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Table 1 The sinc-convolution algorithm

From: Precise two-dimensional D-bar reconstructions of human chest and phantom tank via sinc-convolution algorithm

Step

Operation

1

Specify the bounds of D-bar integral equation

 

μ ( x , k ) = 1 + r ( k ) π , where r ( k ) = 2 R 2 R 2 R 2 R t R exp e k ( x ) 4 k ¯ ( s k ) μ ( x , k ) ¯ d k 1 d k 2 , k = k 1 + i k 2 C , k 0 .

2

1. Decompose convolution integral r = i = 1 4 r i

2. Define mapping functions φ 1 ( z l ) = φ 2 ( z l ) = φ ( z l ) = ln ( z l + 2 R 2 R z l ) , l = M , , N .

3. Compute sinc points z l = φ 1 ( l h ) = 2 R ( 1 + e lh ) ( 1 + e lh ) , l = M , , N .

 

4. Compute derivative of the mapping functions at sinc points φ ' ( z l ) = 4 R ( 2 R + z l ) ( z l 2 R )

3

Use sinc matrices I m i 1 ( s , t ) = 0 s t sin ( π z ) π z d z + 0.5 , f o r s , t = M , . , N ,

 

A 1 = h I m 1 D ( 1 φ ' ( z l ) ) = X 1 S 1 X 1 1 , A 2 = h [ I m 1 ] T D ( 1 φ ' ( z l ) ) = X 2 S 2 X 2 1 .

4

Compute the special “Laplace Transform” of the convolution kernel g ( k ) = 1 k , k = k 1 + i k 2 C G ( u , v ) = 0 0 g ( k 1 , k 2 ) e k 1 u + k 2 v d k 1 d k 2 = v π π 2 ( v u ) + ln ( v u ) 1 + ( v u ) 2 + i u π π 2 ( u v ) + ln ( u v ) 1 + ( u v ) 2 , where Re ( u v ) > 0 , Re ( v u ) > 0.

5

Iteratively

1. Compute each r i , i = 1 , . , 4 , using the separation-of-variables procedure of Table 2.

 

2. Solve equation μ ( k ) = 1 + 1 π ( r 1 + r 2 + r 3 + r 4 ) to find μ(k).