分数维变换域滤波在图像恢复中的应用

傅立叶变换是对信号进行分析的重要数学工具之一．广义上的傅立叶变换，即分数维变换已成为时变信号分析的强有力工具．对原始信号估计的判据常采用均方误差．时间为Ｏ（ＮｌｏｇＮ）的Ｗｉｅｎｅｒ滤波可完成对具有时不变退化模型的信号估计，若退化模型为时变或非平稳的，则需Ｏ（Ｎ ２）的估计时间．这里用在分数维变换域中进行滤波来实现图像恢复，估计时间亦为Ｏ（ＮｌｏｇＮ），且均方误差比在普通傅立叶变换域中的滤波小．实验中，对具有不同信噪比的时变退化模型（ ｃｈｉｒｐ函数污染）的图像进行恢复，结果显示此方法是有效的，且恢复效果随信噪比的提高而改善．

Fractional Fourier Domain Filter for Image Restoration

The FOURIER transform is one of the most frequently used tools in signal analysis. A generalization of the Fourier transform-the fractional Fourier transform-has become a powerful tool for time-varying signal analysis. The mean square error(MSE) is used as design criteria to estimate signal. Wiener filter, which can be implement in O(NlogN) time, is suited for time-invariant degradation models. For time-variant and non-stationary processes, however, the linear estimate requires O(N 2 ). Filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain filtering while requiring O(NlogN) implementation time. The blurred images that have several degradation models with different SNR are restored in the experiments. The results show that the method presented in this paper is valid and that the effect of restoration is improved as SNR is increased.