Research on TV-L1 Optical Flow Model for Image Registration Based on Fractional-order Differentiation
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摘要: 图像的非刚性配准在计算机视觉和医学图像分析中有着重要的作用.TV-L1（全变分L1范数、Total variation-L1）光流模型是解决非刚性配准问题的有效方法，但TV-L1光流模型的正则项是一阶导数，会导致纹理特征等具有弱导数性质的信息模糊.针对该问题，将G-L（Grünwald-Letnikov）分数阶引入TV-L1光流模型，提出基于G-L分数阶微分的TV-L1光流模型，并应用原始-对偶算法求解该模型.新的模型用G-L分数阶微分代替正则项中的一阶导数，由于分数阶微分比整数阶微分具有更好的细节描述能力，并能有效地、非线性地保留具有弱导数性质的纹理特征，从而提高图像的配准精度.另外，通过实验给出了配准精度与G-L分数阶模板参数之间的关系，从而为模板最佳参数的选取提供了依据.尽管不同类型的图像其最佳参数是不同的，但是其最佳配准阶次一般在1 ~2之间.理论分析和实验结果均表明，提出的新模型能够有效地提高图像配准的精度，适合于包含较多弱纹理和弱边缘信息的医学图像配准，该模型是TV-L1光流模型的重要延伸和推广.Abstract: In computer vision and medical image analysis, non-rigid image registration is a challenging task. TV-L1 (Total variation-L1) optical flow model has been proved to be an effective method in the field of non-rigid image registration. It can solve the problem of fuzzy edge caused by smooth displacement fields of Horn-Schunck, but its first-order derivative in regularization term leads to fuzzy texture information with weak derivative property. Aiming at the problem, this paper introduces G-L (Grünwald-Letnikov) fractional differentiation to TV-L1 optical flow model, and proposes a new TV-L1 optical flow model based on fractional differentiation, and then finds the solution of the model using primal-dual algorithm. In this paper we use Grünwald-Letnikov fractional order differential instead of the first-order derivative in the regularization term for its better ability of detail description than first-order's. Then we purposefully control to retain or inhibit the texture information with weak derivative nature, thus improving the registration accuracy. Experimental results show that the proposed method has a better registration accuracy in registration of texture information with weak derivative, and that it can be considered an important extension and generalization of TV-L1 optical flow modes.1) 本文责任编委 张长水
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图 4 验图像(其中第一行为参考图像, 第二行为与参考图像对应的浮动图像)
Fig. 4 Experimental images (The first line are reference images, the second line are floating images corresponding to reference images)
图 5 Lena图像实验(第一行为浮动图像和配准后图像, 第二行为差值图像. (a)为浮动图像, (b) $\sim$ (e)为配准后的图像; (b) TV-L$^{1}$方法; (c)本文方法($\alpha=1.2, k=1$); (d)本文方法($\alpha=1.2, k=2$); (e)本文方法($\alpha=1.2, k=3$); (f) $\sim$ (j)分别为第一行图像与参考图像(图 4(a))的差值图像)
Fig. 5 Lena image (The first line is floating image and registered image, the second line is difference image. (a) Floating image, (b) $\sim$ (e) are registered images, (b) TV-L$^{1}$, (c) Our method ($\alpha=1.2, k=1$), (d) Our method ($\alpha=1.2, k=2$), (e) Our method ($\alpha=1.2, k=3$), (f) $\sim$ (j) are difference images)
图 7 Brain1图像实验(第一行为浮动图像和配准后图像, 第二行为差值图像. (a)为浮动图像, (b) $\sim$ (e)为配准后的图像; (b) TV-L$^{1}$方法; (c)本文方法($\alpha=1.3, k=1$); (d)本文方法($\alpha=1.3, k=2$); (e)本文方法($\alpha=1.3, k=3$); (f) $\sim$ (j)分别为第一行图像与参考图像(图 4(b))的差值图像)
Fig. 7 Brain1 image (The first line is floating image and registered image, the second line is difference image. (a) Floating image, (b) $\sim$ (e) are registered images, (b) TV-L$^{1}$, (c) Our method ($\alpha=1.3, k=1$), (d) Our method ($\alpha=1.3, k=2$), (e) Our method ($\alpha=1.3, k=3$), (f) $\sim$ (j) are difference images)
图 8 Brain2图像实验(第一行为浮动图像和配准后图像, 第二行为差分图像. (a)为浮动图像, (b) $\sim$ (e)为配准后的图像; (b) TV-L$^{1}$方法; (c)本文方法($\alpha=1.3$, $k=1$); (d)本文方法($\alpha=1.3$, $k=2$); (e)本文方法($\alpha=1.3$, $k=3$); (f) $\sim$ (j)分别为第一行图像与参考图像(图 4(c))的差值图像)
Fig. 8 Brain2 image. The first line is floating image and registered image, the second line is difference image ((a) floating image, (b) $\sim$ (e) are registered images, (b) TV-L$^{1}$, (c) Our method ($\alpha=1.3$, $k=1$), (d) Our method ($\alpha=1.3$, $k=2$), (e) Our method ($\alpha=1.3$, $k=3$), (f) $\sim$ (j) are difference images)
表 1 参考图像和配准图像的均方误差(MSE)
Table 1 Mean square error (MSE) of reference image and registered
输入图片 配准前 TV-L $^{1}$ 本文算法($k$ = 1) 本文算法($k$ = 2) 本文算法($k$ = 3) Lena ($\alpha$ = 1.2) 669.33 14.86 10.36 9.17 11.56 Brain1 ($\alpha$ = 1.3) 295.85 27.51 18.20 15.93 20.22 Brain2 ($\alpha$ = 1.3) 813.77 31.02 11.47 9.95 17.89 
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表 2 峰值信噪比(PSNR)
Table 2 Peak signal to noise ratio (PSNR)
输入图片 配准前 TV-L $^{1}$ 本文算法($k$ = 1) 本文算法($k$ = 2) 本文算法($k$ = 3) Lena ($\alpha$ = 1.2) 19.32 35.55 38.18 38.88 37.50 Brain1 ($\alpha$ = 1.3) 22.78 33.73 35.34 36.17 34.89 Brain2 ($\alpha$ = 1.3) 19.03 31.21 37.73 38.15 35.60
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