GEOMETRIC MODELLING AND CALIBRATION OF A VIRTUAL FOUR-HEADED HIGH SPEED CAMERA-MIRROR SYSTEM FOR 3-D MOTION ANALYSIS APPLICATIONS

The present paper gives an overview of the potential of using a mirror system to generate virtual cameras. This enables the determination of 3-D object coordinates using only one camera. Advantages of this strategy are economical aspects and the elimination of the necessity of synchronisation. Obvious disadvantages are in decreasing the active sensor area. To determine highly dynamic processes a high speed camera is compulsory. The integration of such a camera in a mirror system forms an economic tool to determine 3-D motion of dense target fields. The paper describes a 3-D motion analysis data capture tool based an one high speed camera and a four-headed mirror system and presents practical data handling as well as calibration results. 1. Instruction The observation of static objects with photogrammetric techniques can be performed with only one camera taking images sequentially. In contrast, dynamic processes have to be tracked with a synchronized stereo camera system to ensure that homologous image points are captured at the same time. Standard applications in slow dynamic processes and with low point density can be solved with two industrial machine vision cameras. Complex tasks with a high image point density or occlusions are more demanding concerning hardware efforts. Ambiguities in multi-image matching caused by a high image point density may be solved using a third or fourth camera, as shown in [2]. In case of highly dynamic processes, industrial machine vision cameras with a frame rate of 30 frames per second (fps) are insufficient. High speed cameras offer imaging rates of 1000 fps and beyond, but at made higher prices as compared to off-the shell machine vision type cameras. A stereo mirror system forms a viable alternative to create a 3-D particle tracking system on the basis of only one high speed camera. 3-D Particle Tracking Velocimetry (PTV) is a flexible technique for the determination of velocity fields in liquid or gas flows. Therefore, a flow is densely visualised by particles and recorded by a multi high speed camera system. In contrast to other methods, such as PIV, LDA or HDA, which determine flow characteristics at limited spatial or dimensional resolution, 3-D PTV generates full 3-D trajectory information of the observed 3-D flow volume. This information is obtained from tracer particles, which represent the flow. The tracers are detected in all images in all epochs. This leads to 3-D coordinates for each tracer particle at each epoch. The velocity information results from the linking of successively tracked point clouds in consideration of the used camera frame rate. The results of 3-D PTV are not only 3-D velocity vectors in object space with certain temporal resolution, but also 3-D trajectories plus information on the particle distribution and the temporal behaviour. The method of 3-D PTV can be used in a wide range of applications. While tracer particles in liquid or gas are used as additives to describe the flow, in some industrial applications the focus is set on the flight path or trajectories of the objects themselves. 2. System In order to determine 3-D object coordinates, two or more images taken at the same time step are necessary. In dynamic processes a multi camera system with synchronous image capturing is essential. In case of fast gas flow, a high speed camera system is compulsory due to the velocity spectra. In practical applications, four high speed cameras would be expensive and are often not synchroniseable. As shown in [6], the impact of a non synchronised camera system is essential and has to be corrected. A bias of 0.5 ms (0.5 frames at 1000 fps) and a flow velocity of 10 meters per second mean a tracer movement of 5 mm between the two recordings. If there is a high density of tracer particles or occlusions of targets, more than two cameras are required to solve the matching of image points reliably [3]. If the density is even higher, a four camera system is mandatory to solve the ambiguities of image point matching reliably. With a mirror system it is possible to generate virtual cameras. In our case four cameras convergently arranged cameras are generated by using only a single camera and 8 mirrors. 2.1. Camera There are specific camera requirements necessary in order to verify dynamic processes. In addition to the sensor format and quality, the temporal resolution is most important. Successive temporal matching depends on the relation between velocity, camera frame rate and observation area. Figure 1: FASTCAM ultima 1024 (control panel, head and memory) In our case a CMOS high speed camera FASTCAM ultima 1024 (Figure 1) is used. Its image size is 1024 x 1024 pixel with a frame rate up to 500 fps or 512 x 512 pixel with a frame rate of 1000 fps. The camera memory continuously saves image sequences of 1000 or more frames, depending on the resolution. CMOS technology offers a flexible reduction of the active senor area in order to increase frame rates. The accuracy of the used CMOS camera can be compared to high quality industrial cameras (see chapter 2.4.) or SLR cameras. Beside that there is no impact of frame rate to the accuracy of the high speed camera (Table 1). 60 fps 125 fps 250 fps 500 fps 1000 fps 1 IPM σ [μm] 0,23 0,25 0,24 0,27 0,29 2 0 σ [μm] 0,28 0,30 0,22 0,27 0,39 3 XY σ [μm] 4 8 5 4 9 3 Z σ [μm] 8 30 17 8 42 : standard deviation of image point measurement : standard deviation of unit weight : standard deviation of object point coordinates Table 1: Accuracy of FASTCAM ultima 1024 (1000 fps with 1⁄4 sensor area) Furthermore, the illumination is very important to get useful results. In most cases especially in gas flows the particles are very small. In order to follow the flow, high frame rates necessitate short exposure times. To be able to detect tracers in the observation area, a concentrated illumination in the whole observation volume is essential. 2.2. Mirror system In amateur applications stereo lens modules for SLR cameras have been established for many years. Here, the basis is nearly the eye distance of human beings. For measurement applications the mirror system has to allow convergent views at a much larger base. They are used for instance in the automobile industry [7] or in application areas with limited space [5]. The mirror system developed in cooperation with ETH Zurich generates a four camera system. Figure 2 shows the arrangement of four adjustable mirrors and one fix prismatic mirror in front of a camera. All parts are mounted flexibly so that several base lengths and designs may be realised [4]. The obtained image (Figure 3) is split. The results are four images of four virtual cameras. Hence, all images are captured at the same time. Besides this advantage, the system is more economic instead of using four cameras. The disadvantage is that one virtual camera consists of only one fourth of the active sensor area. Figure 2: Mirror arrangement (K. Hoyer, ETH Zurich) 2.3. Modelling and calibration The calibration procedure for the virtual camera system delivers virtual camera orientation parameters. There are several possible strategies for calibrating this system: One uses the original image size (Figure 4) with only one fourth of the sensor area containing information and the principal point in the image centre (Figure 3 left). Another one uses only one fourth of the image size with full information, but with the principal point in the appropriate corner (Figure 3 right). Figure 3: Image handling An additional option is in the handling of the lens distortion parameters. The first method handles each of the four image parts as taken with an individual virtual camera and the according mirror. The second method presumes that all four image parts are taken with one camera (one set of inner orientation parameters), but different mirrors. While the latter one is more rigorous in the restriction of the number of parameters, the first one offers the advantage of being able to partly compensate for effects introduces by mirror surface irregularities. Information area Principal point Principal point Camera Mirror system Monitoring space