Mathematical methods in imaging - superresolution, coregistration, fusion - with applications; fractal-based methods of analysis

我的大部分研究可以被称为数学成像，使用数学来开发图像处理（例如去噪，去模糊）或图像分析的新方法。 它是从早期的研究项目中发展而来的，在该项目中，我探索了分形几何的使用，以近似图像和信号。 例如，我们对“超分辨率”问题感兴趣-用一种或几种分辨率拍摄图像，并将它们的信息结合起来，产生一种更高分辨率的图像。 无论技术发展有多快，专家们总是希望看到更高分辨率的东西。 我目前正在与两种类型的专家合作：当地癌症中心的医学物理学家和我大学的工程师，他们对岩石，混凝土，木材和软骨等多孔介质的特性以及流体如何流过这些介质感兴趣。 但这两组研究人员还面临着另一个问题，即比较从不同成像设备获得的图像，例如，磁共振光谱（MRI），正电子发射断层扫描（PET），计算机断层扫描（CT）。 这些设备中的每一个都捕获被成像的对象的不同属性。挑战是将每个法师提供的独特信息联合收割机组合起来，以更好地全面了解样本/受试者中正在发生的事情。 这就是所谓的“数据融合”。这个问题在医学成像中更加困难，因为病人在不同的时间和不同的机器中根本不会处于完全相同的位置！这就是"图像配准“的问题：适当地扭曲图像（包括内脏），使它们能够相互重叠。 我们一直在努力解决尽可能好地登记当地“感兴趣区域”的重要问题，这是诊断专家特别感兴趣的。我们也重新回到分形成像，发现图像通常是相当“自相似的”-给定一个图像的“A”，通常有许多其他图像可以看起来像“A”。 我们还在研究根据图像的自相似程度来表征图像的方法。

Much of my research could be called mathematical imaging, the use of mathematics to develop new methods of image processing (e.g. denoising, deblurring) or image analysis.  It has evolved from an earlier research programme in which I explored the use of fractal geometry to approximate images and signals.  For example, we are interested in the problem of ``super-resolution'' -- taking images with one or several resolutions and combining their information to produce an image with a higher resolution.  No matter how quickly technology is moving, specialists would always like to see things at even higher resolution.  I am currently working with two such types of specialists:  a medical physicist at a local cancer centre and engineers at my university who are interested in the properties of porous media such as rock, concrete, wood and cartilage, and how fluid flows through these media.  But these two sets of researchers are also faced with another problem, that of comparing images that are obtained from different imaging devices, for example, magnetic resonance spectroscopy (MRI), positron emission tomography (PET), computerized tomography (CT).  Each of these devices captures different properties of what is being imaged. The challenge is to combine the unique information offered from each  mage to get a better total picture of what is going on in the sample/subject.  This is  called ``data fusion.'' The problem is even more difficult in medical imaging since it is not at all the case that a patient will be in exactly the same position at different times and in different machines! This is the problem of ``image registration:''  warping the images - internal organs included - appropriately so that they can overlap each other.  We have been working on the important problem of registering local ``regions of interest'' as well as possible, which is of particular interest to diagosticians. We have also returned to fractal imaging with fresh eyes to find that images are in general quite ``self-similar'' -- given a piece ``A'' of an image, there are generally many other pieces of the image that can be made to look like ``A''.  We are also looking into ways of characterizing images based upon their degrees of self-similarity.