一、简介
全变分（Total variation），也称为全变差，是图象复原中常用的一个名词。本文简要介绍全变分的概念以及在图象去噪中的应用。
1 一维信号的全变分和去噪
1.1 一维连续函数的全变分
一维连续实函数f(x)f(x)在区间[a,b]⊂R[a,b]⊂R上的全变分定义为参数曲线x→f(x),x∈[a,b]x→f(x),x∈[a,b]的弧长。其表达式为
Vba(f)=∫ba|f′(x)|dx
Vab(f)=∫ab|f′(x)|dx
说白了，所谓的“变分”就是|f(x+Δx)−f(x)||f(x+Δx)−f(x)|，对于连续函数Δx→0Δx→0。而全变分是对函数定义的区间而言的，就是将“变分”在区间上累加起来。
一维离散信号的全变分
从上面连续实函数的全变分，我们可以很容易想到它的离散形式。给出信号序列{yi},i=1,…,n{yi},i=1,…,n，它的全变分定义为
V(y)=∑i=1n|yi+1−yi|
V(y)=∑i=1n|yi+1−yi|
用一句话来概括，全变分是前后项之差的绝对值之和。
1.2 一维信号去噪
当我们得到观察信号xixi，希望xixi变得平滑，也就是对xixi去噪。一种很直观的想法就是让信号的全变分变小。全变分对应的物理意义就是输入信号的平滑度。设得到的恢复信号为yiyi，它应该满足两个条件：
yiyi与观察信号xixi的差距不大。这个差距的常用数学表达式就是
E(x,y)=12∑i(xi−yi)2
E(x,y)=12∑i(xi−yi)2
yiyi的全变分不大。
将物理约束转化为数学模型，求解yy等价于求解下面这个优化问题：
minyE(x,y)+λV(y)
minyE(x,y)+λV(y)
其中参数λλ是正常数，用于调节两个约束的作用大小。注意到E(x,y)E(x,y)和V(y)V(y)都是凸函数，这是一个无约束凸优化问题，有很多经典方法可以求解。
2 二维离散信号（图象）的全变分和去噪
图象是典型的二维离散信号，Rudin在1992年将其全变分定义为
V(y)=∑i,j|yi+1,j−yi,j|2+|yi,j+1−yi,j|2−−−−−−−−−−−−−−−−−−−−−−−√
V(y)=∑i,j|yi+1,j−yi,j|2+|yi,j+1−yi,j|2
这个函数是各项同性的，但是不可微，也并不是凸函数。非凸函数的优化求解难度、速度和稳定性都无法与凸函数相比。因此二维全变分有另一种常用定义
V(y)=∑i,j|yi+1,j−yi,j|2−−−−−−−−−−−√+|yi,j+1−yi,j|2−−−−−−−−−−−√=∑i,j|yi+1,j−yi,j|+|yi,j+1−yi,j|
V(y)=∑i,j|yi+1,j−yi,j|2+|yi,j+1−yi,j|2=∑i,j|yi+1,j−yi,j|+|yi,j+1−yi,j|
这个函数是凸函数了。
仿照一维信号的去噪，基于全变分的图象去噪可以看成求解优化问题
minyE(x,y)+λV(y)
minyE(x,y)+λV(y)
其中E(x,y)E(x,y)作为数据误差项定义为
E(x,y)=12∑i,j(xi,j−yi,j)2
E(x,y)=12∑i,j(xi,j−yi,j)2
当VV有凸函数形式时，问题变为无约束凸优化问题，从而容易求解。
二、源代码
function [img_estimated,energyi,ISNRi,energyo,ISNRo]=tvmm_debluring(img_noisy,h,lambda,varargin)
% function [img_estimated]=tvmm_debluring(img_noisy,h,lamdba,...
% 'optional_parameter_name1',value1,'optional_parameter_name2', value2,...);
%
% Total Variation-based image deconvolution with
% a majorization-minimization approach.
%
% Written by: Joao Oliveira, Jose Bioucas-Dias, Mario Figueiredo
% email: joao.oliveira@lx.it.pt
% SITE: www.lx.it.pt/~jpaos/tvmm
% Date: 21/11/2005
%
%
% ========================== INPUT PARAMETERS (required) =================
% Parameter Values
% name and description
% ========================================================================
% img_noisy (double) Noisy blured image of size ny.
% h (double) Blur kernel.
% lambda Regularization parameter (which is multiplied by the
% TV penalty).
%
% ======================== OPTIONAL INPUT PARAMETERS ====================
% Parameter Values
% name and description
% =======================================================================
% boa_iter (double) The number of outer loop iterations.
% Default: 20
% cg_iter (double) The max number of inner CG iterations.
% Default: 200
% soim_iter (double) The max number of inner SOIM iterations.
% Default: 20
% soim_cg_iter (double) The max number of CG iterations in inner SOIM.
% Default: 6
% cg_thrld (double) Conjugate Gradient threshold.
% Default: 1e-5;
% image (double) Original image.
% displayIm ({
'yes','no'}) if 'yes' is passed the restored imaged
% is displayed along the CG iterations
% Default: 'no'
% info_energyi ({
'yes','no'}) if 'yes' is passed the isnr and energy of
% the objective function is displayed inside the inner loop.
% Default: 'no'
% info_energyo ({
'yes','no'}) if 'yes' is passed the energy of
% the objective function is displayed at each outter loop.
% Default: 'no'
% info_ISNRi ({
'yes','no'}) if 'yes' is passed the isnr is displayed.
% Default: 'no'
% Requires: image
% info_ISNRo ({
'yes','no'}) if 'yes' is passed the isnr is displayed.
% Default: 'no'
% Requires: image
% x_0 (double) initial image iteration.
% Default: Wiener filter
% method ({
'cg','soim'}) Selects the method used to solve the linear
% system: cg=conjugate gradient; soim=second order iterative
% method.
% Default: 'cg'
% l_min (double) Minimum eigenvalue of the system for the second
% order iterative method.
% Default: 1e-5
% l_max (double) Maximum eigenvalue of the system for the second
% order iterative method.
% Default: 1
%
% ===================================================================
% The following functions can be provided in order to overwrite the
% internal ones. Recall that, by default, all calculations are
% performed with circular convolutions. (see Technical Report)
% ===================================================================
%
% mult_H (function_handle) Function handle to the function that
% performes H*x. This function must accept two parameters:
% x : image to apply the convolution kernel h
% h : Blur kernel
% mult_Ht (function_handle) Function handle to the function that
% performes Ht*x (Ht = H transpose). This function must
% accept two parameters:
% x : image to apply the convolution kernel h
% h : Blur kernel
% diffh (function_handle) Function handle to the function that
% computes the horizontal differences of an image x.
% diffv (function_handle) Function handle to the function that
% computes the vertical differences of an image x.
% diffht (function_handle) Function handle to the function that
% computes the horizontal differences (transposed) of an
% image x.
% diffh (function_handle) Function handle to the function that
% computes the vertical differences (transposed) of an
% image x.
%
%
% ====================== Output parameters ===============================
% img_estimated Estimated image
% energyi Energy of inner loop iterations.
% Required input arguments: 'info_energyi' and 'image'.
% ISNRi Improvement Signal-to-noise ratio of inner loop iterations.
% Required input arguments: 'info_ISNRi' and 'image'.
% energyo Energy of outer loop iterations.
% Required input arguments: 'info_energyo' and 'image'.
% ISNRo Improvement Signal-to-noise ratio of outer loop iterations.
% Required input arguments: 'info_ISNRo' and 'image'.
%
% ============= EXAMPLES =========================================
% Normal use:
% [img_estimated]=tvmm_debluring(img_noisy,lambda,sigma)
%
% Display restored images along CG iterations:
% [img_estimated]=tvmm_debluring(img_noisy,lambda,sigma,...
% 'displayIm','yes')
%
% Perform ISNR calculations of the outer loop iterations:
% [img_estimated,energyi,ISNRi,energyo,ISNRo]=tvmm_debluring(img_noisy,...
% lambda,sigma,'info_SNRo','yes','image',image);
%
% ======================= DEFAULT PARAMETERS ===================
global mH mHt diffh diffv diffht diffvt
boa_iter = 20; % Number of outer loop iterations
cg_iter = 200; % Number of inner CG iterations
soim_iter = 20; % Number of inner SOIM iterations
soim_cg_iter = 6; % Number of CG iterations inside SOIM
cg_thrld = 1e-5; % CG threshold
displayIm = 0;
image=[];
info_energyi='no';
info_energyo='no';
info_ISNRi='no';
info_ISNRo='no';
info_int = 0;
info_ext = 0;
energyi=0;
ISNRi=0;
energyo=0;
ISNRo=0;
mH=@conv2c;
mHt=@conv2c;
diffh=@f_diffh;
diffv=@f_diffv;
diffht=@f_diffht;
diffvt=@f_diffvt;
method='cg';
x_0=[];
l_min=1e-5;
l_max=1;
% ====================== INPUT PARAMETERS ========================
% Test for number of required parametres
if (nargin-length(varargin)) ~= 3
error('Wrong number of required parameters');
end
% Read the optional parameters
if (rem(length(varargin),2)==1)
error('Optional parameters should always go by pairs');
else
for i=1:2:(length(varargin)-1)
% change the value of parameter
switch varargin{
i}
case 'boa_iter' % Outer loop iterations
boa_iter = varargin{
i+1};
case 'cg_iter' % Inner loop iterations
cg_iter = varargin{
i+1};
case 'soim_iter' % Inner loop iterations
soim_iter = varargin{
i+1};
case 'soim_cg_iter' % CG iterations in Inner loop
soim_cg_iter = varargin{
i+1};
case 'displayIm' % display sucessive xe estimates
if (isequal(varargin{
i+1},'yes'))
displayIm=1;
end
case 'cg_thrld' % CG threshold
cg_thrld = varargin{
i+1};
case 'image' % Original image
image = varargin{
i+1};
case 'info_ISNRi' % display ISNR inside inner loop
info_ISNRi = varargin{
i+1};
case 'info_ISNRo' % display ISNR on outer loop
info_ISNRo = varargin{
i+1};
case 'info_energyi' % display energy inside inner loop
info_energyi = varargin{
i+1};
case 'info_energyo' % display energy on outer loop
info_energyo = varargin{
i+1};
case 'x_0' % Initial image iteration
x_0 = varargin{
i+1};
case 'mult_H' % Function handle to perform H*x
mH = varargin{
i+1};
case 'mult_Ht' % Function handle to perform Ht*x
mHt = varargin{
i+1};
case 'diffh' % External function to perform
diffh = varargin{
i+1}; % horizontal differences
case 'diffv' % External function to perform
diffv = varargin{
i+1}; % vertical differences
case 'diffht' % External function to perform
diffht = varargin{
i+1}; % horizontal differences (transposed)
case 'diffvt' % External function to perform
diffvt = varargin{
i+1}; % vertical differences (transposed)
case 'method' % External function to perform
method = varargin{
i+1}; % vertical differences (transposed)
case 'l_min' % Minimum eigenvalue
l_min = varargin{
i+1};
case 'l_max' % Maximum eigenvalue
l_max = varargin{
i+1};
otherwise
% Hmmm, something wrong with the parameter string
error(['Unrecognized parameter: ''' varargin{
i} '''']);
end;
end;
end
三、运行结果
四、备注
完整代码或者代写添加QQ 912100926