TomoSAR on Detail's Blog

IST

Wed, 04 Mar 2026 00:00:00 +0000

基本原理

迭代软阈值（Iterative Soft Thresholding, IST）利用LASSO回归模型将问题表述为

$$ \arg\min_\mathbf x \lVert{\mathbf y-\Phi\mathbf x}\rVert_2^2+\lambda\lVert \mathbf x\rVert_1, $$

可以得到其迭代方法

初始化：$\mathbf x^{(0)}=0$或$\mathbf x^{(0)}=\Phi ^\text T \mathbf y$
计算梯度下降：$\mathbf r^{(k)}=\mathbf x^{(k)}-t\nabla\lVert\mathbf y-\Phi \mathbf x^{(k)}\rVert_2^2$
迭代优化直至收敛：$\mathbf x^{(k+1)}=S_{\lambda t}(\mathbf r^{(k)})$ 其中$t$为步长，$S_\omega(x)$表示软阈值函数，即 $$ [S_\omega(x)]_i = \left\{ \begin{align*} &x_i-\omega, \quad x_i>\omega\\ &0, \quad \lvert x_i\rvert\le\omega\\ &x_i+\omega, \quad x_i<-\omega \end{align*}\right. $$

推导过程

IST的推导需要使用近端梯度下降（PGD）方法，可以将原问题修改为

$$ \mathbf x = \arg\min_\mathbf x {\frac12\lVert A\mathbf x-\mathbf b\rVert_2^2+\lambda\lVert\mathbf x\rVert_1}, $$

令$f(\mathbf x)=\frac12\lVert A\mathbf x-\mathbf b\rVert_2^2+\lambda\lVert\mathbf x\rVert_1$，对其求导

$$ f^\prime(\mathbf x) = A^\text T (A\mathbf x-\mathbf b)+\lambda\text{sgn}(\mathbf x), $$

求其最小值即令$f^\prime(\mathbf x)=0$。

特殊情况$A=I$

特别的当$A=I$时，$f^\prime(\mathbf x) = (\mathbf x-\mathbf b)+\lambda\text{sgn}(\mathbf x)=0$，其解为软阈值函数

$$ \mathbf x = S_\lambda(\mathbf b) = \left\{ \begin{align*} &b+\lambda, \quad b< -\lambda \\ &0, \quad b\le\lvert\lambda\rvert \\ &b-\lambda, \quad b>\lambda \end{align*}\right.. $$

一般情况

回到一般情况，令$f(\mathbf x) = \lVert A\mathbf x-\mathbf b\rVert^2_2$。对$f(\mathbf x)$在$\mathbf x^{(k)}$处做二阶泰勒展开，令$L=\nabla^2f(\mathbf x)=\frac1t$

$$ \begin{align*} \arg\min_\mathbf x f(\mathbf x) &= \arg\min_\mathbf x{f(\mathbf x^{(k)})+\nabla f(\mathbf x^{(k)})^\text T (\mathbf x -\mathbf x^{(k)})+\frac12(\mathbf x-\mathbf x^{(k)})^\text T\nabla^2f(\mathbf x^{(k)})(\mathbf x-\mathbf x^{(k)})} \\ &= \arg\min_\mathbf x\frac1{2t}\sum_i{t^2[\nabla f(\mathbf x^{(k)})]_i^2+2t[\nabla f(\mathbf x^{(k)})]_i (x_i -x_i^{(k)})+(x_i-x_i^{(k)})^2} \\ &= \arg\min_\mathbf x\frac1{2t}{\lVert\mathbf x-[\mathbf x^{(k)}-t\nabla f(\mathbf x^{(k)})]\rVert^2_2} \\ \end{align*}. $$

此时，令$\mathbf z^{(k)}=\mathbf x^{(k)}-t\nabla f(\mathbf x^{(k)})$，原凸优化问题转换为$A=I$的形式（注意这里是$\mathbf x^{(k+1)}=\text{prox}_{h,t}(\mathbf z^{(k)})$），于是有

$$ \begin{align*} \mathbf x^{(k+1)} &= \arg\min_\mathbf x \frac1{2t}\lVert\mathbf x-\mathbf z\rVert^2_2+\lambda\lVert\mathbf x\rVert_1 \\ &= \arg\min_\mathbf x \frac1{2}\lVert\mathbf x-\mathbf z\rVert^2_2+\lambda t\lVert\mathbf x\rVert_1 \\ &= S_{\lambda t}(\mathbf z^{(k)}) \end{align*}. $$

步长选取

固定步长

假设函数$f(\mathbf x) = \lVert A\mathbf x-\mathbf b\rVert^2_2$满足梯度上的Lipschitz连续。当步长$t\le\frac1L$时，能保证多次迭代后最终收敛至梯度为0的点，$L$表示$\nabla f$的Lipschitz常数。

$$ L = \sup\frac{\lVert\nabla f(a)-\nabla f(b)\rVert_2}{\lVert a-b\rVert_2} $$

令$a=\mathbf x+\mathbf x_0,b=\mathbf x$则

$$ \begin{align*} L &= \sup\frac{\lVert\nabla f(\mathbf x+\mathbf x_0)-\nabla f(\mathbf x)\rVert_2}{\lVert\mathbf x_0\rVert_2} \\ &= \sup\frac{\lVert 2A^\text T [A(\mathbf x+\mathbf x_0)-\mathbf b]-2A^\text T (A\mathbf x-\mathbf b)\rVert_2}{\lVert\mathbf x_0\rVert_2} \\ &= \sup\frac{2\lVert A^\text TA\mathbf x_0\rVert_2}{\lVert\mathbf x_0\rVert_2} \\ &\le\frac{2\lVert\lambda_\text{max}(A^\text TA)\mathbf x_0\rVert_2}{\lVert\mathbf x_0\rVert_2} \\ &=2\lambda_\text{max}(A^\text TA) \end{align*} $$

其中$\lambda_\text{max}(\cdot)$表示取最大特征值，令$t=\frac1L=\frac1{2\lambda_\text{max}(A^\text TA)}$

回溯型步长

当矩阵$A$很大时，不容易计算$A^\text TA$的特征值，或者不知道矩阵$A$时，可以通过回溯搜索的方式估计Lipschitz常数。将$f(\mathbf x)$泰勒展开并放缩得到上界函数

$$ \begin{align*} f(\mathbf x) &={f(\mathbf x_0)+\nabla f(\mathbf x_0)^\text T (\mathbf x -\mathbf x_0)+\frac12(\mathbf x-\mathbf x_0)^\text T\nabla^2f(\mathbf x_0)(\mathbf x-\mathbf x_0)}+o(x^3) \\ &\le{f(\mathbf x_0)+\nabla f(\mathbf x_0)^\text T (\mathbf x -\mathbf x_0)+\frac L2(\mathbf x-\mathbf x_0)^\text T(\mathbf x-\mathbf x_0)} \\ &=Q(\mathbf x,\mathbf x_0;L) \end{align*}, $$

令$\mathbf x=\mathbf x^{(k+1)},\mathbf x_0=x^{(k)}$，不断增大$\hat L$，直到$f(\mathbf x^{(k+1)})\le Q(\mathbf x^{(k+1)},\mathbf x^{(k)};\hat L)$满足。

\begin{algorithm}
\caption{BackTracking}
\begin{algorithmic}
	\STATE 初始化$L_0>0, \eta>1, \mathbf x_0\in\mathbb R^n$
	\REPEAT
		\STATE $\hat L = \eta L_\text{before}$
	\UNTIL{$f(\mathbf x^{(k+1)})\le Q(\mathbf x^{(k+1)},\mathbf x^{(k)};\hat L)$}
\end{algorithmic}
\end{algorithm}

PGD

Wed, 04 Mar 2026 00:00:00 +0000

基本原理

近端梯度下降（Proximal Gradient Descent）是梯度下降方法的一种，其适用于目标函数存在不可微部分的凸优化问题。问题描述为

$$ \mathbf x = \arg\min_\mathbf x {g(\mathbf x)+h(\mathbf x)}, $$

其中，$g(\mathbf x)$为可微凸函数、$h(\mathbf x)$为不可微凸函数。

求解方法

求解该类型凸优化问题可以通过近端映射（Proximal Mapping）和梯度迭代的手段实现。近端映射函数定义为

$$ \text{prox}_{h,t}(\mathbf x) = \arg\min_\mathbf z{\frac1{2t}\lVert\mathbf x-\mathbf z\rVert^2_2+h(\mathbf z)}, $$

梯度迭代方法为

$$ \begin{align*} &\mathbf z^{(k)}=\mathbf x^{(k)}-t_k\nabla g(\mathbf x^{(k)}) \\ &\mathbf x^{(k+1)}=\text{prox}_{h,t_k}(\mathbf z^{(k)}) \end{align*} $$

证明

对$g(\mathbf x)$在$\mathbf x^{(k)}$处做二阶泰勒展开，令$L=\nabla^2g(\mathbf x)=\frac1t$

$$ \begin{align*} \arg\min_\mathbf x g(\mathbf x)+h(\mathbf x) &\approx \arg\min_\mathbf x{g(\mathbf x^{(k)})+\nabla g(\mathbf x^{(k)})^\text T (\mathbf x -\mathbf x^{(k)})+\frac12(\mathbf x-\mathbf x^{(k)})^\text T\nabla^2g(\mathbf x^{(k)})(\mathbf x-\mathbf x^{(k)})}+h(\mathbf x) \\ \mathbf{\hat x}&= \arg\min_\mathbf x{\nabla g(\mathbf x^{(k)})^\text T (\mathbf x -\mathbf x^{(k)})+\frac1{2t}(\mathbf x-\mathbf x^{(k)})^\text T(\mathbf x-\mathbf x^{(k)})}+h(\mathbf x) \\ &= \arg\min_\mathbf x\sum_i{[\nabla g(\mathbf x^{(k)})]_i (x_i -x_i^{(k)})+\frac1{2t}(x_i-x_i^{(k)})^2}+h(\mathbf x) \\ &= \arg\min_\mathbf x\frac1{2t}\sum_i{2t[\nabla g(\mathbf x^{(k)})]_i (x_i -x_i^{(k)})+(x_i-x_i^{(k)})^2}+h(\mathbf x) \\ &= \arg\min_\mathbf x\frac1{2t}\sum_i{t^2[\nabla g(\mathbf x^{(k)})]_i^2+2t[\nabla g(\mathbf x^{(k)})]_i (x_i -x_i^{(k)})+(x_i-x_i^{(k)})^2}+h(\mathbf x) \\ &= \arg\min_\mathbf x\frac1{2t}\sum_i{[x_i-x_i^{(k)}+t[\nabla g(\mathbf x^{(k)})]_i]^2}+h(\mathbf x) \\ &= \arg\min_\mathbf x\frac1{2t}{\lVert\mathbf x-[\mathbf x^{(k)}-t\nabla g(\mathbf x^{(k)})]\rVert^2_2}+h(\mathbf x) \\ \end{align*}. $$

令$\mathbf z^{(k)}=\mathbf x^{(k)}-t\nabla g(\mathbf x^{(k)})$，原问题即可转换为近端映射

$$ \begin{align*} \mathbf{\hat x} &= \arg\min_\mathbf x\frac1{2t}{\lVert\mathbf x-\mathbf z\rVert^2_2}+h(\mathbf x) \\ &=\text{prox}_{h,t}(\mathbf z) \end{align*}. $$

设定合适的参数，即可使用迭代方法求得$\arg\min_\mathbf x g(\mathbf x)+h(\mathbf x)$的结果。

Lipschitz常数

假定目标凸函数满足梯度上的Lipschitz连续。当步长$t\le\frac1L$时，保证多次迭代后能收敛到梯度为0的点。梯度Lipschitz连续表示为

$$ \frac{\lVert\nabla f(a)-\nabla f(b)\rVert_2}{\lVert a-b\rVert_2} \le L(f) \quad \forall a,b\in\text{dom}f, $$

函数$\nabla f$的Lipschitz常数为

$$ L = \sup\frac{\lVert\nabla f(a)-\nabla f(b)\rVert_2}{\lVert a-b\rVert_2} $$

IHT

Tue, 10 Feb 2026 00:00:00 +0000

基本原理

迭代硬阈值算法（Iterative Hard Thresholding, IHT），在提出时称M-Sparse Algorithm。$M$稀疏问题可以表示为

$$ \min_\mathbf y\lVert \mathbf x - \Phi\mathbf y \rVert^2_2\quad \text{s.t.}\quad \lVert\mathbf y\rVert_0\le M, $$

可以得到其迭代算法

$$ \mathbf y^{n+1}=H_M(\mathbf y^n+\Phi^\rm H(\mathbf x-\Phi\mathbf y^n)), $$

其中，$H_M$为非线性算子，其返回$M$维最大幅值

$$ H_M(y_i)= \left\{ \begin{align} 0\quad \lvert y_i\rvert\lt\lambda^{0.5}_M(\mathbf y)\\ y_i\quad \lvert y_i\rvert\ge\lambda^{0.5}_M(\mathbf y) \end{align} \right.. $$

推导过程

替代目标函数（Surrogate Objective Function）

$$ C_M^S(\mathbf{y,z})=\lVert \mathbf{x-\Phi y} \rVert_2^2 - \lVert \mathbf{\Phi y-\Phi z} \rVert_2^2 + \lVert \mathbf{y-z} \rVert_2^2 \qquad \lVert\mathbf\Phi\rVert_2\lt1, $$

当$\mathbf{y=z}$时，该函数即为原目标函数，其余情况下均大于目标函数。$\lVert\mathbf\Phi\rVert_2\lt1$由$0\lt eig(\mathbf I-\mathbf \Phi^\rm{H} \mathbf\Phi)\lt 1$推导出。

替代目标函数变形

$$ C_M^S(\mathbf{y,z})=\sum_i[y_i^2-2y_i(z_i+\phi_i^\rm{T}\mathbf x-\phi_i^\rm{T}\mathbf{\Phi z})]+\lVert\mathbf x\rVert_2^2+\lVert\mathbf z\rVert_2^2+\lVert\mathbf{\Phi z}\rVert_2^2, $$

由于优化目标为$\mathbf y$，而后面三项与其无关。因此，对其优化时忽略这三项不影响结果。

$$ C_M^S(\mathbf{y,z})\propto\sum_i[y_i^2-2y_i(z_i+\phi_i^\rm{T}\mathbf x-\phi_i^\rm{T}\mathbf{\Phi z})]. $$

极值点获取

$$ y_i^*=z_i+\phi_i^\rm H\mathbf x-\phi_i^\rm H\mathbf{\Phi z}, $$

极值点通过简单的配方法即可获得。这时，取得最小值$\sum_i{-(y_i^*)^2}$。

迭代公式获取

考虑到$\lVert\mathbf y\rVert_0\le M$的约束，需要保留$\lVert\mathbf y^*\rVert$的最大$M$项，即使用硬阈值函数。

代码

简单

function [s_est, info] = iht(g, Phi, max_iter, K)
    [~, N] = size(Phi);
    s_est = zeros(N, 1);
    g = g(:);

    g_norm = norm(g);
    L = compute_lipschitz_constant(Phi);
    t = 1 / L;  % 计算步长
    
    residual_tol = 1e-2;
    plateau_tol = 1e-6;
    plateau_patience = 20;
    plateau_count = 0;

    residual = Phi * s_est - g;  % 初始残差
    residual_norm = norm(residual);
    stop_reason = "max_iter";
    iter_done = 0;

    for iter = 1:max_iter
        z = s_est - Phi' * residual * t;    % 计算梯度步长更新
        s_est = hard_thresholding(z, K); % 应用硬阈值处理

        prev_residual_norm = residual_norm;
        residual = Phi * s_est - g;  % 更新残差
        residual_norm = norm(residual);
        iter_done = iter;

        if residual_norm < residual_tol * g_norm
            stop_reason = "residual_tol";
            break;
        end

        improvement = (prev_residual_norm - residual_norm) / max(prev_residual_norm, eps);
        if improvement < plateau_tol
            plateau_count = plateau_count + 1;
        else
            plateau_count = 0;
        end

        if plateau_count >= plateau_patience
            stop_reason = "plateau";
            break;
        end
    end

    if nargout > 1
        info = struct();
        info.iter = iter_done;
        info.residual_norm = residual_norm;
        info.relative_residual = residual_norm / max(g_norm, eps);
        info.stop_reason = stop_reason;
        info.step_size = t;
        info.L = L;
        info.plateau_count = plateau_count;
    end
end

function r = hard_thresholding(x, K)
    % 对向量x进行硬阈值处理，保留最大的K个元素
    r = zeros(size(x));
    [~, idx] = sort(abs(x), 'descend');
    r(idx(1:K)) = x(idx(1:K));
end

function L = compute_lipschitz_constant(Phi)
    ATA = Phi' * Phi;
    L = max(eig(ATA));
end

原文

function [s, err_mse, iter_time]=hard_l0_Mterm(x,A,m,M,varargin)
% hard_l0_Mterm: Hard thresholding algorithm that keeps exactly M elements 
% in each iteration. 
%
% This algorithm has certain performance guarantees as described in [1],
% [2] and [3].
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Usage
%
%   [s, err_mse, iter_time]=hard_l0_Mterm(x,P,m,M,'option_name','option_value')
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Input
%
%   Mandatory:
%               x   Observation vector to be decomposed
%               P   Either:
%                       1) An nxm matrix (n must be dimension of x)
%                       2) A function handle (type "help function_format" 
%                          for more information)
%                          Also requires specification of P_trans option.
%                       3) An object handle (type "help object_format" for 
%                          more information)
%               m   length of s 
%               M   non-zero elements to keep in each iteration
%
%   Possible additional options:
%   (specify as many as you want using 'option_name','option_value' pairs)
%   See below for explanation of options:
%__________________________________________________________________________
%   option_name    |     available option_values                | default
%--------------------------------------------------------------------------
%   stopTol        | number (see below)                         | 1e-16
%   P_trans        | function_handle (see below)                | 
%   maxIter        | positive integer (see below)               | n^2
%   verbose        | true, false                                | false
%   start_val      | vector of length m                         | zeros
%   step_size      | number                                     | 0 (auto)
%
%   stopping criteria used : (OldRMS-NewRMS)/RMS(x) < stopTol
%
%   stopTol: Value for stopping criterion.
%
%   P_trans: If P is a function handle, then P_trans has to be specified and 
%            must be a function handle. 
%
%   maxIter: Maximum number of allowed iterations.
%
%   verbose: Logical value to allow algorithm progress to be displayed.
%
%   start_val: Allows algorithms to start from partial solution.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Outputs
%
%    s              Solution vector 
%    err_mse        Vector containing mse of approximation error for each 
%                   iteration
%    iter_time      Vector containing computation times for each iteration
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Description
%
%   Implements the M-sparse algorithm described in [1], [2] and [3].
%   This algorithm takes a gradient step and then thresholds to only retain
%   M non-zero elements. It allows the step-size to be calculated
%   automatically as described in [3] and is therefore now independent from 
%   a rescaling of P.
%   
%   
% References
%   [1]  T. Blumensath and M.E. Davies, "Iterative Thresholding for Sparse 
%        Approximations", submitted, 2007
%   [2]  T. Blumensath and M. Davies; "Iterative Hard Thresholding for 
%        Compressed Sensing" to appear Applied and Computational Harmonic 
%        Analysis 
%   [3] T. Blumensath and M. Davies; "A modified Iterative Hard 
%        Thresholding algorithm with guaranteed performance and stability" 
%        in preparation (title may change) 
% See Also
%   hard_l0_reg
%
% Copyright (c) 2007 Thomas Blumensath
%
% The University of Edinburgh
% Email: thomas.blumensath@ed.ac.uk
% Comments and bug reports welcome
%
% This file is part of sparsity Version 0.4
% Created: April 2007
% Modified January 2009
%
% Part of this toolbox was developed with the support of EPSRC Grant
% D000246/1
%
% Please read COPYRIGHT.m for terms and conditions.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    Default values and initialisation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



[n1 n2]=size(x);
if n2 == 1
    n=n1;
elseif n1 == 1
    x=x';
    n=n2;
else
   error('x must be a vector.');
end
    
sigsize     = x'*x/n;
oldERR      = sigsize;
err_mse     = [];
iter_time   = [];
STOPTOL     = 1e-16;
MAXITER     = n^2;
verbose     = false;
initial_given=0;
s_initial   = zeros(m,1);
MU          = 0;

if verbose
   display('Initialising...') 
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                           Output variables
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

switch nargout 
    case 3
        comp_err=true;
        comp_time=true;
    case 2 
        comp_err=true;
        comp_time=false;
    case 1
        comp_err=false;
        comp_time=false;
    case 0
        error('Please assign output variable.')        
    otherwise
        error('Too many output arguments specified')
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                       Look through options
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Put option into nice format
Options={};
OS=nargin-4;
c=1;
for i=1:OS
    if isa(varargin{i},'cell')
        CellSize=length(varargin{i});
        ThisCell=varargin{i};
        for j=1:CellSize
            Options{c}=ThisCell{j};
            c=c+1;
        end
    else
        Options{c}=varargin{i};
        c=c+1;
    end
end
OS=length(Options);
if rem(OS,2)
   error('Something is wrong with argument name and argument value pairs.') 
end
for i=1:2:OS
   switch Options{i}
        case {'stopTol'}
            if isa(Options{i+1},'numeric') ; STOPTOL     = Options{i+1};   
            else error('stopTol must be number. Exiting.'); end
        case {'P_trans'} 
            if isa(Options{i+1},'function_handle'); Pt = Options{i+1};   
            else error('P_trans must be function _handle. Exiting.'); end
        case {'maxIter'}
            if isa(Options{i+1},'numeric'); MAXITER     = Options{i+1};             
            else error('maxIter must be a number. Exiting.'); end
        case {'verbose'}
            if isa(Options{i+1},'logical'); verbose     = Options{i+1};   
            else error('verbose must be a logical. Exiting.'); end 
        case {'start_val'}
            if isa(Options{i+1},'numeric') && length(Options{i+1}) == m ;
                s_initial     = Options{i+1};  
                initial_given=1;
            else error('start_val must be a vector of length m. Exiting.'); end
        case {'step_size'}
            if isa(Options{i+1},'numeric') && (Options{i+1}) > 0 ;
                MU     = Options{i+1};   
            else error('Stepsize must be between a positive number. Exiting.'); end
        otherwise
            error('Unrecognised option. Exiting.') 
   end
end

if nargout >=2
    err_mse = zeros(MAXITER,1);
end
if nargout ==3
    iter_time = zeros(MAXITER,1);
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Make P and Pt functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if          isa(A,'float')      P =@(z) A*z;  Pt =@(z) A'*z;
elseif      isobject(A)         P =@(z) A*z;  Pt =@(z) A'*z;
elseif      isa(A,'function_handle') 
    try
        if          isa(Pt,'function_handle'); P=A;
        else        error('If P is a function handle, Pt also needs to be a function handle. Exiting.'); end
    catch error('If P is a function handle, Pt needs to be specified. Exiting.'); end
else        error('P is of unsupported type. Use matrix, function_handle or object. Exiting.'); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Do we start from zero or not?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



if initial_given ==1;
    
    if length(find(s_initial)) > M
        display('Initial vector has more than M non-zero elements. Keeping only M largest.')
    
    end
    s                   =   s_initial;
    [ssort sortind]     =   sort(abs(s),'descend');
    s(sortind(M+1:end)) =   0;
    Ps                  =   P(s);
    Residual            =   x-Ps;
    oldERR      = Residual'*Residual/n;
else
    s_initial   = zeros(m,1);
    Residual    = x;
    s           = s_initial;
    Ps          = zeros(n,1);
    oldERR      = sigsize;
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                 Random Check to see if dictionary norm is below 1 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

        
        x_test=randn(m,1);
        x_test=x_test/norm(x_test);
        nP=norm(P(x_test));
        if abs(MU*nP)>1;
            display('WARNING! Algorithm likely to become unstable.')
            display('Use smaller step-size or || P ||_2 < 1.')
        end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Main algorithm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if verbose
   display('Main iterations...') 
end
tic
t=0;
done = 0;
iter=1;

while ~done
    
    if MU == 0

        %Calculate optimal step size and do line search
        olds                =   s;
        oldPs               =   Ps;
        IND                 =   s~=0;
        d                   =   Pt(Residual);
        % If the current vector is zero, we take the largest elements in d
        if sum(IND)==0
            [dsort sortdind]    =   sort(abs(d),'descend');
            IND(sortdind(1:M))  =   1;    
         end  

        id                  =   (IND.*d);
        Pd                  =   P(id);
        mu                  =   id'*id/(Pd'*Pd);
        s                   =   olds + mu * d;
        [ssort sortind]     =   sort(abs(s),'descend');
        s(sortind(M+1:end)) =   0;
        Ps                  =   P(s);
        
        % Calculate step-size requirement 
        omega               =   (norm(s-olds)/norm(Ps-oldPs))^2;

        % As long as the support changes and mu > omega, we decrease mu
        while mu > (0.99)*omega && sum(xor(IND,s~=0))~=0 && sum(IND)~=0
%             display(['decreasing mu'])
                    
                    % We use a simple line search, halving mu in each step
                    mu                  =   mu/2;
                    s                   =   olds + mu * d;
                    [ssort sortind]     =   sort(abs(s),'descend');
                    s(sortind(M+1:end)) =   0;
                    Ps                  =   P(s);
                    % Calculate step-size requirement 
                    omega               =   (norm(s-olds)/norm(Ps-oldPs))^2;
        end
        
    else
        % Use fixed step size
        s                   =   s + MU * Pt(Residual);
        [ssort sortind]     =   sort(abs(s),'descend');
        s(sortind(M+1:end)) =   0;
        Ps                  =   P(s);
        
    end
        Residual            =   x-Ps;

        
     ERR=Residual'*Residual/n;
     if comp_err
         err_mse(iter)=ERR;
     end
     
     if comp_time
         iter_time(iter)=toc;
     end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Are we done yet?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     
         if comp_err && iter >=2
             if ((err_mse(iter-1)-err_mse(iter))/sigsize<STOPTOL);
                 if verbose
                    display(['Stopping. Approximation error changed less than ' num2str(STOPTOL)])
                 end
                done = 1; 
             elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i mse change',iter ,(err_mse(iter-1)-err_mse(iter))/sigsize)) 
                t=toc;
             end
         else
             if ((oldERR - ERR)/sigsize < STOPTOL) && iter >=2;
                 if verbose
                    display(['Stopping. Approximation error changed less than ' num2str(STOPTOL)])
                 end
                done = 1; 
             elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i mse change',iter ,(oldERR - ERR)/sigsize)) 
                t=toc;
             end
         end
         
    % Also stop if residual gets too small or maxIter reached
     if comp_err
         if err_mse(iter)<1e-16
             display('Stopping. Exact signal representation found!')
             done=1;
         end
     elseif iter>1 
         if ERR<1e-16
             display('Stopping. Exact signal representation found!')
             done=1;
         end
     end

     if iter >= MAXITER
         display('Stopping. Maximum number of iterations reached!')
         done = 1; 
     end
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    If not done, take another round
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
     if ~done
        iter=iter+1; 
        oldERR=ERR;        
     end
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                  Only return as many elements as iterations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if nargout >=2
    err_mse = err_mse(1:iter);
end
if nargout ==3
    iter_time = iter_time(1:iter);
end
if verbose
   display('Done') 
end

MM 优化框架

Fri, 23 Jan 2026 00:00:00 +0000

基本原理

Majorization-Minimization优化框架在各类算法中很常见。在目标函数$J(x)$难优化时，可以寻找一个容易优化的目标函数$G(x)$，当$G(x)$满足一定条件时，$G(x)$的最优解能无限逼近$J(x)$的最优解。

$G(x)$应该满足的条件

容易优化
$G_k(x)\ge J(x)$
$G_k(x_k)=J(x_k)$

优化过程

考虑一个优化问题，$\mathscr X$为闭凸集、$J(\mathbf x)$连续

$$ \min_\mathbf x J(\mathbf x)\quad \text{s.t.} \quad x \in \mathscr X. $$

\begin{algorithm}
\caption{Majorization Minimization}
\begin{algorithmic}
    \STATE 寻找一个可行点$x^0 \in \mathscr X，k \gets 0$
    \REPEAT
        \STATE $\mathbf x^{k+1}=\arg\min_{x\in \mathscr X} G_k(\mathbf{x})$
        \STATE $k\gets k+1$
    \UNTIL{满足一些条件}
\end{algorithmic}
\end{algorithm}

成像预处理

Fri, 21 Nov 2025 00:00:00 +0000

流程图

配准

强度图像金字塔

使用主辅图像的幅值构建影像金字塔，自顶向下逐层对图像进行匹配。¹使得每轮匹配的搜索范围都相对较小，快速接近对应位置。代码中采用4层金字塔，上一层图像为下一层图像在距离向和方位向分别进行2倍下采样得到。匹配窗口为$21\times 21$、每层最大偏移为$15\times 15$、控制点数量为64。图像偏移量为控制点偏移量的均值。控制点匹配采用归一化相关匹配，使用相关性最高的偏移量为该控制点的偏移量。²

$$ R\left( x,y \right) =\frac{\sum_{x^\prime ,y^\prime }{M\left( x^\prime ,y^\prime \right) \cdot S\left( x+x^\prime ,y+y^\prime \right)}}{\sqrt{\sum_{x^\prime ,y^\prime }{M\left( x^\prime ,y^\prime \right) ^2}\cdot \sum_{x^\prime ,y^\prime}{S\left( x+x^\prime ,y+y^\prime \right) ^2}}} $$

最大频谱法精匹配

使用最大频谱法对主辅图像进行匹配，使得主辅图像对齐到亚像元级别以满足后续处理需求。代码中采用的匹配窗口为$15\times 15$、最大偏移为$3\times 3$、控制点数量为64，计算前在图像距离向和方位向上分别进行5倍上采样。控制点偏移量使用最大频谱法得到的信噪比最大的偏移量。³

$$ u_{int}=u_m\cdot u_{s}^{*}=a_ma_s\cdot e^{j\left( \varphi _m-\varphi _s \right)} $$$$ SNR=\frac{f_{max}}{\sum{f_{i,j}}-f_{max}} $$

图像偏移量使用控制点去除奇异值点和信噪比较低的点后，使用三阶多项式拟合。³

$$ \left\{ \begin{array}{l} \Delta x=a_0+a_1\cdot x+a_2\cdot y+a_3\cdot xy+a_4\cdot x^2+a_5\cdot y^2+a_6\cdot x^2y+a_7\cdot xy^2+a_8\cdot x^3+a_9\cdot y^3\\ \Delta y=b_0+b_1\cdot x+b_2\cdot y+b_3\cdot xy+b_4\cdot x^2+b_5\cdot y^2+b_6\cdot x^2y+b_7\cdot xy^2+b_8\cdot x^3+b_9\cdot y^3\\ \end{array} \right. $$

插值

应用偏移量到辅图像上采用复数双线性插值

$$ f\left( x,y \right) =f\left( Q_{11} \right) w_{11}+f\left( Q_{21} \right) w_{21}+f\left( Q_{12} \right) w_{12}+f\left( Q_{22} \right) w_{22} $$$$ \left\{ \begin{align*} w_{11} &= \frac{(x_2-x)(y_2-y)}{(x_2-x_1)(y_2-y_1)}\\ w_{21} &= \frac{(x-x_1)(y_2-y)}{(x_2-x_1)(y_2-y_1)}\\ w_{12} &= \frac{(x_2-x)(y-y_1)}{(x_2-x_1)(y_2-y_1)}\\ w_{22} &= \frac{(x-x_1)(y-y_1)}{(x_2-x_1)(y_2-y_1)} \end{align*} \right. $$

干涉

干涉图由主辅图像共轭相乘得到，相干图由干涉图用$9\times 9$窗口均值滤波并归一化得到。³

$$ \begin{align*} u_{int}&=u_m\cdot u_{s}^{*}=a_ma_s\cdot e^{j\left( \varphi _m-\varphi _s \right)}=a_{int}e^{j\varphi} \\ \varphi &=\varphi _\text{flat}+\varphi _\text{topo}+\varphi _\text{atm}+\varphi _\text{noise} \end{align*} $$$$ \gamma =\frac{|E\left[ M\cdot S^* \right] |}{\sqrt{E\left[\mid M\mid ^2 \right] E\left[\mid S\mid ^2 \right]}} $$

去平地效应

斜距$R$和入射角$\theta$在SML文件中得到，使用线性插值对斜距$R$进行插值。$\Delta 𝑅$为当前像元与第一个像元斜距之差。 ³ $R_0$为第一个像元的斜距，$R_{eq}$为干涉相位为0时对应的斜距。

$$ \varphi _\text{flat}=-\frac{4\pi}{\lambda}\cdot \frac{B_{\bot}\Delta \text{R}}{Rtan\left( \theta \right)}+\varphi _\text{const} $$$$ \varphi_\text{const}=-\frac{4\pi}{\lambda}\cdot \frac{B_{\bot}(R_{0}-R_{\text{eq}})}{R_{0}tan\left( \theta \right)} $$

去平地效应后的干涉图相位:

$$ \begin{align*} \varphi _1& =\varphi +\frac{4\pi}{\lambda}\cdot \frac{B_{\bot}\Delta \text{R}}{Rtan\left( \theta \right)}\\ & =\varphi _\text{topo}+\varphi _\text{atm}+\varphi _\text{const}+\varphi _\text{noise}\\ \end{align*} $$

去DEM相位

使用轨道数据将DEM数据逐点从WGS84坐标系转换到雷达距离-方位坐标系，使用线性插值将得到的数据插值到整个雷达距离-方位坐标系。通过高程相位的公式得到 ³

$$ \varphi _\text{dem}=-\frac{4\pi}{\lambda}\cdot \frac{B_{\bot}\text{H}_\text{dem}}{R\sin \left( \theta \right)} $$

去DEM相位后的干涉图相位:

$$ \begin{align*} \varphi _2& =\varphi _1-\varphi _\text{dem}\\ & =\varphi _\text{topo}-\varphi _\text{dem}+\varphi _\text{atm}+\varphi _\text{const}+\varphi _\text{noise}\\ & =-\frac{4\pi}{\lambda}\cdot \frac{B_{\bot}\Delta H}{R\sin \left( \theta \right)}+\varphi _\text{atm}+\varphi _\text{const}+\varphi _\text{noise} \quad\quad (\Delta H = H - H_\text{dem})\\ \end{align*} $$

滤波

使用Goldstein滤波，将干涉图分成相互重叠的小块。对每个小块做傅里叶变换，得到其频谱$Z(u,v)$ , 用其对干涉图进行处理。$S[\cdot]$ 为平滑算子，使用$3\times 3$的均值滤波。Goldstein滤波窗口大小为$32\times 32$ , 步长为8 。⁴⁵

$$ H(u,v) = S[ \lvert Z(u,v)\rvert ]^{1-\bar \gamma}\cdot Z(u,v) $$

忽略残余噪声，滤波后干涉图相位:

$$ \varphi_3 = =-\frac{4\pi}{\lambda}\cdot \frac{B_{\bot}\Delta H}{R\sin \left( \theta \right)}+\varphi _\text{atm}+\varphi _\text{const}\\ $$

大气相位校正

经过去平地效应和去DEM相位后，相位还剩余高频的相对高程相位与低频的大气相位和固定相位。对相位使用高通滤波器去除大气相位和固定相位。⁶ 假设$E[\Delta H]=0$

$$ \varphi _3=k\Delta H+\varphi _\text{atm}+\varphi _\text{const} \quad\quad ( k=-\frac{4\pi B_{\bot}}{\lambda Rsin(\theta)}) $$

去大气相位后的干涉图相位:

$$ \varphi _4=HPF( \varphi _3) =k\Delta H $$

靳国旺. InSAR获取高精度DEM关键处理技术研究[D]. 解放军信息工程大学, 2007. ↩︎
OpenCV计算机视觉库. 模板匹配[EB/OL], (2025-7-3). https://docs.opencv.ac.cn/4.12.0/de/da9/tutorial_template_matching.html ↩︎
杨红磊, 彭军还, 康志忠. InSAR技术原理及实践[M]. 北京：科学出版社, 2021. ↩︎ ↩︎ ↩︎ ↩︎ ↩︎
BARAN I, STEWART M P, KAMPES B M, 等. A modification to the goldstein radar interferogram filter[J]. IEEE Transactions on Geoscience and Remote Sensing, 2003, 41(9): 2114-2118. ↩︎
Nome2s. InSARFilter[CP/OL], (2025-10-21). https://github.com/Nome2s/InSARFilter. ↩︎
FORNARO G, LOMBARDINI F, SERAFINO F. Three-dimensional multipass SAR focusing: experiments with long-term spaceborne data[J]. IEEE Transactions on Geoscience and Remote Sensing, 2005, 43(4): 702-714. ↩︎

压缩感知

Wed, 19 Nov 2025 00:00:00 +0000

利用信号的稀疏性作为先验知识，完成对信号的压缩。

基本公式

$x$ 为信号
$s$ 为稀疏系数，$x=\Psi s$
$y$ 为观测值，$y = \Phi\Psi s = Rs$
$\Phi$ 为观测矩阵
$\Psi$ 为稀疏矩阵

如何重建

$L_0$最小范数(理论解)

$$ \min_s \Vert s\Vert_0\quad s.t. \quad y=Rs $$

$L_1$最小范数

求解$L_0$最小范数是NP-hard问题，但如果信号足够稀疏（$M \gg O(Klog(N/K))$），可以使用$L_1$最小范数得到相同结果。

$$ \min_s \Vert s\Vert_1\quad s.t. \quad y=Rs $$

存在噪声的情况下

没有K的先验知识

$$ \hat s = \arg\min_s{\{\Vert y-Rs\Vert_2^2+\lambda_K\Vert s\Vert_1\}} $$

$\lambda_K$为拉格朗日乘子，取决于采样点数$N$和噪声水平$\sigma_\varepsilon$

有K的先验知识

$$ \hat s= \min_s \Vert s\Vert_1\quad s.t. \quad \Vert y-Rs\Vert_2 \lt \sigma_\varepsilon $$

性质

非相关性

通过最大化每次测量的信息效率，分散信号特征到所有采样点中。

$$ \mu(\mathrm{\bf{R}})=\mu(\bf{\Phi},\bf{\Psi})=\sqrt{N}\operatorname*{max}_{1\leq k,j\leq N}\frac{\vert\langle\varphi_{k},\psi_{j}\rangle\vert}{\Vert\varphi_{k}\Vert_2\Vert\psi_{j}\Vert_2} $$

典型测量基$\Phi$ 和稀疏基$\Psi$组合 1. 傅里叶矩阵+单位矩阵：当信号自身具有稀疏性时 2. 噪声基+小波基 3. 随机矩阵+任意固定基

限制等距条件(RIP)

保证测量矩阵对稀疏信号的几何结构保持稳定

$$ \left(1-\delta_{s}\right)\Vert \nu\Vert_{2}^{2}\ \leq\ \Vert\mathbf{R}\,\nu\Vert_{2}^{2}\ \leq\ \left(1+\delta_{s}\right)\Vert\nu\Vert_{2}^{2} $$

典型重建方法

基于松弛理论

凸优化

凸优化算法利用$L_1$范数最小化来求解$L_0$问题等效的凸优化问题问题，问题如下

$$ \hat s = \arg\min_s{\lVert s\rVert_1} \qquad \text{s.t.} \quad \lVert Rs-y\rVert_2\le\varepsilon. $$

该类算法可以分为内点法（Interior Point Method, IPM）和一阶算法。内点法的典型算法有基追踪（BP）、$L_1$正则化最小二乘（$L_1$-LS）、$L_1$-magic等。它们属于重构精度较高的高阶算法，但是它们对高阶问题的求解效率很低。一阶算法的代表有梯度投影（GPSR）、软阈值迭代（IST, TwIST, FIST）、Bregman、不动点延拓（FPC, FPC-AC）、基于Nesterov理论的算法（NESTA）、一阶锥求解器（TFOCS）、交替方向法（ADM/ADMM）等。这类方法对高维问题的求解非常有效。 $L_1$范数最小化算法可以用来重构稀疏性较差的信号，对加性噪声具有良好的鲁棒性。

非凸优化

非凸优化与之类似，主要的研究方向是基于迭代重加权原理和基于$L_q(0<q<1)$正则化理论。基于迭代重加权原理的算法由优化最小法（Majorization Minimization）框架推导而来。基本思想是利用与未知信号相关的加权矩阵求解一个优化问题序列。典型的算法有迭代重加权最小二乘（IRLS）和FOCUSS等。基于$L_q(0<q<1)$正则化理论的算法所要求解的问题如下

$$ \hat s = \arg\min_s \{{\lVert Rs-y\rVert_2}+\lambda\lVert s\rVert^q_q\}. $$

典型的算法有$L_{1/2}$阈值迭代算法和$L_{2/3}$阈值迭代算法。相比于凸优化算法，非凸优化算法通常对稀疏信号的重构精度更高，然而若参数选择不合适极易收敛到局部极小值，导致信号重构结果出现偏差。

基于贪婪追踪

贪婪追踪算法是一类搜索算法，此类算法主要包含两个步骤，即支集选择和系数更新。现有各种贪婪追踪算法主要由匹配追踪（MP）和正交匹配追踪（OMP）发展而来。典型的算法有OMP、IHT等，此类算法优势是在稀疏性较强时，能迅速重建所需信号。当信号稀疏性较弱或测量存在噪声时，运算效率低、重建结果与真实值之间会存在很大偏差

基于贝叶斯

贝叶斯重构算法是通过将信号稀疏性与先验信息关联，从概率论角度求解稀疏信号的一类算法。典型的有基于稀疏性贝叶斯学习和基于图模型的信息传递算法。