[Tikhonov] image super-resolution reconstruction based on Tikhonov regularization

1. Software version

matlab2013b

2. System principle

        For the problem of image super-resolution reconstruction , It often involves solving large-scale equations , And the dimension of the equation is often very large . Therefore, the iterative algorithm is often used in the solution algorithm of regularization method . Here we will mainly explain the iteration Tikhonov The regularization method is taken as an example , This method utilizes some prior properties of regular parameters , At the same time, the smoothness condition is imposed on the exact solution , A posteriori selection strategy is used to improve the convergence rate and determine the regularization parameters .

First, the convergence analysis . Get the restoration result directly

  But the solution of inverse matrix is very complicated , In this paper, iterative descent algorithm is used to solve the restored image , Iterative process meets

  fitting (2) Take norm on both sides to get

  According to the definition of matrix spectral norm , type (3) Can be written as

  type (4) in , Represents the maximum eigenvalue of the matrix . According to the triangle inequality

  Will type (5) Substituting (4) Formula

  therefore ,α Meet the conditions α<1 Time algorithm convergence , To achieve the search for the global optimal solution .

in summary ,α The relation shall be satisfied 0<α<1 To ensure that the restoration result is the global optimal solution .

In this paper, the spatial domain iteration method is used to calculate , Then the iterative format of the regularization function of image restoration is ：

3. Part of the source code

function [object,yy,err,Iter_Max]=func_Tikhonov(I1);

Images = I1;
Fact_cos_sin = func_cos_sin(Images);
H            = func_fuzzy_gauss(I1,1);

% Parameter initialization 
Iter_Max     = 30;
[Rs,Cs,Ks]   = size(Images); 
fft_operator = fft2([0,-1,0;-1,4,-1;0,-1,0],Rs,Cs);
fft_images   = fft2(Images,Rs,Cs);
fft_nimages  = fft2(I1,Rs,Cs);
FFT_H        = fft2(H,Rs,Cs);
FFT_H        = FFT_H.*Fact_cos_sin;
CONJ_FFT_H   = conj(FFT_H);
det_FFT_H    = abs(FFT_H);
det_FFT_p    = abs(fft_operator); 
I2           = I1;  
Y            = I1;   
mark         = 0;
% iterative process 
for k=1:Iter_Max
    %  Find regularization parameters  AL
    A  = H;
    B  = imfilter(Y,[0,-1,0;-1,4,-1;0,-1,0],'replicate');
    N  = imfilter(I2,A,'replicate');
    a1 = norm((double(Y)-double(N)),2).^2;
    a2 = norm(double(Y),2).^2;
    AL = log(a1/a2/1e2+1);     
    
    FFT_X1     = fft2(I2,Rs,Cs);
    FFT_X1_t   = CONJ_FFT_H.*FFT_X1;  
    FFT_object = FFT_X1_t+CONJ_FFT_H.*fft_images-FFT_X1_t.*(det_FFT_H.^2+AL*det_FFT_p.^2);
    object     = abs(ifft2(FFT_object));
    
    object_max = max(max(object));  % calibration 
    object_min = min(min(object));
    object     =(object-object_min)*255/(object_max-object_min); 
    object     = uint8(object);
    I3         = object;
    t          = I2;
    I2         = I3;
    diff       =(norm(double(I3)-double(t),2).^2)/((norm(double(t),2)).^2);
    yy(k)      = AL;
    err(k)     = diff;
end

4. Simulation conclusion

The simulation effect of this part is as follows ：

 A05-19