Aims

The aims of the course are to:

• Teach some mathematical techniques that have wide applicability to many areas of engineering.

Objectives

As specific objectives, by the end of the course students should be able to:

• Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
• Calculate the least squares solution of a set of linear equations.
• Understand how to apply Principal Component Analysis (PCA) to a problem.
• Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
• Represent linear iterative schemes using linear algebra and understand what influences the rate of convergence.
• Understand the definitions and application areas of Stochastic Processes.
• Understand the principle of Markov Chains.
• Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
• Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
• Understand the algorithms of the different gradient search methods.
• Solve unconstrained problems using appropriate search methods.
• Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
• Understand how Markov Chain-based algorithms can be used to give reasonable solutions to global optimisation problems.

Content

Linear Algebra provides important mathematical tools that are not only essential to solve many technical and computational problems, but also help in obtaining a deeper understanding of many areas of engineering.  Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.

Linear Algebra (5L, Prof G Wells)

• Revision of IB material
• Matrix norms, condition numbers, conditions for convergence of iterative schemes
• Positive definite matrices
• Singular Value Decomposition (SVD), pseudo-inverse of a matrix and least squares solutions of Ax = b
• Principal Component Analysis
• Markov matrices and applications

Stochastic Processes (5L, Dr H Ge)

• Definition of a stochastic process, Markov assumption (with examples), the Chapman-Kolmogorov (CK) equation, conversion of a particular CK integral equation into a differential equation (for the case of Brownian motion)
• The general Fokker-Planck equation with particular examples (Brownian motion, Ornstein-Uhlenbeck process)
• Introduction to sampling Gibbs sampler, Metropolis Hastings, Importance sampling with applications.

Optimization (6L, Prof M Girolami)

• Introduction: Formulation of optimization problems; conditions for local and global minimum in one, two and multi-dimensional problems
• Unconstrained Optimization: gradient search methods (Steepest Descent, Newton’s Method, Conjugate Gradient Method)
• Linear programming (Simplex Method)
• Constrained Optimization: Lagrange and Karush-Kuhn-Tucker (KKT) multipliers; penalty and barrier functions

This syllabus contributes to the following areas of the UK-SPEC standard:

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Last modified: 22/01/2025 13:56

Aims

The aims of the course are to:

• Teach some mathematical techniques that have wide applicability to many areas of engineering.

Objectives

As specific objectives, by the end of the course students should be able to:

• Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
• Calculate the least squares solution of a set of linear equations.
• Understand how to apply Principal Component Analysis (PCA) to a problem.
• Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
• Represent linear iterative schemes using linear algebra and understand what influences the rate of convergence.
• Understand the definitions and application areas of Stochastic Processes.
• Understand the principle of Markov Chains.
• Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
• Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
• Understand the algorithms of the different gradient search methods.
• Solve unconstrained problems using appropriate search methods.
• Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
• Understand how Markov Chain-based algorithms can be used to give reasonable solutions to global optimisation problems.

Content

Linear Algebra provides important mathematical tools that are not only essential to solve many technical and computational problems, but also help in obtaining a deeper understanding of many areas of engineering.  Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.

Linear Algebra (4L, Prof G Wells)

• Revision of IB material
• Matrix norms, condition numbers, conditions for convergence of iterative schemes
• Positive definite matrices
• Singular Value Decomposition (SVD), pseudo-inverse of a matrix and least squares solutions of Ax = b
• Principal Component Analysis
• Markov matrices and applications

Stochastic Processes (5L, Prof S Godsill)

• Definition of a stochastic process, Markov assumption (with examples), the Chapman-Kolmogorov (CK) equation, conversion of a particular CK integral equation into a differential equation (for the case of Brownian motion)
• The general Fokker-Planck equation with particular examples (Brownian motion, Ornstein-Uhlenbeck process)
• Introduction to sampling Gibbs sampler, Metropolis Hastings, Importance sampling with applications.

Optimization (7L, Luca Magri)

• Introduction: Formulation of optimization problems; conditions for local and global minimum in one, two and multi-dimensional problems
• Unconstrained Optimization: gradient search methods (Steepest Descent, Newton’s Method, Conjugate Gradient Method)
• Linear programming (Simplex Method)
• Constrained Optimization: Lagrange and Karush-Kuhn-Tucker (KKT) multipliers; penalty and barrier functions
• Global optimisation: Simulated Annealing

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 08/03/2020 17:41

Aims

The aims of the course are to:

• Develop an understanding of the materials issues associated with man-made and naturally-derived materials for medical purposes. Specific case studies will be considered in addition to the general framework.

Objectives

As specific objectives, by the end of the course students should be able to:

• Identify the mechanism by which medical devices and implants come to market.
• Know about the classes of materials used in medical materials and the associated reasons.
• Understand the requirements for materials used in the body and assess potential for implant-body interactions.
• Perform quantitative evaluations of drug delivery.
• Identify appropriate implants and tissue engineering approaches for tissue and body function replacements.
• Understand bioethics and safety regulations associated with medical devices and implants.

Content

Introductory concepts (1L)

Biomaterials as integral parts of medical devices (1L)

Biocompatibility; sterilisation techniques (1L)

• Sterilisation techniques
• Choosing a technique

Sector analysis and regulatory affairs (1.5L)

• Market analysis
• Role of standards
• EU and US approval process

Advanced medical devices and biomaterials of the future (0.5L, non-examinable)

Orthopaedic Implants - Hip Replacement (2L)

• Types of implant fixation
• Materials in hip implants
• Surface engineering approaches
• In vivo loading of hip joint

Cardiovascular Stents (2L)

• Balloon expandable & self expanding stents
• Materials in ​stents
• Stent mechanics and design

Synthetic polymers for tissue engineering applications (2L)

• Introduction to polymers
• Synthetic biodegradable polymers 

Host response to implants (1L)

• Wound repair
• Innate immunity
• The biological response to biomaterials

Using cells in tissue engineering (1L)

• What happens when biomaterials fail
• Cell therapy
• Combining cells with scaffolds
• Working with biology - implant integration and vascularisation

Naturally derived polymers for tissue engineering application (1L)

Drug delivery and diffusion (2L)

• Drug delivery systems
• Diffusion controlled systems in drug delivery

This syllabus contributes to the following areas of the UK-SPEC standard:

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Last modified: 23/07/2018 10:40

Aims

The aims of the course are to:

• Introduce state-of-the-art techniques for the acquisition, representation and visualisation of structured 3D data.

Objectives

As specific objectives, by the end of the course students should be able to:

• Explain the principles of operation of CT, nuclear medicine and diagnostic ultrasound and magnetic resonance imaging.
• Be aware of the advantages and risks associated with these techniques and understand the types of diagnostic problems that each can address.
• Be aware of other types of data to which segmentation and visualisation algorithms can be applied (eg. CAD models).
• Understand the different ways to represent 3D data and appreciate the advantages and disadvantages of each technique.
• Know how to extract surfaces from volumetric data.
• Be aware of the range of computer graphics algorithms and hardware used to visualise 3D data.
• Understand how surfaces can be rendered using suitable illumination and reflection models.
• Know how to visualise voxel arrays directly using volume rendering techniques.

Content

The main application area considered in the module is diagnostic medical imaging: 3D data is acquired using one of the popular imaging modalities (e.g. CT), represented as a voxel array or segmented into surfaces, then visualised using advanced computer graphic techniques. While medical imaging is the focus of the course, many of the techniques used to segment, represent and visualise the 3D data sets are generic and can equally be applied to other types of data, such as CAD models.

Medical Image Acquisition (5L, Prof Richard Prager)

• X-rays and the Radon transform
• Tomographic reconstruction algorithms in both the spatial and frequency domains
• Emission computed tomography
• SPECT and PET
• Iterative reconstruction algorithms
• 2D and 3D ultrasound
• Introduction to Magnetic Resonance Imaging

Extracting information from 3D data (6L, Dr Graham Treece)

Polygonal representations and efficient storage

• Parametric curves and surfaces
• Subdivision and display of parametric surfaces

Surfaces from sampled data

• Thresholding, morphological operators and contours
• Surface extraction - marching cubes

Interpolating sampled data

• Interpolation of isotropic data
• Distance transforms and interpolation of non-isotropic data
• Unstructured data - RBFs and Delaunay triangulation

Direct surface capture

• Laser stripe scanners
• Space encoding: the cubicscope

3D Graphical Rendering (5L, Dr Andrew Gee)

• Viewing systems: viewpoints and projection
• Reflection and illumination models: the Phong reflection model
• Surface rendering: incremental shading techniques, hidden surface removal using Z-buffers
• Shadows and textures
• Ray tracing
• Volume rendering
• Computer graphics hardware

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 16/05/2018 13:46