用正则化解的跨度求解非负最小二乘反问题及其在磁共振弛豫测量中的应用
Solution of Non-negative Least Squares Inverse Problems Using a Span of Regularized Solutions, with Application to Magnetic Resonance Relaxometry
摘要：
我们提出了一个从根本上新的正则化方法的解决方案的Fredholm积分方程的第一类，其中我们将相应的解决方案的范围内的Tikhonov正则化到最终结果。该方法确定的解决方案在一个更大的函数空间，跨越这一组正则化的解决方案，比传统的正则化方法。这些解决方案中的每一个都被正则化到不同的程度。实际上，我们将具有更大正则化程度的解的稳定性与那些正则化程度较低的解的分辨率相结合。相比之下，当前方法涉及选择单个或在某些情况下几个正则化参数，其定义最佳正则化程度。由于所识别的解在一组不同正则化的解的跨度内，因此我们将此方法称为\textit{span of regularizations}或SpanReg。我们通过采用高斯基的非负最小二乘分析证明了SpanReg的性能，并证明了与传统方法相比，双峰高斯分布函数的改进恢复。我们还表明，这种方法表现出减少依赖的最终结果的最优正则化参数的选择。我们进一步说明的方法与应用程序的髓鞘水分数映射在人脑中的实验磁共振成像弛豫数据。我们期望SpanReg作为一种有效的反问题正则化新方法具有广泛的应用前景。
英文摘要：
We present a fundamentally new regularization method for the solution of the Fredholm integral equation of the first kind, in which we incorporate solutions corresponding to a range of Tikhonov regularizers into the end result. This method identifies solutions within a much larger function space, spanned by this set of regularized solutions, than is available to conventional regularizaton methods. Each of these solutions is regularized to a different extent. In effect, we combine the stability of solutions with greater degrees of regularization with the resolution of those that are less regularized. In contrast, current methods involve selection of a single, or in some cases several, regularization parameters that define an optimal degree of regularization. Because the identified solution is within the span of a set of differently-regularized solutions, we call this method \textit{span of regularizations}, or SpanReg. We demonstrate the performance of SpanReg through a non-negative least squares analysis employing a Gaussian basis, and demonstrate the improved recovery of bimodal Gaussian distribution functions as compared to conventional methods. We also demonstrate that this method exhibits decreased dependence of the end result on the optimality of regularization parameter selection. We further illustrate the method with an application to myelin water fraction mapping in the human brain from experimental magnetic resonance imaging relaxometry data. We expect SpanReg to be widely applicable as an effective new method for regularization of inverse problems.
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