On the optimizing weighting problem in active image processing

以前，我们已经提出了一个概念的积极（调制）的图像处理方法，这是一个众所周知的调制光谱学的扩展，为2维图像。对于主动图像处理和调制光谱，积分操作是在将双极加权函数与观察到的图像/信号相乘之后进行的，与主动调制同步。为了应用加权函数的乘法，两种不同的加权实现是可能的。首先是传统的数值加权，然后是停留时间加权。本文定义了一个最优加权问题，即如何使这两种加权方法的组合达到最大信噪比。在研究的第一阶段{1994}，理论分析已经针对噪声功率与调制或信号无关的情况。利用施瓦茨不等式得到了该方程的精确解。也就是说，我们获得最大的S/N比时，只有停留时间加权是在这个简单的情况下实施。在下一阶段{1995}，我们已经应用了计算机模拟的两个加权实现，以评估在更复杂的情况下，噪声功率是依赖于调制或信号的S/N比。同时，我们继续得到了这类优化问题的精确解。然后，我们成功地获得了这种复杂情况的精确解。这是“执行数值加权，因为执行数值加权后的噪声功率与调制无关”，然后“应用停留时间加权以满足整个加权函数”。这个重要的结果已经被前面的模拟很好地证实了。

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.