在無參數混和效應模型中小波的彈性收縮

Abstract

這篇論文以貝氏及經驗貝氏觀點探討無參數混和效應模型﹙NPMEM﹚，運用小波收縮在一維訊號及二維影像上去除雜訊。當參數的比例已知時，這個方法即是最佳的線性不偏預測﹙BLUP﹚，所以稱它為BLUPWAVE。當比例未知時，我們提出非線性的估計方法。為了進一步使非線性的估計方法更調適及更具彈性，我們運用交叉確認法﹙GCV﹚從資料來選擇不同層次的門檻來去除雜訊。此外，我們同時考慮選擇主要分解層次及小波基底的平滑度。在模擬研究中，我們比較了BLUPWAVE和soft thresholding兩種方法在一維訊號、二維影像去除雜訊後的標準平均平方誤差及壓縮比。我們也討論了BLUPWAVE及soft及hard thresholding的壓縮比的理論性質。

This article investigates the performance of a wavelet shrinkage method for signals and images based on the perspective of Bayes and empirical Bayes for nonparametric mixed-effects models (NPMEM). This is called BLUPWAVE because it is also the best linear unbiased prediction (BLUP) when the ratio of parameters for NPMEM is known. When the ratio is unknown, a nonlinear estimator guided by the oracle of BLUP has been derived. To make this nonlinear estimator adaptive and the data-driven selection of the level/subband dependent thresholds by generalized cross validation (GCV) is proposed. Furthermore, simultaneous selection of the primary resolution level and smoothness of wavelet basis is also discussed. The simulation studies of this adaptive BLUPWAVE and the soft thresholding by GCV for 1D signals and 2D images are compared by the standardized average squared error (SASE) in denoising and the compression ratio in compression. The theoretical comparison of compression ratios of BLUPWAVE vs. hard and soft thresholding are also discussed.