Modelling oedemous limbs and venous ulcers using partial differential equations

Background Oedema, commonly known as tissue swelling, occurs mainly on the leg and the arm. The condition may be associated with a range of causes such as venous diseases, trauma, infection, joint disease and orthopaedic surgery. Oedema is caused by both lymphatic and chronic venous insufficiency, which leads to pooling of blood and fluid in the extremities. This results in swelling, mild redness and scaling of the skin, all of which can culminate in ulceration. Methods We present a method to model a wide variety of geometries of limbs affected by oedema and venous ulcers. The shape modelling is based on the PDE method where a set of boundary curves are extracted from 3D scan data and are utilised as boundary conditions to solve a PDE, which provides the geometry of an affected limb. For this work we utilise a mixture of fourth order and sixth order PDEs, the solutions of which enable us to obtain a good representative shape of the limb and associated ulcers in question. Results A series of examples are discussed demonstrating the capability of the method to produce good representative shapes of limbs by utilising a series of curves extracted from the scan data. In particular we show how the method could be used to model the shape of an arm and a leg with an associated ulcer. Conclusion We show how PDE based shape modelling techniques can be utilised to generate a variety of limb shapes and associated ulcers by means of a series of curves extracted from scan data. We also discuss how the method could be used to manipulate a generic shape of a limb and an associated wound so that the model could be fine-tuned for a particular patient

Modelling of oedemous limbs and venous ulcers using partial differential equations

BACKGROUND: Oedema, commonly known as tissue swelling, occurs mainly on the leg and the arm. The condition may be associated with a range of causes such as venous diseases, trauma, infection, joint disease and orthopaedic surgery. Oedema is caused by both lymphatic and chronic venous insufficiency, which leads to pooling of blood and fluid in the extremities. This results in swelling, mild redness and scaling of the skin, all of which can culminate in ulceration. METHODS: We present a method to model a wide variety of geometries of limbs affected by oedema and venous ulcers. The shape modelling is based on the PDE method where a set of boundary curves are extracted from 3D scan data and are utilised as boundary conditions to solve a PDE, which provides the geometry of an affected limb. For this work we utilise a mixture of fourth order and sixth order PDEs, the solutions of which enable us to obtain a good representative shape of the limb and associated ulcers in question. RESULTS: A series of examples are discussed demonstrating the capability of the method to produce good representative shapes of limbs by utilising a series of curves extracted from the scan data. In particular we show how the method could be used to model the shape of an arm and a leg with an associated ulcer. CONCLUSION: We show how PDE based shape modelling techniques can be utilised to generate a variety of limb shapes and associated ulcers by means of a series of curves extracted from scan data. We also discuss how the method could be used to manipulate a generic shape of a limb and an associated wound so that the model could be fine-tuned for a particular patient

Wire mesh design

We present a computational approach for designing wire meshes, i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties of wire meshes to the geometric model of Chebyshev nets. This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods

Efficient 3D data representation for biometric applications

YesAn important issue in many of today's biometric applications is the development of efficient and accurate techniques for representing related 3D data. Such data is often available through the process of digitization of complex geometric objects which are of importance to biometric applications. For example, in the area of 3D face recognition a digital point cloud of data corresponding to a given face is usually provided by a 3D digital scanner. For efficient data storage and for identification/authentication in a timely fashion such data requires to be represented using a few parameters or variables which are meaningful. Here we show how mathematical techniques based on Partial Differential Equations (PDEs) can be utilized to represent complex 3D data where the data can be parameterized in an efficient way. For example, in the case of a 3D face we show how it can be represented using PDEs whereby a handful of key facial parameters can be identified for efficient storage and verification

Double Bubbles Minimize

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as described in the article. You can obtain this code by viewing the source of this articl

An investigation on the framework of dressing virtual humans

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Realistic human models are widely used in variety of applications. Much research has been carried out on improving realism of virtual humans from various aspects, such as body shapes, hair, and facial expressions and so on. In most occasions, these virtual humans need to wear garments. However, it is time-consuming and tedious to dress a human model using current software packages [Maya2004]. Several methods for dressing virtual humans have been proposed recently [Bourguignon2001, Turquin2004, Turquin2007 and Wang2003B]. The method proposed by Bourguignon et al [Bourguignon2001] can only generate 3D garment contour instead of 3D surface. The method presented by Turquin et al. [Turquin2004, Turquin2007] could generate various kinds of garments from sketches but their garments followed the shape of the body and the side of a garment looked not convincing because of using simple linear interpolation. The method proposed by Wang et al. [Wang2003B] lacked interactivity from users, so users had very limited control on the garment shape.This thesis proposes a framework for dressing virtual humans to obtain convincing dressing results, which overcomes problems existing in previous papers mentioned above by using nonlinear interpolation, level set-based shape modification, feature constraints and so on. Human models used in this thesis are reconstructed from real human body data obtained using a body scanning system. Semantic information is then extracted from human models to assist in generation of 3 dimensional (3D) garments. The proposed framework allows users to dress virtual humans using garment patterns and sketches. The proposed dressing method is based on semantic virtual humans. A semantic human model is a human body with semantic information represented by certain of structure and body features. The semantic human body is reconstructed from body scanned data from a real human body. After segmenting the human model into six parts some key features are extracted. These key features are used as constraints for garment construction.Simple 3D garment patterns are generated using the techniques of sweep and offset. To dress a virtual human, users just choose a garment pattern, which is put on the human body at the default position with a default size automatically. Users are allowed to change simple parameters to specify some sizes of a garment by sketching the desired position on the human body.To enable users to dress virtual humans by their own design styles in an intuitive way, this thesis proposes an approach for garment generation from user-drawn sketches. Users can directly draw sketches around reconstructed human bodies and then generates 3D garments based on user-drawn strokes. Some techniques for generating 3D garments and dressing virtual humans are proposed. The specific focus of the research lies in generation of 3D geometric garments, garment shape modification, local shape modification, garment surface processing and decoration creation. A sketch-based interface has been developed allowing users to draw garment contour representing the front-view shape of a garment, and the system can generate a 3D geometric garment surface accordingly. To improve realism of a garment surface, this thesis presents three methods as follows. Firstly, the procedure of garment vertices generation takes key body features as constraints. Secondly, an optimisation algorithm is carried out after generation of garment vertices to optimise positions of garment vertices. Finally, some mesh processing schemes are applied to further process the garment surface. Then, an elaborate 3D geometric garment surface can be obtained through this series of processing. Finally, this thesis proposes some modification and editing methods. The user-drawn sketches are processed into spline curves, which allow users to modify the existing garment shape by dragging the control points into desired positions. This makes it easy for users to obtain a more satisfactory garment shape compared with the existing one. Three decoration tools including a 3D pen, a brush and an embroidery tool, are provided letting users decorate the garment surface by adding some small 3D details such as brand names, symbols and so on. The prototype of the framework is developed using Microsoft Visual Studio C++,OpenGL and GPU programming

Toroidal membrane vesicles in spherical confinement

We investigate the morphology of a toroidal fluid membrane vesicle confined inside a spherical container. The equilibrium shapes are assembled in a geometrical phase diagram as a function of scaled area and reduced volume of the membrane. For small area the vesicle can adopt its free form. When increasing the area, the membrane cannot avoid contact and touches the confining sphere along a circular contact line, which extends to a zone of contact for higher area. The elastic energies of the equilibrium shapes are compared to those of their confined counterparts of spherical topology to predict under which conditions a topology change is favored energetically.Comment: 16 pages, 7 figure

A study of early stopping, ensembling, and patchworking for cascade correlation neural networks

The constructive topology of the cascade correlation algorithm makes it a popular choice for many researchers wishing to utilize neural networks. However, for multimodal problems, the mean squared error of the approximation increases significantly as the number of modes increases. The components of this error will comprise both bias and variance and we provide formulae for estimating these values from mean squared errors alone. We achieve a near threefold reduction in the overall error by using early stopping and ensembling. Also described is a new subdivision technique that we call patchworking. Patchworking, when used in combination with early stopping and ensembling, can achieve an order of magnitude improvement in the error. Also presented is an approach for validating the quality of a neural network’s training, without the explicit use of a testing dataset