Modelling a Lamb Hind Leg

Transcription

1 Modelling a Lamb Hind Leg Joanne P. Crocombe Ross D. Clarke MIRINZ Food Technology & Research East Street (Ruakura Campus), PO Box 617 HAMILTON, NEW ZEALAND Andrew J. Pullan Department of Engineering Science University of Auckland, Private Bag AUCKLAND, NEW ZEALAND Keywords: 3-d Geometric Modelling, Lamb Leg Model, Data Fitting, Cubic Hermite Elements. ABSTRACT: An anatomically accurate geometric model of a lamb hind leg has been constructed and a framework to describe the interrelationships between the components of the model developed. This is the first step towards development of a model of the full carcass, which will eventually also include joint kinematics and other mechanical and electrical properties. Such models will enable researchers and engineers to better understand and visualise the way the different components of an animal carcass interrelate. A lamb carcass was imaged using x-ray computed tomography (CT) to provide accurate anatomical information. Twodimensional x-ray images were generated at 3 mm intervals showing the bone, fat, and muscle at that point. To model the hind leg of the lamb carcass, three-dimensional data sets first had to be constructed from the two-dimensional images. An automated marching cubes algorithm was used to build three-dimensional data sets of the outer leg surface and bones. The muscles, however, had to be digitised by hand, as it was difficult to detect the boundaries between the muscles. A bicubic Hermite boundary element mesh was then fitted to each data set to give a full mathematical description of each surface. The models were fitted using an iterative nonlinear optimisation process. Each of the major muscle groups and bones was modelled, along with the outer surface of the leg. In addition a modelling framework was developed using graph theory to describe how the hind leg muscles, bones, and fat interrelate. The framework can be easily expanded as the model size and complexity grow. This modelling approach has a wide range of applications within the meat industry, from improved measurement and control systems that produce added-value product to virtual carcass disassembly for marketing and training. 1. Introduction This paper describes the construction of a 3-d geometric model of a lamb hind leg, including the bones and major muscle groups, from x-ray computed tomography (CT) images and the design of a modelling framework that describes how the various model components interrelate. The CT images were digitised using both automated and manual techniques. Surface models made up of bicubic Hermite elements were then fitted to the resulting 3-d data sets using a non-linear optimisation process. The models of the major muscle groups, bones, and outer surface provide an accurate representation of the lamb hind leg anatomy. While much work has been done in the medical and animation fields on accurately modelling the human anatomy [1,2,3,4] this is the first application of such techniques to a carcass for the meat industry that we are aware of. Extending this work to the creation of a full carcass model within the framework will not only enhance understanding but also lead to the development of more intelligent processing systems. 2. Data Collection 2.1 CT Images of the Carcass X-ray computed tomography (CT) is a medical scanning technique used to image the internal anatomy of patients. A CT scanner uses a rotating x-ray tube to generate two-dimensional slices through the body. The spatially varying x-ray attenuation coefficient is measured and from this, tissues that vary in density (e.g. fat and muscle) can be displayed in different shades of grey for visualisation of the internal structures. A whole lamb carcass was CT scanned in 3 mm thick slices at 3 mm intervals along the length of the carcass, producing over 360 two-dimensional images through the carcass such as those shown in Figure 1. Each image shows the bone, fat, and muscle regions at a particular point along the length of the carcass. High density areas like bone show up as white while the darker shades of grey represent lower density fat. 2.2 Creating 3-D Data Sets In order to create accurate three-dimensional models of the various structures within the lamb hind leg, three-dimensional data sets first had to be built up from the series of the twodimensional CT images. The first method used was the marching cubes algorithm [5,6]. This technique produced

2 three-dimensional surfaces of the bones and outer of lamb leg anatomy is required to accurately identify each muscle in sequential x-ray images. surface of the hind leg from the CT images. (c) Shoulders. Figure 1: Three CT images through the carcass at various points. The marching cubes algorithm uses intensity values within the images to determine a surface. A cube is created from eight pixels: four from each of two adjacent slices or images. The algorithm then determines how the surface intersects the cube by assigning a one to a cube vertex where the intensity value exceeds or equals the predefined value for the surface or a zero if the intensity value is below that predefined value. The surface intersects those cube edges where one vertex is outside the surface (value of one) and the other is inside (value of zero). The topology of the surface within the cube is then determined and the cube moves to the next position. The resulting 3-d triangulated surface models of the bones and outer surface of the lamb hind leg are shown in Figure 2. Changing the target intensity value allows the algorithm to first find the outer leg surface then the bone surface, which has a much higher intensity value. These triangulated surfaces consist of over small triangular elements. The vertices of these triangular elements provide an accurate three-dimensional data set to which a computationally efficient mathematical model with much fewer elements can be fitted. The changes in intensity across muscle boundaries are difficult to detect even with the human eye. Therefore, an automated edge detection algorithm could not be used to determine the 3-d surfaces of the individual muscle groups. A hand digitisation process was used instead. In this process, for each muscle or muscle group, the two-dimensional x-ray images were stepped through and points defining the outline of a particular muscle were selected with the mouse. The two-dimensional data sets were then combined to produce a three-dimensional data set for a given muscle or muscle group. A thorough understanding Figure 2: Bones and outer surface of the leg modelled using the marching cubes algorithm. 3. Model Development The next stage in the process was to fit geometric models to the three-dimensional data sets to fully describe the surfaces of the muscles, bones, and outer leg surface. The boundary element method was used for this, with bicubic Hermite basis functions. 3.1 Bicubic Hermite Models In this modelling method, the surface is broken down into a number of smaller sub-regions or elements. The threedimensional shape of each small surface region can then be simply described using relatively low order polynomials (cubic in this case). Each surface element is defined by the three-dimensional coordinates of its four corner points and the curvature at each of those points. Continuity of slope and position is maintained across element boundaries. Figure 3 shows two surface elements, each defined by four nodes. The element on the left is defined by nodes 1, 2, 4, and 5, and any point within the element can be defined in terms of those nodes with the bicubic Hermite equations that describe the three-dimensional element shape. Nodes 2 and 5 are common to both surface elements and ensure continuity across the boundary. Bicubic Hermite elements provide a powerful means of describing curvilinear surfaces and allow relatively complicated geometries to be modelled with a minimal number of elements [1].

3 4 1 5 Figure 3: Two bicubic Hermite elements. 3.2 Model Fitting The bicubic Hermite surfaces were fitted to the measured data sets using a non-linear optimisation process. The fitting procedure minimises the perpendicular distance between the data points and the model surface, to give a better approximation to the actual surface. For this iterative fitting procedure, a measure of error is defined for each data point as the smallest distance from the data point to the current surface mesh. The sum of the squares of the individual errors gives an overall measure of error, which is minimised by the fitting process. Non-linear constraints are added to the problem in order to maintain the C 1 continuity of slope over the mesh during the fitting. Smoothing terms are also incorporated to account for noisy or unevenly distributed data. These smoothing terms must be carefully adjusted during the fitting process in order to obtain the desired smoothness over the mesh while still closely fitting the data points. The non-linear optimisation then iteratively alters the coordinate values and curvature of the mesh points in order to minimise the error in the fit to the given data set. The fitting was carried out in two stages. Several two-dimensional slices were extracted from the threedimensional data set and two-dimensional fits were carried out as illustrated in Figure 4. Joining these two-dimensional meshes together gave a good initial mesh for the three-dimensional fit that was then carried out to accurately model the full threedimensional surface. (a) (b) Figure 4: (a) Initial 2-D mesh and data with a small number of error projections shown. (b) Final fitted mesh Results A bicubic Hermite surface was fitted to each individual data set for the lamb hind leg. Figures 5 to 8 show the final models. The root mean squared error and average absolute error of these fits are given in Table 1. These errors are within the measurement error for the muscle data. The larger and more complicated components such as the bones and gluteobiceps have the largest errors. These errors could be reduced by increasing the total number of elements used to model the given surface or perhaps through further three-dimensional fitting iterations. At this stage, however, the level of accuracy achieved it sufficient for the proposed applications. The bicubic Hermite elements allow the surfaces to be modelled using only a small number of elements. The muscle models are made up of 24 surface elements, while 54 elements were used to describe the more complicated tibia and femur bones. This gives greater computational efficiency when compared to a triangulation approach where hundreds of elements may be required to model a muscle to the same level of accuracy. These models were also converted to VRML (Virtual Reality Modelling Language) to allow the model to be easily viewed and manipulated on a standard PC. Bone or Muscle Group Root Mean Squared Error (mm) Average Absolute Error (mm) Shank Gastrocnemius Semitendinous Semimembranous Adductor Gracilis Quadriceps Gluteobiceps Femur Tibia Patella Tarsal Bones Table 1: Errors for each fitted model.4. Modelling Framework The modelling framework at this stage is essentially a way of viewing how the various bones and muscles interrelate in space. A graph theory approach was used. Each part of the leg was represented by a node in the graph and the graph arcs connected adjacent objects within the leg and were directed from the outermost object inwards. The resulting graph for the lamb leg is shown in Figure 9. This method of viewing the way everything fits together will become more useful as the model is expanded to include the whole carcass, and the structure and interrelationships become more complex. It is also planned that the framework will grow to include more information within each node, such as the physical properties of the object, and will eventually be linked to the model so they can feed each other information.

4 Figure 5: Lamb leg bones. anatomy and will also be of great benefit to the meat industry. Combining the carcass model with on-line measurement techniques to customise the model to an individual carcass will allow more accurate prediction of yield and improved carcass breakdown, to get maximum returns for each carcass. Increased automation of the carcass grading and carcass disassembly processes will be possible, giving a more consistent product with reduced processing costs. The model could also be used for training and marketing purposes, using virtual butchery to demonstrate exactly where and how the carcass should be cut to produce the desired set of meat cuts. The applications for this type of modelling approach are wide ranging and the lamb leg model demonstrates this potential. Figure 6: Five of the major leg muscle groups. Figure 9. Graph showing how the various parts of the leg interrelate Figure 7: Lamb leg with all the major muscle groups shown. Figure 8: Outer surface of the lamb leg over bones and muscles. 5. Future Work Modelling the lamb hind leg is only the beginning. We are currently working on modelling all of the major muscles and bones in the entire lamb carcass. This anatomically accurate carcass model will enable improved visualisation and understanding of ovine 6. References 1. Bradley, C. P., A. J. Pullan, and P. J. Hunter, Geometric modelling of the human torso using cubic Hermite elements. Annals Biomedical Engineering. Vol. 25. pp (1997). 2. Hirsch, B. E., J. K. Udupa, and D. Roberts. Threedimensional reconstruction of the foot from computed tomography scans. Journal of the American Podiatric Medical Association. Vol. 79. pp (1989). 3. Kalra, P., P. Beylot, P. Gingins, N. Magnenat-Thalmann, P. Volino, P. Hoffmeyer, J. Fasel, and F. Terrier. Topological modeling of human anatomy using medical data. Proc. Computer Animation `95, IEEE Computer Society. pp (1995). 4. Scheepers, F., R. E. Parent, W. E. Carlson, and S. F. May. Anatomy-based modeling of the human musculature. Computer Graphics (SIGGRAPH 97 Proceedings), (1997). 5. Lorensen, W.E. and H.E. Cline, Marching cubes: A high resolution 3D surface construction algorithm. Computer Graphics. Vol 21. No. 4. pp (1987). 6. Schroeder, W., K. Martin, and W.E. Lorensen, The Visualization Toolkit (2 nd ed.). Prentice Hall PTR, New Jersey (1998).

5 7. Author Biographies Joanne Crocombe studied at the University of Auckland in the Engineering Science Department receiving her BE(Hons) degree in 1996 and ME degree in She is currently a researcher for MIRINZ Food Technology & Research. Her research interests include 3-d mathematical modelling, biomechanics, and the application of modelling techniques to improve meat industry processing systems. Ross Clarke received a BE(Hons I) degree in Electrical and Electronic Engineering in 1986 and a ME degree in Electrical and Electronic Engineering in 1987, both from the University of Canterbury. He is currently a leader of the Measurement and Electronic Technology Team at MIRINZ Food Technology & Research Ltd. His research interests include the application of mathematical modelling techniques to the estimation of carcass composition and yield distribution to assist in the optimisation of carcass breakdown, the application of non-invasive sensing techniques and technologies for online measurement of carcass and meat cut quality attributes, and the automation of meat product tracking and handling. Andrew Pullan received a BSc(Hons) degree in mathematics in 1985 and a PhD degree in Engineering in 1988; both form the University of Auckland. He is currently a senior lecturer in the Department of Engineering Science at the University of Auckland. His research interests include the application of boundary and finite element techniques to engineering problems, especially those arising in the biomedical area. He has developed computational human torso models and has been collaborating with MIRINZ since 1995.