频域的布特沃斯低通滤波_机器视觉 03.3 频域高通滤波（锐化）

- 日理万妓 2022-10-25 14:08 192阅读 0赞

![feb53d320ece69815134ce26da7392d5.png][]

## I 理想高通滤波器 ##

传递函数：

![equation?tex=H%28u%2Cv%29%3D%5Cbigg%5C%7B+%5Cmatrix%7B+0+%26+dis%28u%2Cv%29%3CD\_0%5C%5C+1%26dis%28u%2Cv%29%3ED\_0+%7D+][equation_tex_H_28u_2Cv_29_3D_5Cbigg_5C_7B_5Cmatrix_7B_0_26_dis_28u_2Cv_29_3CD_0_5C_5C_1_26dis_28u_2Cv_29_3ED_0_7D]

理想高通滤波器与理想低通滤波器刚好相反，把小于截止频率D0的频率部分全部置0。这会把图像中灰度变化缓慢的低频部分去除，保留下了灰度变化快的区域，比如边线，噪声等。

![1f39a9c0b8cad16f83afca63baaae160.png][]

理想理想低通滤波器传递函数图

特点：和ILPF一样，也是因为截断位置尖锐的截断，会出现振铃现象。详情见上一节。

## II 布特沃斯高通滤波器 ##

传递函数：

![equation?tex=H%28u%2Cv%29%3D%5Cfrac%7B1%7D%7B1%2B%5Cleft+%5B%5Cfrac+%7BD\_0%7D%7Bdis%28u%2Cv%29%7D%5Cright%5D%5E%7B2n%7D%7D][equation_tex_H_28u_2Cv_29_3D_5Cfrac_7B1_7D_7B1_2B_5Cleft_5B_5Cfrac_7BD_0_7D_7Bdis_28u_2Cv_29_7D_5Cright_5D_5E_7B2n_7D_7D]

比值处刚好相反。

特征和低通一样：①低阶无震铃，高阶有振铃；②想比与高斯，分水岭效果更好。

## III 高斯高通滤波器 ##

传递函数：

![equation?tex=H%28u%2Cv%29%3D1-e%5E%7B-dis%5E2\_%7B%28u%2Cv%29%7D%2F2D\_0%5E2%7D][equation_tex_H_28u_2Cv_29_3D1-e_5E_7B-dis_5E2_7B_28u_2Cv_29_7D_2F2D_0_5E2_7D]

优点：完全没有振铃现象；

缺点：曲线太平缓，截断效果不明显；

![414afdcfb328f5d16fb866b1f7daf099.png][]

上：高斯高通滤波器；下：布特沃斯高通滤波器

## IV 前三种滤波器的结果 ##

![2681539c3cbac47f04073988a5473aab.png][]

ad：理想高通；be：高斯高通；ef：布特沃斯高通

从滤波结果可以看出，由于高通滤波器把直流分量

![equation?tex=H%280%2C0%29%3D0][equation_tex_H_280_2C0_29_3D0] ,所以平均灰度值等于0，图像本应有很多负数，为了显示都截断成0了，所以整体色调很暗。

布特沃斯和高斯都没有振铃现象。

## V 频域拉普拉斯滤波 ##

拉普拉斯的传递函数是

![equation?tex=H%28u%2Cv%29%3D-4%5Cpi%5E2dis%5E2%28u%2Cv%29][equation_tex_H_28u_2Cv_29_3D-4_5Cpi_5E2dis_5E2_28u_2Cv_29] .

空间域的拉普拉斯图像为

![equation?tex=%5Cnabla+f%5E2%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D+%5C%7BH%28u%2Cv%29F%28u%2Cv%29%5C%7D+][equation_tex_5Cnabla_f_5E2_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5C_7BH_28u_2Cv_29F_28u_2Cv_29_5C_7D]

用拉普拉斯做图像增强：

![equation?tex=g%28x%2Cy%29%3Df%28x%2Cy%29%2Bc%2A%5Cnabla%5E2f%28x%2Cy%29][equation_tex_g_28x_2Cy_29_3Df_28x_2Cy_29_2Bc_2A_5Cnabla_5E2f_28x_2Cy_29]

![3e88c070d6b9a27f6a8f4ec1b7a48b4e.png][]

## VI 频域钝化掩模 ##

掩模的计算：

![equation?tex=g\_%7BMask%7D%28x%2Cy%29%3Df%28x%2Cy%29-f\_%7BLP%7D%28x%2Cy%29][equation_tex_g_7BMask_7D_28x_2Cy_29_3Df_28x_2Cy_29-f_7BLP_7D_28x_2Cy_29]

其中的平滑图像是来自

![equation?tex=f\_%7BLP%7D%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D%5C%7B+H\_%7BLP%7D%28u%2Cv%29F%28u%2Cv%29+%5C%7D][equation_tex_f_7BLP_7D_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5C_7B_H_7BLP_7D_28u_2Cv_29F_28u_2Cv_29_5C_7D]

锐化的结果为：

![equation?tex=g\_%7Bsharp%7D%28x%2Cy%29%3Df%28x%2Cy%29%2Bc%2Ag\_%7BMask%7D%28x%2Cy%29][equation_tex_g_7Bsharp_7D_28x_2Cy_29_3Df_28x_2Cy_29_2Bc_2Ag_7BMask_7D_28x_2Cy_29]

上面过程的推导也可以直接在频域中进行

![equation?tex=g\_%7Bsharp%7D%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D%5Cleft%5C%7B+%5Cbigg%5B1%2Bc%2A%281-H\_%7BLP%7D%28u%2Cv%29%29%5Cbigg%5DF%28u%2Cv%29+%5Cright%5C%7D][equation_tex_g_7Bsharp_7D_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5Cleft_5C_7B_5Cbigg_5B1_2Bc_2A_281-H_7BLP_7D_28u_2Cv_29_29_5Cbigg_5DF_28u_2Cv_29_5Cright_5C_7D]

上式就是**频域钝化掩模**的表达。

如果换做频域高通滤波器来锐化，即**高频强调滤波器**

![equation?tex=g\_%7Bsharp%7D%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D%5Cleft%5C%7B+%5Cbigg%5B1%2Bc%2AH\_%7BHP%7D%28u%2Cv%29%5Cbigg%5DF%28u%2Cv%29+%5Cright%5C%7D][equation_tex_g_7Bsharp_7D_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5Cleft_5C_7B_5Cbigg_5B1_2Bc_2AH_7BHP_7D_28u_2Cv_29_5Cbigg_5DF_28u_2Cv_29_5Cright_5C_7D]

![aff18fe4b11d8fbf2324ad7b1a69669f.png][]

a. 原图； b. 高斯高通滤波器；c：高频强调滤波器；d.对c做直方图均衡。

[feb53d320ece69815134ce26da7392d5.png]: /images/20221024/dab10bf4fdc842329ecce77191a19ac3.png
[equation_tex_H_28u_2Cv_29_3D_5Cbigg_5C_7B_5Cmatrix_7B_0_26_dis_28u_2Cv_29_3CD_0_5C_5C_1_26dis_28u_2Cv_29_3ED_0_7D]: https://www.zhihu.com/equation?tex=H%28u%2Cv%29%3D%5Cbigg%5C%7B+%5Cmatrix%7B+0+%26+dis%28u%2Cv%29%3CD_0%5C%5C+1%26dis%28u%2Cv%29%3ED_0+%7D+
[1f39a9c0b8cad16f83afca63baaae160.png]: /images/20221024/d99ea29b22e043b2bf57a1fd9b74a141.png
[equation_tex_H_28u_2Cv_29_3D_5Cfrac_7B1_7D_7B1_2B_5Cleft_5B_5Cfrac_7BD_0_7D_7Bdis_28u_2Cv_29_7D_5Cright_5D_5E_7B2n_7D_7D]: https://www.zhihu.com/equation?tex=H%28u%2Cv%29%3D%5Cfrac%7B1%7D%7B1%2B%5Cleft+%5B%5Cfrac+%7BD_0%7D%7Bdis%28u%2Cv%29%7D%5Cright%5D%5E%7B2n%7D%7D
[equation_tex_H_28u_2Cv_29_3D1-e_5E_7B-dis_5E2_7B_28u_2Cv_29_7D_2F2D_0_5E2_7D]: /images/20221024/e90083bc3c3b43edbd898ec70dd23304.png
[414afdcfb328f5d16fb866b1f7daf099.png]: /images/20221024/4db17f16d9b646d2b645e54826e34d0a.png
[2681539c3cbac47f04073988a5473aab.png]: /images/20221024/a5ff4a0ac7dd43a9b4aed763f2410206.png
[equation_tex_H_280_2C0_29_3D0]: /images/20221024/55c84463bb124ea7863291173cdfb0d3.png
[equation_tex_H_28u_2Cv_29_3D-4_5Cpi_5E2dis_5E2_28u_2Cv_29]: /images/20221024/6a5d2a113e504aecbcb4753ef968ad30.png
[equation_tex_5Cnabla_f_5E2_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5C_7BH_28u_2Cv_29F_28u_2Cv_29_5C_7D]: https://www.zhihu.com/equation?tex=%5Cnabla+f%5E2%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D+%5C%7BH%28u%2Cv%29F%28u%2Cv%29%5C%7D+
[equation_tex_g_28x_2Cy_29_3Df_28x_2Cy_29_2Bc_2A_5Cnabla_5E2f_28x_2Cy_29]: /images/20221024/2b54e32b4ac741eb81c1f35b13bb7722.png
[3e88c070d6b9a27f6a8f4ec1b7a48b4e.png]: /images/20221024/9c72835694744ad396f3e4ac734160af.png
[equation_tex_g_7BMask_7D_28x_2Cy_29_3Df_28x_2Cy_29-f_7BLP_7D_28x_2Cy_29]: /images/20221024/52ad6559d80340998fc3eddf51252cbf.png
[equation_tex_f_7BLP_7D_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5C_7B_H_7BLP_7D_28u_2Cv_29F_28u_2Cv_29_5C_7D]: https://www.zhihu.com/equation?tex=f_%7BLP%7D%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D%5C%7B+H_%7BLP%7D%28u%2Cv%29F%28u%2Cv%29+%5C%7D
[equation_tex_g_7Bsharp_7D_28x_2Cy_29_3Df_28x_2Cy_29_2Bc_2Ag_7BMask_7D_28x_2Cy_29]: /images/20221024/68d2ac9361394ce0bfb08f725502f188.png
[equation_tex_g_7Bsharp_7D_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5Cleft_5C_7B_5Cbigg_5B1_2Bc_2A_281-H_7BLP_7D_28u_2Cv_29_29_5Cbigg_5DF_28u_2Cv_29_5Cright_5C_7D]: https://www.zhihu.com/equation?tex=g_%7Bsharp%7D%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D%5Cleft%5C%7B+%5Cbigg%5B1%2Bc%2A%281-H_%7BLP%7D%28u%2Cv%29%29%5Cbigg%5DF%28u%2Cv%29+%5Cright%5C%7D
[equation_tex_g_7Bsharp_7D_28x_2Cy_29_3D_5Cmathscr_7BF_7D_5E_7B-1_7D_5Cleft_5C_7B_5Cbigg_5B1_2Bc_2AH_7BHP_7D_28u_2Cv_29_5Cbigg_5DF_28u_2Cv_29_5Cright_5C_7D]: https://www.zhihu.com/equation?tex=g_%7Bsharp%7D%28x%2Cy%29%3D%5Cmathscr%7BF%7D%5E%7B-1%7D%5Cleft%5C%7B+%5Cbigg%5B1%2Bc%2AH_%7BHP%7D%28u%2Cv%29%5Cbigg%5DF%28u%2Cv%29+%5Cright%5C%7D
[aff18fe4b11d8fbf2324ad7b1a69669f.png]: /images/20221024/579007af725940c0984fa3eae813ce2c.png