最近看到一篇文章講IMAGE DECOMPOSITION,裏面提到了將圖像分爲Texture layer和Structure layer,測試了很多方法,對於那些具有非常強烈紋理的圖像,總覺得用TV去燥的方法分離的結果都比其他的方法都要好(比如導向、雙邊),比如下圖:
再比如:
可見TV可以把紋理很好的提取出來。
現在應該能找到很多的TV代碼,比如IPOL上就有,詳見 http://www.ipol.im/pub/art/2013/61/。
我在其他地方也見過一些,比如這裏: http://yu-li.github.io/paper/li_eccv14_jpeg.zip,他是藉助於FFT實現的,當然少不了多次迭代,速度也是比較慢的。
我還收藏了很久前一位朋友寫的M代碼,但是現在我不知道把他QQ或者微信弄到哪裏去了,也不知道他會不會介意我把他的代碼分享出來。
function dualROF()
clc
f0=imread('rr.bmp');
f0=f0(:,:,1);
[m,n]=size(f0);
f0=double(f0);
lamda=30; % smoothness paramter, the larger the smoother
tao=.125; % fixed do not change it.
p1=zeros(m,n);
p2=zeros(m,n);
tic
for step=1:100
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
div_p=div(p1,p2);
cx=Fx(div_p-f0/lamda);
cy=Fy(div_p-f0/lamda);
abs_c=sqrt(cx.^2+cy.^2);
p1=(p1+tao*cx)./(1+tao*abs_c);
p2=(p2+tao*cy)./(1+tao*abs_c);
end
u=f0-lamda*div_p;
toc
figure; imagesc(f0); colormap(gray); axis off; axis equal;
figure; imagesc(u); colormap(gray); axis off; axis equal;
% Compute divergence using backward derivative
function f = div(a,b)
f = Bx(a)+By(b);
% Forward derivative operator on x with boundary condition u(:,:,1)=u(:,:,1)
function Fxu = Fx(u)
[m,n] = size(u);
Fxu = circshift(u,[0 -1])-u;
Fxu(:,n) = zeros(m,1);
% Forward derivative operator on y with boundary condition u(1,:,:)=u(m,:,:)
function Fyu = Fy(u)
[m,n] = size(u);
Fyu = circshift(u,[-1 0])-u;
Fyu(m,:) = zeros(1,n);
% Backward derivative operator on x with boundary condition Bxu(:,1)=u(:,1)
function Bxu = Bx(u)
[~,n] = size(u);
Bxu = u - circshift(u,[0 1]);
Bxu(:,1) = u(:,1);
Bxu(:,n) = -u(:,n-1);
% Backward derivative operator on y with boundary condition Bxu(1,:)=u(1,:)
function Byu = By(u)
[m,~] = size(u);
Byu = u - circshift(u,[1 0]);
Byu(1,:) = u(1,:);
Byu(m,:) = -u(m-1,:);
M的代碼,代碼量不大,那是因爲Matlab的向量化確實很厲害,但是這個代碼還是很慢的,256*256的灰度圖迭代100次都要700ms了。
這裏拋開一些優化不說,用這個circshift會造成很大的性能損失,我們稍微分析下就能看到用這個地方其實就是簡單的水平或者垂直方向的差分,完全沒有必要這樣寫。
直接按照代碼的意思用C語言把他們展開並不做其他的優化可得到大概下面這種不怎麼好的代碼:
int IM_DualTVDenoising(unsigned char *Src, unsigned char *Dest, int Width, int Height, int Stride, float Lamda = 20 , int Iter = 20)
{
int Channel = Stride / Width;
if ((Src == NULL) || (Dest == NULL)) return IM_STATUS_NULLREFRENCE;
if ((Width <= 0) || (Height <= 0)) return IM_STATUS_INVALIDPARAMETER;
if ((Channel != 1) && (Channel != 3) && (Channel != 4)) return IM_STATUS_INVALIDPARAMETER;
if (Channel == 1)
{
float tao = 0.125; // fixed do not change it.
float InvLamda = 1.0 / Lamda;
float *p1 = (float *)malloc(Width * Height * sizeof(float));
float *p2 = (float *)malloc(Width * Height * sizeof(float));
float *div_p = (float *)malloc(Width * Height * sizeof(float));
float *cx = (float *)malloc(Width * Height * sizeof(float));
float *cy = (float *)malloc(Width * Height * sizeof(float));
float *Temp = (float *)malloc(Width * Height * sizeof(float));
int X, Y;
float q1, q2, q, abs_c;
float *LineP1, *LineP2, *LineP3, *LineP4;
unsigned char *LinePS, *LinePD;
for (int Z = 0; Z < Iter; Z++)
{
//Div(p1, p2, div_p);
for (Y = 0; Y < Height; Y++)
{
LineP1 = p1 + Y * Width; //Fx(Temp, cx);
LineP2 = cx + Y * Width;
LineP2[0] = LineP1[0];
for (X = 1; X < Width; X++)
{
LineP2[X] = LineP1[X] - LineP1[X - 1];
}
LineP2[Width - 1] = -LineP1[Width - 2];
}
memcpy(cy, p2, Width * sizeof(float));
for (Y = 1; Y < Height; Y++)
{
LineP1 = (float *)(p2 + (Y - 1)* Width);
LineP2 = (float *)(p2 + Y * Width); //Fy(Temp, cy);
LineP3 = (float *)(cy + Y * Width);
for (X = 0; X < Width; X++)
{
LineP3[X] = LineP2[X] - LineP1[X];
}
}
LineP1 = (float *)(p2 + (Height - 2) * Width);
LineP2 = (float *)(cy + (Height - 1) * Width);
for (X = 0; X < Width; X++)
{
LineP2[X] = -LineP1[X];
}
for (Y = 0; Y < Height; Y++)
{
LineP1 = (float *)(cx + Y * Width);
LineP2 = (float *)(cy + Y * Width); //Fy(Temp, cy);
LineP3 = (float *)(div_p + Y * Width);
for (X = 0; X < Width; X++)
{
LineP3[X] = LineP1[X] + LineP2[X];
}
}
for (Y = 0; Y < Height; Y++)
{
LineP1 = (float *)(div_p + Y * Width);
LineP2 = (float *)(Temp + Y * Width);
LinePS = Src + Y * Stride;
for (X = 0; X < Width; X++)
{
LineP2[X] = LineP1[X] - LinePS[X] * InvLamda;
}
}
for (Y = 0; Y < Height; Y++)
{
LineP1 = (float *)(Temp + Y * Width); //Fx(Temp, cx);
LineP2 = (float *)(cx + Y * Width);
for (X = 0; X < Width - 1; X++)
{
LineP2[X] = LineP1[X + 1] - LineP1[X];
}
LineP2[Width - 1] = 0;
}
for (Y = 0; Y < Height - 1; Y++)
{
LineP1 = (float *)(Temp + Y * Width);
LineP2 = (float *)(Temp + (Y + 1) * Width); //Fy(Temp, cy);
LineP3 = (float *)(cy + Y * Width);
for (X = 0; X < Width; X++)
{
LineP3[X] = LineP2[X] - LineP1[X];
}
}
memset(Temp + (Height - 1) * Width, 0, Width * sizeof(float));
for (Y = 0; Y < Height; Y++)
{
LineP1 = (float *)(p1 + Y * Width);
LineP2 = (float *)(p2 + Y * Width);
LineP3 = (float *)(cx + Y * Width);
LineP4 = (float *)(cy + Y * Width);
for (X = 0; X < Width; X++)
{
abs_c = sqrt(LineP3[X] * LineP3[X] + LineP4[X] * LineP4[X]);
abs_c = 1 / (1 + tao * abs_c);
LineP1[X] = (LineP1[X] + tao * LineP3[X]) * abs_c;
LineP2[X] = (LineP2[X] + tao * LineP4[X]) * abs_c;
}
}
}
for (Y = 0; Y < Height; Y++)
{
LineP1 = (float *)(div_p + Y * Width);
LinePS = Src + Y * Stride;
LinePD = Dest + Y * Stride;
for (X = 0; X < Width; X++)
{
LinePD[X] = IM_ClampToByte((int)(LinePS[X] - Lamda * LineP1[X]));
}
}
free(p1);
free(p2);
free(div_p);
free(cx);
free(cy);
free(Temp);
}
else
{
}
}
算法明顯佔用很大的內存,而且看起來彆扭,不過速度還是槓槓的,256*256的灰度圖迭代100次都要30ms了。反編譯看了下代碼,編譯器對代碼做了很好的SIMD指令優化。
上面的C語言還是可以繼續優化的,這就需要大家自己的認真的去研讀代碼深層次的邏輯關係了,實際上可以只要上面的一半的臨時內存的,而且很多計算可以集中在一個循環裏完成,可以手動內嵌SIMD指令,或者直接使用編譯器的優化能力,基本上這樣的簡單的算法邏輯編譯器編譯後的速度不會比我們手寫的SIMD指令慢,有的時候還是會快一些,不得不佩服那些寫編譯器的大牛。優化後的速度大概在14ms左右。
研究TV算法需要很好的數學功底,以前朋友曾經給我寄過一本書,裏面都是微分方面的數學公式,看的我嚇死了,不過TV算法似乎有很多很好的應用,也曾經流行過一段時間,可惜現在深度學習一出來,很多人都喜歡這種直接從海量數據中建造黑盒模型,而對那些有着很明顯的數學邏輯的算法嗤之以鼻了,真有點可惜。
以前在基於總變差模型的紋理圖像中圖像主結構的提取方法 一文中曾提到那個論文附帶的Matlab代碼沒有什麼意義,因爲他很難轉換成C的代碼,即時轉換成功了,也處理不了大圖,但是本文這裏的TV算法總的來說在內存佔用或者速度方面都還令人滿意。
在去噪效果上,這個算法還算可以:
本文Demo下載地址: http://files.cnblogs.com/files/Imageshop/SSE_Optimization_Demo.rar, 算法位於Denoise --> TV Denoising下。