在近期內對於活細胞的三維結構的研究中，共軛焦雷射顯微鏡(CLSM)已經發展成為一項優秀的顯微成像技術。共軛焦雷射顯微鏡能夠產生出具有高解析度與高對比的影像，這是其相較於其他顯微技術最大的突破。藉由放置一個針孔於物鏡與細胞的共軛焦平面上，共軛焦顯微鏡能夠阻擋絕大部分的失焦光線，避免其進入成像感測器，從而得到更加清晰的焦平面影像。在取得多重深度的焦平面影像後，我們就能夠藉此重構出細胞樣本內的三維體積。然而，共軛焦顯微鏡仍有其限制。由物體與點擴散函數(PSF)之摺積所產生的影像扭曲對於光軸(Z方向)上的影像清晰度造成很大的影響，而在橫軸上的影響則由物鏡聚焦的效果而變得輕微許多。這樣的缺陷可能會限制我們取得的三維結構影像的可靠性。因此，還原點擴散函數造成的影像扭曲在三維結構的觀測上相當重要。
我們提出了一種奠基於三維總變異量與三維點擴散函數模型的三維反摺積方法，用以還原已經扭曲模糊的三維影像體積。在此演算法中，我們需要先決定要使用何種形態的點擴散函數的理論模型，而此模型必須能夠準確描述共軛焦顯微鏡系統的光學特性。之後，當我們最小化受到三維總變異量限制的目標函式的時後，這樣的過程能夠被看做是一種反摺積方法，用以還原點擴散函數的影響。使用三維總變異量做為限制函式而非使用其他種類的限制函式，其原因在於以往的研究中，三維總變異量限制函式能夠在還原影像的過程中，保存重要的特徵資訊，像是清晰的邊緣，這是其他限制函式所無法達成的。而這些特徵資訊通常在細胞影像的研究中非常關鍵。然而，三維總變異量反摺積是一種非線性及不可微的方法，而最小化這樣的函式通常需要大量的運算。因此在我們的反摺積方法中，使用了「半二次函式」的概念來分離變數，藉此達成加速運算的效果。

Recently in the study of three-dimensional structure of a living cell, confocal laser scanning microscope (CLSM) has developed to become an excellent technique to the research of biological specimens. CLSM can acquire images with higher resolution and better contrast. It generates clearer images at various depths by blocking out-of-focus light through a spatial pinhole. With the obtained multiple focal plane images, biologists are able to reconstruct the three dimensional volume of the specimen. However, there are still several limitations on the performance of CLSM. The aliasing effect caused by point spread function (PSF) along the optical axis is much worse than that along the lateral axis. This disadvantage may restrain the spatial reliability of the reconstructed three dimensional volume data of the specimen. Therefore, recovering images from such aliasing effect has become an important goal.
We propose a 3-D deconvolution method based on total variation regularizer and a parametric theoretical PSF model to reconstruct the corrupted 3-D image volume. In this algorithm, we first determine a theoretical PSF model that can best estimate the system optical properties of CLSM. Then by minimizing an objective function regularized by total variation regularizer, this process could be taken as an inverse procedure of PSF. The reason to use total variation regularizer rather than other regularizer is that it preserves important features while recovering the image. Since total variation deconvolution is non-linear and non-differentiable, it is very expensive in computation to minimize the objective function. Therefore, the concept of half-quadratic function is applied to split variables, so as to accelerate the computation.

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