目标跟踪VOT-ECO中，cubic_spline_fourier函数推导与源码解释

客官°小女子只卖身不卖艺 2022-05-08 14:04 153阅读 0赞

读visual object tracking --　eco (Efficient Convolution Operator (ECO) tracker.) 算法的源码，一时好奇，想自己推导一下这个cubic spline kernel的计算函数。

作者：[Ｍartin-danelljan][artin-danelljan]　原项目地址：[https://github.com/martin-danelljan/ECO][https_github.com_martin-danelljan_ECO]

原来的matlab是这样写的，

function bf = cubic_spline_fourier(f, a)
    
    % The continuous Fourier transform of a cubic spline kernel.
    
    bf = -(- 12*a + 12*exp(-pi*f*2i) + 12*exp(pi*f*2i) + 6*a*exp(-pi*f*4i) + ...
        6*a*exp(pi*f*4i) + f.*(pi*exp(-pi*f*2i)*12i) - f.*(pi*exp(pi*f*2i)*12i) + ...
        a*f.*(pi*exp(-pi*f*2i)*16i) - a*f.*(pi*exp(pi*f*2i)*16i) + ...
        a*f.*(pi*exp(-pi*f*4i)*4i) - a*f.*(pi*exp(pi*f*4i)*4i) - 24)./(16*f.^4*pi^4);
    
    bf(f == 0) = 1;

首先是自己走了些弯路，不知道作者到底用的哪个cubic spline, 找了一通，最后发现是这个函数(很普通，但没找到之前让好一番好猜，再后来才发现作者提供的supplementary material中有说明)，详情可参考[wikimedia][] ：

[https://en.wikipedia.org/wiki/Bicubic\_interpolation][wikimedia]

[![f(x) =\\left\\\{\\begin\{matrix\} 0 & x<-2 \\\\ -a x^3-5 a x^2-8 a x-4 a & -2<x\\leq -1 \\\\ (-a-2) x^3-(a+3) x^2+1 & -1<x\\leq 0 \\\\ (a+2) x^3-(a+3) x^2+1 & 0<x\\leq 1 \\\\ a x^3-5 a x^2+8 a x-4 a & 1<x\\leq 2 \\\\ 0 & x>2 \\end\{matrix\}\\right.][f_x_ _left_begin_matrix_ 0 _ x_-2 _ -a x_3-5 a x_2-8 a x-4 a _ -2_x_leq -1 _ _-a-2_ x_3-_a_3_ x_2_1 _ -1_x_leq 0 _ _a_2_ x_3-_a_3_ x_2_1 _ 0_x_leq 1 _ a x_3-5 a x_2_8 a x-4 a _ 1_x_leq 2 _ 0 _ x_2 _end_matrix_right.]][wikimedia]

对该函数作傅立叶变换，得到  
![\\frac\{\\sqrt\{\\frac\{2\}\{\\pi \}\} e^\{-2 i t\} \\left(a \\left(-1+e^\{2 i t\}\\right) \\left(4 i e^\{i t\} t+i t+i e^\{2 i t\} (t+3 i)+3\\right)+3 e^\{i t\} \\left(-1+e^\{i t\}\\right) \\left(i t+i e^\{i t\} (t+2 i)+2\\right)\\right)\}\{t^4\}][frac_sqrt_frac_2_pi _ e_-2 i t_ _left_a _left_-1_e_2 i t_right_ _left_4 i e_i t_ t_i t_i e_2 i t_ _t_3 i_3_right_3 e_i t_ _left_-1_e_i t_right_ _left_i t_i e_i t_ _t_2 i_2_right_right_t_4]

表达式有点乱，因为最终和作者给出的相差一个（不影响计算结果的）系数，我这里把这个系数去掉

![\\sqrt\{\\frac\{1\}\{2\\pi \}\}][sqrt_frac_1_2_pi]

显然前面的t相当于我们平常说的![2\\omega t][2_omega t]，代进去，再整理一下顺序就和作者提供的完全一样了　:-)，

![\\frac\{1\}\{16\\pi^4t^4\}(-24 - 12 a + 12 e^\{\_\{-2\\pi i t\}\} + 12e^\{\_\{2\\pi i t\}\} + 6a e^\{\_\{-4\\pi i t\}\} + 6a e^\{\_\{4\\pi i t\}\} + 12\\pi ite^\{\_\{-2\\pi i t\}\} - 12\\pi it e^\{\_\{2\\pi i t\}\} + 16\\pi ait e^\{\_\{-2\\pi i t\}\} - 16\\pi ait e^\{\_\{2\\pi i t\}\} + 4\\pi ait e^\{\_\{-4\\pi i t\}\} - 4\\pi ait e^\{\_\{4\\pi i t\}\})][frac_1_16_pi_4t_4_-24 - 12 a _ 12 e_-2_pi i t_ _ 12e_2_pi i t_ _ 6a e_-4_pi i t_ _ 6a e_4_pi i t_ _ 12_pi ite_-2_pi i t_ - 12_pi it e_2_pi i t_ _ 16_pi ait e_-2_pi i t_ - 16_pi ait e_2_pi i t_ _ 4_pi ait e_-4_pi i t_ - 4_pi ait e_4_pi i t]

另外，作者最后面加了一个负号，我看了下，如果没有这个负号，所以结果都是负值，所以必须转成正数。  
其实在计算时，最后只用到了函数的实部，取出来直接计算的话就是这样的

![\\frac\{1\}\{16\\pi^4t^4\}(-12a + 24cos(2\\pi t) + 12a cos(4\\pi t)+ 24\\pi t sin(2t\\pi) + 32a\\pi t sin(2t\\pi) + 8a\\pi t sin(4t\\pi) - 24)][frac_1_16_pi_4t_4_-12a _ 24cos_2_pi t_ _ 12a cos_4_pi t_ 24_pi t sin_2t_pi_ _ 32a_pi t sin_2t_pi_ _ 8a_pi t sin_4t_pi_ - 24]

如果要看Ｃ++源码的话，可以去这里看：[https://github.com/nicewsyly/ECO/blob/master/ECO/interpolator.cpp][https_github.com_nicewsyly_ECO_blob_master_ECO_interpolator.cpp]  
这里我把他贴出来，如下

cv::Mat interpolator::cubic_spline_fourier(cv::Mat f, float a)
    {
    	if (f.empty())
    		return cv::Mat();
    
    	cv::Mat bf(f.size(), CV_32FC1), temp1(f.size(), CV_32FC1), temp2(f.size(), CV_32FC1),
    		temp3(f.size(), CV_32FC1), temp4(f.size(), CV_32FC1);
    	std::transform(f.begin<float>(), f.end<float>(), temp1.begin<float>(), mat_cos2);
    	std::transform(f.begin<float>(), f.end<float>(), temp2.begin<float>(), mat_cos4);
    
    	std::transform(f.begin<float>(), f.end<float>(), temp3.begin<float>(), mat_sin2);
    	std::transform(f.begin<float>(), f.end<float>(), temp4.begin<float>(), mat_sin4);
    
    	bf = -1 * (-12 * a * cv::Mat::ones(f.size(), CV_32FC1) + 24 * temp1 +
    		12 * a * temp2 + CV_PI * 24 * f.mul(temp3) +
    		CV_PI * a * 32 * f.mul(temp3) + CV_PI * 8 * a * f.mul(temp4) -
    		24 * cv::Mat::ones(f.size(), CV_32FC1));
    
    	cv::Mat L(f.size(), CV_32FC1);
    	cv::pow(f, 4, L);
    	cv::divide(bf, 16 * L * cv::pow(CV_PI, 4), bf);
    	bf.at<float>(bf.rows / 2, bf.cols / 2) = 1;
    
    	return bf;
    }

嗯，希望看到的童鞋可以省点时间。

Ｌatex写长公式好累，眼都花了！

[artin-danelljan]: https://github.com/martin-danelljan
[https_github.com_martin-danelljan_ECO]: https://github.com/martin-danelljan/ECO
[wikimedia]: https://en.wikipedia.org/wiki/Bicubic_interpolation
[f_x_ _left_begin_matrix_ 0 _ x_-2 _ -a x_3-5 a x_2-8 a x-4 a _ -2_x_leq -1 _ _-a-2_ x_3-_a_3_ x_2_1 _ -1_x_leq 0 _ _a_2_ x_3-_a_3_ x_2_1 _ 0_x_leq 1 _ a x_3-5 a x_2_8 a x-4 a _ 1_x_leq 2 _ 0 _ x_2 _end_matrix_right.]: https://private.codecogs.com/gif.latex?f%28x%29%20%3D%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%200%20%26%20x%3C-2%20%5C%5C%20-a%20x%5E3-5%20a%20x%5E2-8%20a%20x-4%20a%20%26%20-2%3Cx%5Cleq%20-1%20%5C%5C%20%28-a-2%29%20x%5E3-%28a&plus;3%29%20x%5E2&plus;1%20%26%20-1%3Cx%5Cleq%200%20%5C%5C%20%28a&plus;2%29%20x%5E3-%28a&plus;3%29%20x%5E2&plus;1%20%26%200%3Cx%5Cleq%201%20%5C%5C%20a%20x%5E3-5%20a%20x%5E2&plus;8%20a%20x-4%20a%20%26%201%3Cx%5Cleq%202%20%5C%5C%200%20%26%20x%3E2%20%5Cend%7Bmatrix%7D%5Cright.
[frac_sqrt_frac_2_pi _ e_-2 i t_ _left_a _left_-1_e_2 i t_right_ _left_4 i e_i t_ t_i t_i e_2 i t_ _t_3 i_3_right_3 e_i t_ _left_-1_e_i t_right_ _left_i t_i e_i t_ _t_2 i_2_right_right_t_4]: https://private.codecogs.com/gif.latex?%5Cfrac%7B%5Csqrt%7B%5Cfrac%7B2%7D%7B%5Cpi%20%7D%7D%20e%5E%7B-2%20i%20t%7D%20%5Cleft%28a%20%5Cleft%28-1&plus;e%5E%7B2%20i%20t%7D%5Cright%29%20%5Cleft%284%20i%20e%5E%7Bi%20t%7D%20t&plus;i%20t&plus;i%20e%5E%7B2%20i%20t%7D%20%28t&plus;3%20i%29&plus;3%5Cright%29&plus;3%20e%5E%7Bi%20t%7D%20%5Cleft%28-1&plus;e%5E%7Bi%20t%7D%5Cright%29%20%5Cleft%28i%20t&plus;i%20e%5E%7Bi%20t%7D%20%28t&plus;2%20i%29&plus;2%5Cright%29%5Cright%29%7D%7Bt%5E4%7D
[sqrt_frac_1_2_pi]: https://private.codecogs.com/gif.latex?%5Csqrt%7B%5Cfrac%7B1%7D%7B2%5Cpi%20%7D%7D
[2_omega t]: https://private.codecogs.com/gif.latex?2%5Comega%20t
[frac_1_16_pi_4t_4_-24 - 12 a _ 12 e_-2_pi i t_ _ 12e_2_pi i t_ _ 6a e_-4_pi i t_ _ 6a e_4_pi i t_ _ 12_pi ite_-2_pi i t_ - 12_pi it e_2_pi i t_ _ 16_pi ait e_-2_pi i t_ - 16_pi ait e_2_pi i t_ _ 4_pi ait e_-4_pi i t_ - 4_pi ait e_4_pi i t]: https://private.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7B16%5Cpi%5E4t%5E4%7D%28-24%20-%2012%20a%20&plus;%2012%20e%5E%7B_%7B-2%5Cpi%20i%20t%7D%7D%20&plus;%2012e%5E%7B_%7B2%5Cpi%20i%20t%7D%7D%20&plus;%206a%20e%5E%7B_%7B-4%5Cpi%20i%20t%7D%7D%20&plus;%206a%20e%5E%7B_%7B4%5Cpi%20i%20t%7D%7D%20&plus;%2012%5Cpi%20ite%5E%7B_%7B-2%5Cpi%20i%20t%7D%7D%20-%2012%5Cpi%20it%20e%5E%7B_%7B2%5Cpi%20i%20t%7D%7D%20&plus;%2016%5Cpi%20ait%20e%5E%7B_%7B-2%5Cpi%20i%20t%7D%7D%20-%2016%5Cpi%20ait%20e%5E%7B_%7B2%5Cpi%20i%20t%7D%7D%20&plus;%204%5Cpi%20ait%20e%5E%7B_%7B-4%5Cpi%20i%20t%7D%7D%20-%204%5Cpi%20ait%20e%5E%7B_%7B4%5Cpi%20i%20t%7D%7D%29
[frac_1_16_pi_4t_4_-12a _ 24cos_2_pi t_ _ 12a cos_4_pi t_ 24_pi t sin_2t_pi_ _ 32a_pi t sin_2t_pi_ _ 8a_pi t sin_4t_pi_ - 24]: https://private.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7B16%5Cpi%5E4t%5E4%7D%28-12a%20&plus;%2024cos%282%5Cpi%20t%29%20&plus;%2012a%20cos%284%5Cpi%20t%29&plus;%2024%5Cpi%20t%20sin%282t%5Cpi%29%20&plus;%2032a%5Cpi%20t%20sin%282t%5Cpi%29%20&plus;%208a%5Cpi%20t%20sin%284t%5Cpi%29%20-%2024%29
[https_github.com_nicewsyly_ECO_blob_master_ECO_interpolator.cpp]: https://github.com/nicewsyly/ECO/blob/master/ECO/interpolator.cpp