基于核方法的医学图像重建
Medical Image Reconstruction Using Kernel Based Methods
AI总结：
针对医学图像重建问题，提出基于正定核函数的正则化方法，利用Hermite-Birkhoff插值思想，通过选择不同核和参数实现灵活重建，并与经典算法进行比较。
摘要：
图像重建问题在于从函数的Radon变换Rf出发找到f的近似。该问题出现在医学成像领域，当人们试图从X射线断层扫描重建身体内部结构时。经典方法基于反投影公式，该公式在已知所有Rf值的情况下给出Radon变换的解析逆。在实际应用中，只给出Rf的离散值集，因此只能得到f的近似。另一类称为ART的方法可用于解决重建问题。遵循ART中的思想，我们尝试将Hermite-Birkhoff插值应用于重建问题。由于核基函数的Radon变换可能无穷大，因此需要正则化技术。我们提出的方法基于正定核函数，并且由于可以选择不同的核和参数而非常灵活。我们研究了该方法的行为，并将其与经典算法进行了比较。
英文摘要：
The image reconstruction problem consists in finding an approximation of a function f starting from its Radon transform Rf. This problem arises in the ambit of medical imaging when one tries to reconstruct the internal structure of the body, starting from its X-ray tomography. The classical approach to this problem is based on the Back-Projection Formula. This formula gives an analytical inversion of the Radon transform, provided that all the values of Rf are known. In applications only a discrete set of values of Rf is given, thus, one can only obtain an approximation of f. Another class of methods, called ART, can be used to solve the reconstruction problem. Following the ideas contained in ART, we try to apply the Hermite-Birkhoff interpolation to the reconstruction problem. It turns out that, since the Radon transform of a kernel basis function can be infinity, a regularization technique is needed. The method we present here is then based on positive definite kernel functions and it is very flexible thanks to the possibility to choose different kernels and parameters. We study the behavior of the methods and compare them with classical algorithms.
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