Hermansen F, Lammertsma A A
Cyclotron Unit, MRC Clinical Sciences Centre, Royal Postgraduate Medical School, Hammersmith Hospital, London, UK.
Phys Med Biol. 1995 Nov;40(11):1909-20. doi: 10.1088/0031-9155/40/11/010.
A general theory is presented for minimizing noise in linear dimension reduction of sequences of medical images when the factors and the covariance matrix and mean of the noise are given. A dimension reduction is optimal when all diagonal elements in the covariance matrix of the noise in the signal (factor) space are minimized. This occurs when the noise in the signal space is uncorrelated with the residual noise. Expressions are given for the resulting covariance matrix of the noise in the signal space. Many optimal inner products exist, which all result in the same optimal dimension reduction. Given any pair of inner products for signal space and residual space, a combined inner product exists that is also optimal. If the covariance matrices of the noise in different pixel vectors are not multiples of each other, different pixel vectors may have different optimal inner products. The averaging process in generating images from tomographic projections tends to make the covariance matrices more uniform.
当噪声的因子、协方差矩阵和均值已知时，提出了一种用于最小化医学图像序列线性降维中噪声的通用理论。当信号（因子）空间中噪声协方差矩阵的所有对角元素最小时，降维是最优的。当信号空间中的噪声与残余噪声不相关时，就会出现这种情况。给出了信号空间中噪声的结果协方差矩阵的表达式。存在许多最优内积，它们都导致相同的最优降维。对于信号空间和残余空间的任何一对内积，都存在一个同样最优的组合内积。如果不同像素向量中噪声的协方差矩阵不是彼此的倍数，则不同像素向量可能具有不同的最优内积。从断层投影生成图像的平均过程往往会使协方差矩阵更加均匀。
