---
title: "Compressed sensing"
chunk: 4/6
source: "https://en.wikipedia.org/wiki/Compressed_sensing"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:40:18.609631+00:00"
instance: "kb-cron"
---

===== Iterative model using a directional orientation field and directional total variation =====
To prevent over-smoothing of edges and texture details and to obtain a reconstructed CS image which is accurate and robust to noise and artifacts, this method is used. First, an initial estimate of the noisy point-wise orientation field of the image 
  
    
      
        I
      
    
    {\displaystyle I}
  
, 
  
    
      
        
          
            
              d
              ^
            
          
        
      
    
    {\displaystyle {\hat {d}}}
  
, is obtained. This noisy orientation field is defined so that it can be refined at a later stage to reduce the noise influences in orientation field estimation. A coarse orientation field estimation is then introduced based on structure tensor, which is formulated as: 
  
    
      
        
          J
          
            ρ
          
        
        (
        ∇
        
          I
          
            σ
          
        
        )
        =
        
          G
          
            ρ
          
        
        ∗
        (
        ∇
        
          I
          
            σ
          
        
        ⊗
        ∇
        
          I
          
            σ
          
        
        )
        =
        
          
            (
            
              
                
                  
                    J
                    
                      11
                    
                  
                
                
                  
                    J
                    
                      12
                    
                  
                
              
              
                
                  
                    J
                    
                      12
                    
                  
                
                
                  
                    J
                    
                      22
                    
                  
                
              
            
            )
          
        
      
    
    {\displaystyle J_{\rho }(\nabla I_{\sigma })=G_{\rho }*(\nabla I_{\sigma }\otimes \nabla I_{\sigma })={\begin{pmatrix}J_{11}&J_{12}\\J_{12}&J_{22}\end{pmatrix}}}
  
. Here, 
  
    
      
        
          J
          
            ρ
          
        
      
    
    {\displaystyle J_{\rho }}
  
 refers to the structure tensor related with the image pixel point (i,j) having standard deviation 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
. 
  
    
      
        G
      
    
    {\displaystyle G}
  
 refers to the Gaussian kernel 
  
    
      
        (
        0
        ,
        
          ρ
          
            2
          
        
        )
      
    
    {\displaystyle (0,\rho ^{2})}
  
 with standard deviation 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
. 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 refers to the manually defined parameter for the image 
  
    
      
        I
      
    
    {\displaystyle I}
  
 below which the edge detection is insensitive to noise. 
  
    
      
        ∇
        
          I
          
            σ
          
        
      
    
    {\displaystyle \nabla I_{\sigma }}
  
 refers to the gradient of the image 
  
    
      
        I
      
    
    {\displaystyle I}
  
 and 
  
    
      
        (
        ∇
        
          I
          
            σ
          
        
        ⊗
        ∇
        
          I
          
            σ
          
        
        )
      
    
    {\displaystyle (\nabla I_{\sigma }\otimes \nabla I_{\sigma })}
  
 refers to the tensor product obtained by using this gradient.
The structure tensor obtained is convolved with a Gaussian kernel 
  
    
      
        G
      
    
    {\displaystyle G}
  
 to improve the accuracy of the orientation estimate with 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 being set to high values to account for the unknown noise levels. For every pixel (i,j) in the image, the structure tensor J is a symmetric and positive semi-definite matrix. Convolving all the pixels in the image with 
  
    
      
        G
      
    
    {\displaystyle G}
  
, gives orthonormal eigen vectors  ω and υ of the 
  
    
      
        J
      
    
    {\displaystyle J}
  
 matrix. ω points in the direction of the dominant orientation having the largest contrast and υ points in the direction of the structure orientation having the smallest contrast.  The orientation field coarse initial estimation 
  
    
      
        
          
            
              d
              ^
            
          
        
      
    
    {\displaystyle {\hat {d}}}
  
 is defined as 
  
    
      
        
          
            
              d
              ^
            
          
        
      
    
    {\displaystyle {\hat {d}}}
  
 = υ.  This estimate is accurate at strong edges. However, at weak edges or on regions with noise, its reliability decreases.
To overcome this drawback, a refined orientation model is defined in which the data term reduces the effect of noise and improves accuracy while the second penalty term with the L2-norm is a fidelity term which ensures accuracy of initial coarse estimation.
This orientation field is introduced into the directional total variation optimization model for CS reconstruction through the equation: 
  
    
      
        
          min
          
            
              X
            
          
        
        ‖
        ∇
        
          X
        
        ∙
        d
        
          ‖
          
            1
          
        
        +
        
          
            λ
            2
          
        
         
        ‖
        Y
        −
        Φ
        
          X
        
        
          ‖
          
            2
          
          
            2
          
        
      
    
    {\displaystyle \min _{\mathrm {X} }\lVert \nabla \mathrm {X} \bullet d\rVert _{1}+{\frac {\lambda }{2}}\ \lVert Y-\Phi \mathrm {X} \rVert _{2}^{2}}
  
. 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathrm {X} }
  
 is the objective signal which needs to be recovered. Y is the corresponding measurement vector, d is the iterative refined orientation field and 
  
    
      
        Φ
      
    
    {\displaystyle \Phi }
  
 is the CS measurement matrix. This method undergoes a few iterations ultimately leading to convergence.
  
    
      
        
          
            
              d
              ^
            
          
        
      
    
    {\displaystyle {\hat {d}}}
  
 is the orientation field approximate estimation of the reconstructed image 
  
    
      
        
          X
          
            k
            −
            1
          
        
      
    
    {\displaystyle X^{k-1}}
  
 from the previous iteration (in order to check for convergence and the subsequent optical performance, the previous iteration is used). For the two vector fields represented by  
  
    
      
        
          X
        
      
    
    {\displaystyle \mathrm {X} }
  
 and 
  
    
      
        d
      
    
    {\displaystyle d}
  
, 
  
    
      
        
          X
        
        ∙
        d
      
    
    {\displaystyle \mathrm {X} \bullet d}
  
 refers to the multiplication of respective horizontal and vertical vector elements of 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathrm {X} }
  
 and 
  
    
      
        d
      
    
    {\displaystyle d}
  
 followed by their subsequent addition. These equations are reduced to a series of convex minimization problems which are then solved with a combination of variable splitting and augmented Lagrangian (FFT-based fast solver with a closed form solution) methods. It (Augmented Lagrangian) is considered equivalent to the split Bregman iteration which ensures convergence of this method.  The orientation field, d is defined as being equal to 
  
    
      
        (
        
          d
          
            h
          
        
        ,
        
          d
          
            v
          
        
        )
      
    
    {\displaystyle (d_{h},d_{v})}
  
, where 
  
    
      
        
          d
          
            h
          
        
        ,
        
          d
          
            v
          
        
      
    
    {\displaystyle d_{h},d_{v}}
  
 define the horizontal and vertical estimates of 
  
    
      
        d
      
    
    {\displaystyle d}
  
.