基於低秩漢克爾方法實現鹽和胡椒去噪

Abstract

在圖像處理中，由於各種原因，例如相機傳感器中的像素故障或硬體中的記 憶體位置錯誤，圖像通常會受到脈衝噪聲的破壞。脈衝噪聲有一種常見的類型， 稱為鹽和胡椒雜訊。當圖像被鹽和胡椒雜訊破壞時，噪聲像素僅取最大值或最小 值，從而導致圖像上出現白點和黑點。 目前已有一些處理鹽和胡椒雜訊的方法，這些方法包括基於中值濾波器的方 法、基於低秩矩陣補齊的方法、TV-L1 方法以及基於湮滅濾波器的方法。這些方 法在低強度的鹽和胡椒雜訊下都有著不錯的表現，但基於中值濾波器的方法及基 於低秩矩陣補齊的方法在高水平的鹽和胡椒雜訊下其效能並不令人滿意。 在我們的模擬中，對於高噪聲水平，基於湮滅濾波器的方法可以實現最高的 效能。作為交換，它們在過程中需要最多的時間，因為它們處理了大型漢克爾矩 陣 H (X) 的最佳化問題。 對於基於湮滅濾波器的方法，我們開發了一個停止標準，以獲得更好的去噪 性能。如果沒有停止標準，它通常會花更多的計算資源而獲得更低的效能。此外， 我們提出一種方法將最佳化問題分別作處理，對於矩陣 H (X)，我們將其分割成 p × q 塊，並分別對其作低秩最佳化。透過不同 p × q 值的選擇，在不同的情況下， 我們可以在峰值信噪比與時間成本之間做權衡。 我們對於漢克爾矩陣的結構進行了詳細的解釋，還介紹了運算子 H (·) 的逆運算。為了將基於湮滅濾波器的方法應用於彩色圖像，我們為彩色影像定義了 H3D(X)，並解釋如何構造 H3D(X)。

In image processing, images are usually corrupted by impulse noise due to a wide variety of reasons, such as malfunctioning pixels in camera sensors or faulty memory locations in hardware. The impulse noise has a common type called salt-and-pepper noise (SPN). When images are corrupted by SPN, the noisy pixels take only the maximum value or the minimum value, contributing to white and black dots on images. There are some known methods for dealing with SPN. These methods include me- dian filter-based methods, low-rank matrix completion-based methods, TV-L1 method, and annihilating filter-based methods. These methods have a good performance on noise reduction for the low level of SPN. But the performance of median filter-based methods and low-rank matrix completion-based methods are not satisfactory for the high level of SPN noise. In our simulation, for high noise levels, annihilating filter-based methods achieve the highest performance. In exchange, they need the most time in the process, because they solve optimization problems for a large Hankel matrix H (X). For annihilating filter-based methods, a stop criterion is developed for a good perfor- mance of denoising. Without the stop criterion, it often uses more computing resources to obtain lower performance. Further, we propose an approach that solves the optimization problem separately, for matrix H (X), we split it into p × q pieces and apply low-rank op- timization to each piece. With different choices of p × q, we can make a trade-off between PSNR value and time cost in some cases. The interpretation of the Hankel matrix structure is explained in detail. We also in- troduce the inverse operation of the operator H (·). To apply annihilating filter-based methods to a color image, we define H3D(X) for a color image, and we explain how to construct H3D(X).