估计神经纤维方向密度函数(fODF)是解析人体大脑白质中交叉神经纤维与纤维束追踪的基础。传统的球面反卷积方法基于最小二乘和高斯噪声假设,难以适应MRI幅度图像中的Rician噪声,尤其在低信噪比下易产生偏差;且fODF须满足非负性约束。本文以高阶张量表征fODF,结合平方和多项式和半正定规划将非负约束转化为凸约束,并建立Rician最大似然估计模型。针对含修正贝塞尔函数的似然项,采用EM算法将原问题化为每轮迭代中带半正定约束的凸子问题,再引入交替方向乘子法(ADMM)高效求解,进而提出Rician高阶张量球面反卷积方法(R-HOT-CSD)。合成数据、模体与人体实验表明,该方法在低信噪比下可有效抑制虚假峰,提升角分辨率与估计精度。
Estimating the fiber orientation density function (fODF) is fundamental for resolving crossing fibers in the human brain white matter and for performing fiber tractography. Conventional spherical deconvolution methods based on least-squares fitting and Gaussian noise assumptions are inadequate for handling Rician noise in magnitude MRI images, especially at low signal-to-noise ratio (SNR), where they tend to introduce biases; moreover, the fODF must satisfy a nonnegative constraint. In this paper, we represent the fODF using high-order tensors, and combine sum-of-squares (SOS) polynomials with semidefinite programming (SDP) to convert the nonnegative constraint into a convex constraint. Meanwhile, we establish a Rician maximum likelihood estimation model. For the likelihood term that contains the modified Bessel function, we adopt the expectation-maximization (EM) algorithm to transform the original problem into a convex subproblem with semidefinite constraints in each iteration, and then introduce the alternating direction method of multipliers (ADMM) for efficient solution, thereby proposing the Rician high-order tensor spherical deconvolution (R-HOT-CSD) method. Experiments on synthetic data, phantom data, and in vivo human brain data demonstrate that the proposed method can effectively suppress spurious peaks and improve angular resolution and estimation accuracy under low-SNR conditions.
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