The forward problem of the scattering phenomena is expressed by the following Equation (1) using the linear Born approximation.

E s c ( r P ) ≈ ∭ V Q [ k s 2 ( r Q ) − k b 2 ] G ^ ( r P , r Q ) E i n c ( r Q ) d r Q (1)

Here, E s c , G ^ , E i n c , k_s and k_b denote the scattering field, dyadic Green function, incident field, wave number of the object, and that of the background, respectively. Then, V_Q, position vector r_p and r_Q denote a volume to be inspected, position of the observation and scattering.

Assuming planar scanning, the contrast distribution of a region represented by a plurality of cross sections parallel to the scanning plane is estimated. E j s c ( x ′ , y ′ , ω l ) , j = x , y , z denotes j-polarization component of the scattering field at r_p. Each component of the scattered field at r p = ( x ′ , y ′ , 0 ) at frequency ω l is expressed by,

E j s c ( x ′ , y ′ , ω l ) ≈ ∭ z y x f ( x , y , z , ω l ) ⋅ ∑ i = x , y , z a i j ( x − x ′ , y − y ′ , z , ω l ) d x d y d z (2)

Here,

f ( x , y , z , ω l ) = k s 2 ( x , y , z , ω l ) − k b 2 ( z , ω l ) (3)

a i j ( x , y , z , ω l ) = E i i n c ( x , y , z , ω l ) G 0 i j ( x , y , z , ω l ) (4)

The two-dimensional Fourier transform on both sides of Equation (2) is

E j s c ( k x , k y , ω l ) ≈ ∫ z F ˜ ( k x , k y , z , ω l ) ∑ i = x , y , z A ˜ i j ( k x , k y , z , ω l ) d z (5)

Here, F ˜ ( k x , k y , z , ω l ) and A ˜ i j ( k x , k y , z , ω l ) are the two-dimensional Fourier transforms of f ( x , y , z , ω l ) and a i j ( x , y , z , ω l ) . Furthermore, k_x and k_y are the Fourier variables corresponding to x and y.

We approximate the integral in Equation (5) with the discrete sum of z.

E j s c ( k x , k y , ω l ) = ∑ n = 1 N z F ˜ ( k x , k y , z n , ω l ) ∑ i = x , y , z A ˜ i j ( k x , k y , z , ω l ) Δ z (6)

where, Δz is the distance between two adjacent reconstruction planes. Considering the frequency dependence of the medium, it is assumed that the contrast function can be expressed by,

f ( x , y , z , ω ) = h ( x , y , z ) φ ( ω ) (7)

Taking the two-dimensional Fourier transform of (x, y) on both sides of Equation (7) with respect to x and y,

F ˜ ( κ , z , ω ) = H ˜ ( κ , z ) φ ( ω ) (8)

where, H ˜ ( κ , z ) is the two-dimensional Fourier transform of h ( x , y , z ) with respect to (x, y). Writing (6) and (8) for the all measured frequencies provides N_ω equations for each spatial frequency pair κ = ( k x , k y ) .

{ y 1 = a 11 x 1 + ⋯ + a 1 N z x N z                                   ⋮ y N ω = a N ω 1 x 1 + ⋯ + a N ω N z x N z (9)

Here,

y l = E ˜ j s c ( κ , ω l ) (10)

a l n = φ ( ω l ) ∑ i = x , y , z A ˜ i j ( κ , z n , ω l ) (11)

x n = H ˜ ( κ , z n ) (12)

We obtain the least-squares solution of this system of equations and find H ˜ ( κ , z n ) , n = 1 , 2 , ⋯ , N z for each spatial frequency pair κ = ( k x , k y ) .

In general, the incident field and Green’s function data are obtained by simulation, but due to modeling and numerical calculation errors, it is difficult to be implemented practically. It has been proposed to obtain products of the incident field and Green’s function specific to the measurement system by measuring a known calibration object (CO). This method uses the concept of point scattering function (PSF), in which the response due to an arbitrary target is convolved with the response due to point scatter (CO).

When a sufficiently small non-dispersion medium CO exists at (0, 0, z_n), the contrast function of Equation (3) is expressed by,

E j s c ( x ′ , y ′ , ω l ) ≈ ∑ n = 1 N z Δ z n ∬ x y f ( x , y , z n ) ⋅ E j s c , c o ( x ′ − x , y ′ − y , ω l ; z n ) d y d x (15)

Therefore, the two-dimensional Fourier transform of E j s c ( x ′ , y ′ , ω l ) , j = x , y , z is expressed by,

E ˜ j s c ( k x , k y , ω l ) ≈ ∑ n = 1 N z Δ z n F ˜ ( k x , k y , z n ) E ˜ j s c , c o ( k x , k y , ω l ; z n ) (16)

E ˜ j s c , c o ( k x , k y , ω l ; z n ) denotes the two-dimensional Fourier transform of E j s c , c o ( x , y , ω l ; z n ) .

Appling Equation (16) to the data at all the frequencies ω l ( l = 1 , ⋯ , N ω ) , a system of equations at each spatial frequency pair κ = ( k x , k y ) is constructed.

[ E ˜ j s c ( κ , ω 1 ) ⋮ E ˜ j s c ( κ , ω N ω ) ] = [ E ˜ j s c , c o ( κ , ω 1 ; z 1 ) Δ z ⋯ E ˜ j s c , c o ( κ , ω 1 ; z N z ) Δ z ⋮ ⋱ ⋮ E ˜ j s c , c o ( κ , ω N ω ; z 1 ) Δ z ⋯ E ˜ j s c , c o ( κ , ω N ω ; z N z ) Δ z [ F ˜ ( κ , z 1 ) ⋮ F ˜ ( κ , z N z ) ] ] (17)

After solving Equation (17) for all values of κ , the inverse Fourier transform of F ˜ ( κ , z n ) , n = 1 , 2 , ⋯ , N z is performed, and the contrast function f ( x , y , z n ) is reconstructed.

The measurement frequency range is 2 - 16 GHz, and the distance between the scanning plane and breast is 50 mm. Scattered fields were obtained in 2 mm

steps, within the plane with a 100 × 100 mm range. The CO has a size, relative permittivity, and conductivity of 2 mm³, 25, and 0.75 [S/m], respectively, and was moved from 3 to 45 mm in steps of 3 mm at (x, y) = (0, 0) mm to obtain the measured PSF. Since no matching liquid is used, the relative dielectric constant and conductivity of background set to 1 and 0 [S/m], respectively.

Figure 2 presents a tomographic image of the real part of the complex permittivity

of the numerical breast phantom and a reconstructed tomographic image. Despite the absence of matching fluid, the breast tissue structure is accurately reconstructed. The time required for calculation and display of the reconstructed image was only 1.8 s on a personal computer equipped with an i7-6700k CPU and 32GB RAM.

The measurement frequency range is 1.1 - 21 GHz and the distance between the scanning plane and head is 50 mm. Scattered fields were obtained in 2 mm steps within the plane, with a 228 × 228 mm range. The CO with a size, relative permittivity and conductivity of 2 mm³, 60, and 2, respectively, was moved from 2 to 100 mm in 2 mm steps at (x, y) = (0, 0) mm to obtain the measured PSF. In this application, it is difficult to reconstruct the image without the matching liquid. The relative dielectric constant and conductivity of the background was set to 25 and 0.75, respectively.

Figure 3 depicts a tomographic image of the real part of the complex permittivity of the numerical head phantom and a reconstructed tomographic image. The inner tissue of the head is accurately reconstructed. The time required for the calculation and display of the reconstructed image was 7.7 s.