Olhede Sofia C
Department of Mathematics, Imperial College London, London SW7 2AZ UK.
IEEE Trans Image Process. 2007 Jun;16(6):1522-37. doi: 10.1109/tip.2007.896633.
A new threshold rule for the estimation of a deterministic image immersed in noise is proposed. The full estimation procedure is based on a separable wavelet decomposition of the observed image, and the estimation is improved by introducing the new threshold to estimate the decomposition coefficients. The observed wavelet coefficients are thresholded, using the magnitudes of wavelet transforms of a small number of "replicates" of the image. The "replicates" are calculated by extending the image into a vector-valued hyperanalytic signal. More than one hyperanalytic signal may be chosen, and either the hypercomplex or Riesz transforms are used, to calculate this object. The deterministic and stochastic properties of the observed wavelet coefficients of the hyperanalytic signal, at a fixed scale and position index, are determined. A "universal" threshold is calculated for the proposed procedure. An expression for the risk of an individual coefficient is derived. The risk is calculated explicitly when the "universal" threshold is used and is shown to be less than the risk of "universal" hard thresholding, under certain conditions. The proposed method is implemented and the derived theoretical risk reductions substantiated.
提出了一种用于估计淹没在噪声中的确定性图像的新阈值规则。完整的估计过程基于观测图像的可分离小波分解，并且通过引入新阈值来估计分解系数以改进估计。利用图像的少量“副本”的小波变换幅度对观测到的小波系数进行阈值处理。“副本”通过将图像扩展为向量值超解析信号来计算。可以选择多个超解析信号，并使用超复数或里斯变换来计算此对象。确定了在固定尺度和位置索引下超解析信号的观测小波系数的确定性和随机性属性。为所提出的过程计算了一个“通用”阈值。推导了单个系数的风险表达式。当使用“通用”阈值时，明确计算风险，并表明在某些条件下该风险小于“通用”硬阈值处理的风险。实现了所提出的方法，并证实了推导的理论风险降低。
