On the optimizing weighting problem in active image processing

之前，我们提出了主动（调制）图像处理方法的概念，这是众所周知的调制光谱学的扩展，用于二维图像。对于主动图像处理和调制光谱，将观测图像/信号的双极加权函数乘以与主动调制同步后进行积分运算。要应用加权函数的乘法，可以使用两种不同的加权实现。首先是传统的数值加权，其次是停留时间加权。两种加权实现可以同时使用，但这种组合会导致处理后图像的信噪比{S/N}不同。在本研究中，我们定义了一个优化加权问题，以获得这些加权实现的优化组合，从而获得最大的信噪比。在研究的第一阶段{1994}，理论分析的目标是噪声功率与调制或信号无关的情况。我们用施瓦茨不等式得到了一个精确解。也就是说，在这种简单的情况下，当只实现停留时间加权时，我们获得了最大的信噪比。在下一阶段{1995}，我们应用了两种加权实现的计算机模拟来评估更复杂情况下的信噪比，其中噪声功率取决于调制或信号。同时继续得到了这类优化问题的精确解。然后我们成功地得到了这种复杂情况的精确解。这是“实现数值加权，因为数值加权后的噪声功率与调制无关”，然后是“应用停留时间加权来满足整个加权函数”。这一重要的结果已经被前面的模拟清楚地证实了。

Previously, we have proposed a concept of the active (modulation) image processing method which is an extension of well known modulation spectroscopy, for 2-dimensional images. For both the active image processing and the modulation spectroscopy, integration operation is carried out after multiplication of a bipolar weighting funciton to observed image/signal, synchronized to the active modulation. To apply multiplication of the weighting function, two different weighting implementations are possible. First is a conventional numerical weighting, and next is a time-of-stay weighting. And both weighting implementations can be used together, while such combinaion results in different signal to noise {S/N} ratio for the processed images.In the present study, we have defined an optimizing weighting problem as to obtain optimized combination of these weighting implementations resulting maximum S/N ratio. At the first stage {1994} of the study, theoretical analysis has been aimed to the case in which the noise power is independent on the modulation or the signal. And we have got an exact solution using Schwarz's inequality. Namely, we obtain maximum S/N ratio when only the time-of-stay weighting is implemented in this simple case.In the next stage {1995}, we have applied a computer simulation of both weighting implementations to evaluate the S/N ratio under more complicated cases in which the noise power is dependent on the modulation or the signal. At same time we have continued to get an exact solution of such optimizing problem. And then we have successfully obtain an exact solution for such complicated case. This is `To implement the numerical weighting, as the noise power after the implementation of the numerical weighing is independent on the modulation', and then `To apply the time-of stay weighting to satisfy the whole weighting function'. This important result has been clearly well confirmed by the previous simulation.