätningsmetoder är fasuppvikning. Vi introducerar en uppvikningsmetod med regularisering genom att använda information från ett CAD-objekt. Formmätningsmetoden som vi använder oss av här baserar sig på två-våglängds holografi. Vår fasuppvikningsmetod funkar oberoende av diskoninuiteter men mätobjektet får inte avvika alltför mycket i form jämfört med CAD-objektet. En metod för att snabbt få fram den behövda forminformationen från CAD-objektet är också utvecklad. För att få fram lämplig forminformation från datapunkter kan en parametrisk kurva eller yta, t.ex. NURBS, anpassas till dessa punkter. Ett delproblem som uppstår vid NURBS-anpassning vid användandet av Gauss-Newton metoden är studerad. Beräkningsaspekter för att få fram en sökriktning är diskuterade. Vi behandlar också metoder för NURBS anpassning som baserar sig på en teknik för separabla icke-linjära minstakvadratproblem. Denna teknik använder sig av variabelprojektioner för att separera beräkningarna av de linjära parametrarna från beräkningarna av de icke-linjära parametrarna.

The evaluation of surface points and derivatives of NURBS surfaces for parameter values that are regularly distributed in a rectangular structure is considered. Because of the regularity, parts of the computations can be done on just a small portion of all parameter values and computed data is stored and reused for many other parameter values. Hence, the evaluation of NURBS surfaces can be performed faster when the regularity is used. We are making a complexity analysis of the number of floating point operations, which is required for the evaluations. To get knowledge about how the evaluations perform in practice, we are doing a numerical experiment where we are measuring the runtime to obtain the output both by using ordinary evaluation of the NURBS surface and by making use of the regular structure. Making use of the regularity gives significantly faster output.

We consider the computational problem of finding the point in 3D-space on a transformed surface corresponding to a coordinate pair given in a perspective mapping. The transformation is a rigid body transformation that is assumed to be small and vary. Initially, it is unknown but when it becomes known, the output must be accurate and quickly returned. Therefore, the computations are adapted for those conditions. Preprocessed shape information about the surface is computed in a perspective mapping where the surface is in an original position. We are discussing algorithms for solving the considered problem.

The update of the rigid body transformation in the iterative closest point (ICP) algorithm is considered. The ICP algorithm is used to solve surface registration problems where a rigid body transformation is to be found for fitting a set of data points to a given surface. Two regions for constraining the update of the rigid body transformation in its parameter space to make it reliable are introduced. One of these regions gives a monotone convergence with respect to the value of the mean square error and the other region gives an upper bound for this value. Point-to-plane distance minimization is then used to obtain the update of the transformation such that it satisfies the used constraint.

Registration of point sets is done by finding a rotation and translation that produces a best fit between a set of data points and a set of model points. We use robust M-estimation techniques to limit the influence of outliers, more specifically a modified version of the iterative closest point algorithm where we use iteratively re-weighed least squares to incorporate the robustness. We prove convergence with respect to the value of the objective function for this algorithm. A comparison is also done of different criterion functions to figure out their abilities to do appropriate point set fits, when the sets of data points contains outliers. The robust methods prove to be superior to least squares minimization in this setting.

The problem of finding a rigid body transformation, which aligns a set of data points with a given surface, using a robust M-estimation technique is considered. A refined iterative closest point (ICP) algorithm is described where a minimization problem of point-to-plane distances with a proposed constraint is solved in each iteration to find an updating transformation. The constraint is derived from a sum of weighted squared point-to-point distances and forms a natural trust region, which ensures convergence. Only a minor number of additional computations are required to use it. Two alternative trust regions are introduced and analyzed. Finally, numerical results for some test problems are presented. It is obvious from these results that there is a significant advantage, with respect to convergence rate of accuracy, to use the proposed trust region approach in comparison with using point-to-point distance minimization as well as using point-to-plane distance minimization and a Newton- type update without any step size control.

We consider a subproblem in parameter estimation using the Gauss-Newton algorithm with regularization for NURBS curve fitting. The NURBS curve is fitted to a set of data points in least-squares sense, where the sum of squared orthogonal distances is minimized. Control-points and weights are estimated. The knot-vector and the degree of the NURBS curve are kept constant. In the Gauss-Newton algorithm, a search direction is obtained from a linear overdetermined system with a Jacobian and a residual vector. Because of the properties of our problem, the Jacobian has a particular sparse structure which is suitable for performing a splitting of variables. We are handling the computational problems and report the obtained accuracy using different methods, and the elapsed real computational time. The splitting of variables is a two times faster method than using plain normal equations.

We consider the problem of matching sets of 3D points from a measured surface to the surface of a corresponding computer-aided design (CAD) object. The problem arises in the production line where the shape of the produced items is to be compared on-line with its pre-described shape. The involved registration problem is solved using the iterative closest point (ICP) method. In order to make it suitable for on-line use, i.e., make it fast, we pre-process the surface representation of the CAD object. A data structure for this purpose is proposed and named Distance Varying Grid tree. It is based on a regular grid that encloses points sampled from the CAD surfaces. Additional finer grids are added to the vertices in the grid that are close to the sampled points. The structure is efficient since it utilizes that the sampled points are distributed on surfaces, and it provides fast identification of the sampled point that is closest to a measured point. A local linear approximation of the surface is used for improving the accuracy. Experiments are done on items produced for the body of a car. The experiments show that it is possible to reach good accuracy in the registration and decreasing the computational time by a factor 700 compared with using the common kd-tree structure.

We are describing a fully automatic in-line shape inspection system for controlling the shape of moving objects on a conveyor belt. The shapes of the objects are measured using a full-field optical shape measurement method based on photogrammetry. The photogrammetry system consists of four cameras, a flash, and a triggering device. When an object to be measured arrives at a given position relative to the system, the flash and cameras are synchronously triggered to capture images of the moving object. From the captured images a point-cloud representing the measured shape is created. The point-cloud is then aligned to a CAD-model, which defines the nominal shape of the measured object, using a best-fit method and a feature-based alignment method. Deviations between the point-cloud and the CAD-model are computed giving the output of the inspection process. The computational time to create a point-cloud from the captured images is about 30 seconds and the computational time for the comparison with the CAD-model is about ten milliseconds. We report on recent progress with the shape inspection system.

A concept for targetless, computer-aided design (CAD)-based, close-range photogrammetry for online shape inspection is introduced. The shape of an object, which is arbitrarily located on a conveyor belt, is to be measured and compared with its nominal shape as defined by a CAD model. For most manufactured objects, deviations are only measured at a few given comparison points. These deviations can be estimated using local photogrammetry based on a priori geometrical information given by the CAD model and the comparison points. Our method results in faster output with higher precision, because we do not generate a shape representation of the entire measured object using typical photogrammetric methods. Images depicting the object from convergent angles are captured by an array of cameras in a precalibrated network, and the CAD model is matched and aligned, within the projective geometry of the camera network, to the depicted object in the images without the use of targets. An algorithm for solving this virtual projective targetless shape matching problem is presented.

This paper discusses the possibility of evaluating the shape of a free-form object in comparison with its shape prescribed by a CAD model. Measurements are made based on a single-shot recording using dualwavelength holography with a synthetic wavelength of 1.4 mm. Each hologram is numerically propagated to different focus planes and correlated. The result is a vector field of speckle displacements that is linearly dependent on the local distance between the measured surface and the focus plane. From these speckle displacements, a gradient field of the measured surface is extracted through a proportional relationship. The gradient field obtained from the measurement is then aligned to the shape of the CAD model using the iterative closest point (ICP) algorithm and regularization. Deviations between the measured shape and the CAD model are found from the phase difference field, giving a high precision shape evaluation. The phase differences and the CAD model are also used to find a representation of the measured shape. The standard deviation of the measured shape relative the CAD model varies between 7 and 19 μm, depending on the slope.

The aim of this work is to evaluate the shape of a free form object using single shot digital holography. The digital holography results in a gradient field and wrapped phase maps representing the shape of the object. The task is then to find a surface representation from this data which is an inverse problem. To solve this inverse problem we are using regularization with additional shape information from the CAD-model of the measured object.

In automotive industry there is an interest of controlling the shape of a large number of identical components on-line in the manufacturing process. We propose a method to do this by capturing a digital hologram of the object and then using information from its computer aided design (CAD) model to calculate the shape and determine the agreement between the manufactured object and the CAD-model. The holographic recording of the object is done using dual wavelengths with a synthetic wavelength of approximately 400 μm. The optical measurement results in a wrapped phase map with the phase values in the interval [−π, π]. Each phase interval represents a depth distance on the object of about 0.2 mm. The phase unwrapping is done iteratively using information from the CADmodel. This implies that it is possible to measure large discontinuities on the surface of the measured object. The method also gives a point-to-point correspondence between the measurement and the CAD-model which is vital for tolerance control.

We describe a method to verify the shape of manufactured objects by using their design model. A non-contact measuring method that consists of a stereo-camera system and a single projected fringe pattern is used. The method acquires one image from each camera. Additional shape information from the design model is also used. This surface-measurement method gives an accuracy of about 45 µm. Deviations from the design model within ±1.6 mm can be correctly detected. The measured surface representation is matched to the design model using the ICP-method. Fast performance has been considered adapting the method for on-line use.

We consider the problem of fitting a non-uniform rational B-spline (NURBS) curve to a set of data points by determining the control points and the weights using techniques aimed for solving separable least squares problems. The main technique under consideration is the variable projection method which utilises that the NURBS model depends linearly on its control points but non-linearly on the weights. The variable projection method can be used with the Gauss-Newton algorithm but also with Newton's algorithm. We investigate the efficiency of the different algorithms when fitting NURBS and observe that the variable projection methods do not perform as well as reported for its use on, e.g., exponential fitting problems.

This study focuses on utilizing image techniques for river velocity measurement, with a specific emphasis onnatural surface floating patterns. Employing a multi-camera system, we conducted 3D measurements on riversurfaces, including surface velocity and water surface reconstruction. A pattern-based tracking approach hasbeen adopted to improve the performance of image measurements on different types of natural floating tracers.The study employs the following approaches: 3D Lagrangian Pattern Tracking Velocimetry (3D-LPTV), 2DLagrangian Pattern Velocimetry (2D- LPTV), and Large-scale Particle Image Velocimetry (LSPIV), for surfacevelocity estimation. The outcomes revealed that all three approaches yielded consistent results in terms ofaveraged velocity. However, the LSPIV method produced about two times higher uncertainty in measured velocitiescompared to the other methods. A strategy to assess the quality of river surface patterns in velocityestimation is presented. Specifically, the sum of squared interrogation area intensity gradient (SSIAIG) was foundto be strongly correlated with measurement uncertainty. Additionally, a term related to the peak sidelobe ratio(PSR) of the cross-correlation map was found as an effective constraint, ensuring the image-tracking processachieves high reliability. The precision of measurements increases corresponding to the increase of image intensitygradient and PSR.

This study describes a multi-camera photogrammetric approach to measure the 3D velocityof free surface flow. The properties of the camera system and particle tracking velocimetry (PTV)algorithm were first investigated in a measurement of a laboratory open channel flow to prepare forfield measurements. The in situ camera calibration methods corresponding to the two measurementsituations were applied to mitigate the instability of the camera mechanism and camera geometry.There are two photogrammetry-based PTV algorithms presented in this study regarding differenttypes of surface particles employed on the water flow. While the first algorithm uses the particletracking method applied for individual particles, the second algorithm is based on correlation-basedparticle clustering tracking applied for clusters of small size particles. In the laboratory, referencedata are provided by particle image velocimetry (PIV) and laser Doppler velocimetry (LDV). Thedifferences in velocities measured by photogrammetry and PIV, photogrammetry and LDV are 0.1%and 3.6%, respectively. At a natural river, the change of discharges between two measurement timesis found to be 15%, and the corresponding value reported regarding mass flow through a nearbyhydropower plant is 20%. The outcomes reveal that the method can provide a reliable estimation of3D surface velocity with sufficient accuracy.

In an on line shape measurement in disturbed environment, use of many wavelengths in order to avoid phase ambiguity may become a problem as it is necessary to acquire all holograms simultaneously due to environmental disturbances. Therefore to make the shape data available the different holograms have to be extracted from a single recorded image in spectral domain. Appropriate cut areas in the Fourier method are therefore of great importance for decoding information carried by different wavelengths. Furthermore using different laser sources, induces aberration and pseudo phase changes which must be compensated. To insure any phase change is only because of the object shape, calibration is therefore indispensable. For this purpose, effects of uncontrolled carrier frequency filtering are discussed. A registration procedure is applied using minimum speckle displacements to find the best cut area to extract and match the interference terms. Both holograms are numerically propagated to a focus plane to avoid any unknown errors. Deviations between a reference known plate and its measurement are found and used for calibration. We demonstrate that phase maps and speckle displacements can be recovered free of chromatic aberrations. To our knowledge, this is the first time that a single shot dual wavelength calibration is reported by defining a criteria to make the spatial filtering automatic avoiding the problems of manual methods. The procedure is shown to give shape accuracy of 35μm with negligible systematic errors using a synthetic wavelength of 1.1 mm.

The objective of this paper is to describe a fast and robust automatic single-shot dual-wavelength holographic calibration method that can be used for online shape measurement applications. We present a model of the correction in two terms for each lobe, one to compensate the systematic errors caused by off-axis angles and the other for the curvature of the reference waves, respectively. Each hologram is calibrated independently without a need for an iterative procedure or information of the experimental set-up. The calibration parameters are extracted directly from speckle displacements between different reconstruction planes. The parameters can be defined as any fraction of a pixel to avoid the effect of quantization. Using the speckle displacements, problems associated with phase wrapping is avoided. The procedure is shown to give a shape accuracy of 34 μm using a synthetic wavelength of 1.1 mm for a measurement on a cylindrical test object with a trace over a field of view of 18  mm×18  mm.

We present a calibration method which allows single shot dual wavelength online shape measurement in a disturbed environment. Effects of uncontrolled carrier frequency filtering are discussed as well. We demonstrate that phase maps and speckle displacements can be recovered free of chromatic aberrations. To our knowledge, this is the first time that a single shot dual wavelength calibration is reported by defining a criteria to make the spatial filtering automatic avoiding the problems of manual methods. The procedure is shown to give shape accuracy of 35 µm with negligible systematic errors using a synthetic wavelength of 1.1 mm.

In order to measure the shape of a large number of identical components in a manufacturing industry we propose a method where digital holography is used to capture an image of the object and then the shape of the object is achieved by using information from the CAD-model. The holographic recording of the object is done using dual wavelengths giving a synthetic wavelength of about 400 μm. This gives a phase map where the phase intervals represent a depth distance on the object of about 0.2 mm. To find the shape of the object the phase map has to be unwrapped. Since the surface contains discontinuities we use information from the CAD-model of the measured object and unwrap the phase iteratively. The result becomes a digital point representation of the measured surface that can either be used just as a description of the object shape or as a way to describe how well the object has been manufactured compared to the CAD-model. The measurement process that is proposed is adapted for on-line purposes; hence it is fast and reliable.

This paper presents an overview of a measurement system that is designed for in-line shape inspection of metal sheet components in real-time utilizing pre-calibrated close-range photogrammetry and a CAD-model. The system is currently designed to measure parts thrown out on a conveyor belt that moves at about 1 m s⁻¹ at a frequency of 0.5 Hz or less without fixturing. Detected features on the components in the camera images are used to align the CAD-model, describing the nominal shape, to the images from which required deviations are computed using photogrammetry. The measurement volume of the current system is 500 × 800 × 200 mm³ and absolute measurements are performed with an accuracy in the order of 0.1 mm. The in-line functionalities of the system have been verified at several real production sites in Sweden. In this paper, the basic components of the system are described together with a few results from real tests.

Defocused laser speckle photography is used as a tool to measure the heat responses in a titanium component during laser heating. The evolution of the response is compared with a set of preprocessed Finite Element Simulations of the corresponding process with the aim to verify the simulation model and to find the simulation settings that best resemble the experimental results. The titanium component consists of a 300 x 100 mm2 substrate of thickness 3.2 mm on which a 200 x 30 x 11 mm3 ridge is built up using the laser metal deposition by wire process. The component is heated on the top of the ridge by a 300 W laser for 10 s and the deformation of the subtrate is followed throughout the heating-cooling cycle. The simulated deformation gradient is shown to resemble the measured response, and the magnitude of the response indicates that about 70 % of the laser power transferres into heat in the metal.