2.3. Terahertz imaging
2.3.2. Hadamard imaging
2. Fundamentals
rays of a specific projection angle in one iteration to perform the correction of the guessed image. As a result, the algorithm achieves fast convergence with a good quality of the reconstructed image.
An additional benefit that all the algebraic reconstruction algorithms share, is the opportunity to incorpo-rate a priori information about the image and ray distortions like refraction effects into the reconstruction process[114]. This is of particular interest in the terahertz frequency domain, since refraction, diffraction and reflection effects have a huge impact on the measured projections[119, 120]. The reconstructed images often show distortions or even are poorly correlated to the physical shape of the object. The main reason for the distortions of the reconstructed images originate in the used reconstruction algorithm, which actually has been developed for x-ray tomography, i.e. wavelengths between 5 and 60 pm, thus much smaller than the object dimensions. Moreover, the refractive index of matter differs very little from unity in the x-ray spectral domain, hence the algorithm assumes straight rays, which penetrate the sample object and get attenuated depending on the spatially distributed absorption coefficient and the optical path length through the object. This circumstance is not given in case of tomographic terahertz imaging, because refraction of the rays, losses due to reflection and diffraction effects occur and strongly influence the measured projections. For this purpose, a lot of efforts are done to improve the quality of reconstructed images in terahertz tomography. Both new concepts for the field of tomographic terahertz imaging have to be developed, and improvements of the experimental set-ups, like for example refractive index matching[121], or special reconstruction algorithms[119, 122]are needed. Also, a good under-standing of the impact of refraction, reflection and diffraction effects on terahertz tomography is required in the latter case, in order to include these effects to the tomographic reconstruction.
An important component of the tomographic imaging set-up is the detection scheme, which must be capable to measure the spatially resolved projections of the sample object. For this purpose, imaging schemes involving raster scanning of the object or multi-pixel detectors are often used. Because multi-pixel detector arrays are not widely available in the terahertz frequency domain, commonly the projection is obtained by raster scanning the object. An alternative concept is given by the single-pixel imaging technique, which allows the measurement of spatially resolved projections without raster scanning of the object and by using a non-spatially resolving ’bucket’ detector. This concept will be introduced in more detail in the following section.
2.3. Terahertz imaging
spatial patterned radiation as illustrated in Fig. 2.12. Mathematically this can be expressed in the following system of equations
M1=P~1·f~ M2=P~2·f~
... Mn=P~n·f~
M~ =P· ~f (2.20)
whereM1toMn denote the measured intensities at the bucket detector and f~is the mask of the object, which is illuminated by the nth pattern P~n. P represents the pattern matrix, in which the nth row is represented by the transposed pattern P~nT. The transmission mask of the object is received by the multiplication of Eq. 2.20 with the inverse ofP
P· ~f =M~ (2.21)
P−1·P· ~f =P−1·M~ (2.22)
~f =P−1·M~. (2.23)
The system of Eqs. 2.20 can only be solved by applying Eq. 2.23 if the number of measurements are equal or larger than the total number of pixels.
Pattern mask
Sample objectBucket detector
...
1st pattern 2nd pattern ... nth pattern
Figure 2.12.: A multitude of different patterns are used to illuminate a sample object while the bucket detector measures the total intensity transmitted through the object under illumination of one pattern. An image of the object can be calculated from these n intensities, although a detector with no spacial resolution is used.
The kind of employed spatial patterns is variable and depends on the specific implementation. One possibility is the application of random patterns, which are often used in the framework of compressed sensing[123, 124]. Compressive sensing offers the possibility to perform a reconstruction of the image with a reduced number of measurements. The number of needed measurements can be reduced to 25−30% of the total number of pixels[124, 125], which enables short measurement times. This of course is achieved by a trade-off between the number of measurements and the quality of the reconstructed image. By increasing the number of measurements, at first the image quality increase quite fast and with a higher amount of measurements slowly converges as shown in[124]. However, Eq. 2.23 cannot be used to reconstruct the image of the object. Instead computationally more complex optimization algorithms like the minimization of the total variation (min-TV)[126]must be used.
2. Fundamentals
(
1 1 1 1 ... 1 1 -1 1 -1 ... -1 1. H+ & H- Pattern:...
...
1 1 -1 -1 ... 1 1 -1 -1 1 ... -1
)
2. H+ & H- Pattern:
31. H+ & H- Pattern:
32. H+ & H- Pattern:
...
32. H+ Pattern
...
32. H- Pattern
...
1. H+ Pattern
...
1. H- Pattern
transparent
pixel opaque
pixel (a)
(b)
reconstruction
original trans- mission maskreconstructed trans- mission mask
10 20 30
0 5 15
intensity at the bucket detector
pattern number M+ M
-0
10 20 30
10
M = M - M+- -5
transparent region
x - position opaque region
Figure 2.13.: (a) Illustration of the pattern generation instructions of a32×32Hadamard matrix used for single-pixel imaging. Each row of the Hadamard matrix represents two patterns. The H+ patterns are created by transparent regions represented by+1entries and opaque regions for the −1 entries, while this is reversed for the H− patterns. (b) Simulation of the Hadamard single-pixel imaging scheme. The measurement vectors M~+ and M~− are calculated using Eq.2.20 from the original transmission mask and afterwards the transmission mask is reconstructed using Eq. 2.23.
A set of patterns which enable the image reconstruction described in Eq.2.23 can be derived from the Hadamard matrices[127]. This group of matrices are squared matrices containing only the values 1 and−1. Further, the matrices are orthogonal, meaning that the rows/columns of these matrices are all orthogonal to each other. As a consequence, the inverse of a Hadamard matrix is given by its transpose divided by its dimension. Besides the faster computational time and possibility to exactly calculate the image using Eq. 2.23, in general single-pixel imaging based on Hadamard patterns achieves higher image
2.3. Terahertz imaging
quality[125]on the cost of measurement time. A Hadamard matrix of the dimension 2ncan be created by using the following recursive construction rule
H1=1, H2=
1 1 1 −1
, . . . , Hn=
Hn−1 Hn−1 Hn−1 −Hn−1
.
As can be seen a Hadamard matrix is composed of the entries 1 and−1, but the available intensity values to form radiation patterns can only take values from 0 (no radiation) to 1 (full intensity). Therefore, the Hadamard matrix must be split into two matrices containing only the entries 0 and 1. This is done by adding and subtracting the Hadamard matrixHn×n to the matrix of ones1nn
M~+ M~−
=
0.5·(1nn+H)·f~ 0.5·(1nn−H)·f~
=
H+·f~ H−·f~
. (2.24)
Now, the measurement vectorM~ is given by the difference between the two newly defined measurement vectorsM~+andM~−
M~+−M~−= 0.5·(1nn+H)· ~f
− 0.5·(1nn−H)· ~f
(2.25)
=0.5·(1nn+H−1nn+H)·f~ (2.26)
=H·f~ (2.27)
=M~ (2.28)
Therefore, 2nmeasurements are needed to achieve Hadamard pattern based single-pixel imaging with a resolution of npixels. The patterns projected on the sample object are formed by the rows of the Hadamard matrix. This is shown in Fig. 2.13(a) for a 32×32 dimensional Hadamard matrix in the one-dimensional case. TheH+patterns are created by transparent regions represented by+1 entries and opaque regions for the−1 entries, while this is reversed for theH− patterns. For example, the first row ofHonly consists of ones and therefore the firstH+pattern is fully transparent whereas theH− pattern is fully opaque. The object under investigation is then illuminated with patterned radiation defined by the rows of the Hadamard matrix one by one. The integrated intensitiesM~+(blue) andM~− (red) for a transmission mask as depicted in Fig. 2.13(b) are calculated using Eq. 2.24 and are shown in the right side of the figure. The sample mask is either fully transparent, which corresponds to a transmission value of 1, or fully opaque, which corresponds to a transmission value of 0. Since the first pattern of H+ is fully transparent the first intensity in M~+ corresponds to the maximum possible transmission through the sample mask. Contrary, the firstM~− value is zero, since the pattern is fully opaque. Afterwards only smaller differences in the values ofM~+andM~− are recognizable, which is also visible in the measurement vector M~ shown in the bottom right of Fig. 2.13(b). The reconstruction is performed using Eq. 2.23 and the resulting reconstructed transmission mask is shown in the left bottom of Fig. 2.13(b). The reconstructed transmission mask is reproduced very well.
This chapter has introduced basic methods for the generation and detection of terahertz radiation.
These methods involve electrical, optical or electro-optical technology to close the terahertz gap of the electromagnetic spectrum. These developments pave the way for terahertz applications as the introduced tomographic and single-pixel imaging. The emphasis of terahertz generation and detection has been on
2. Fundamentals
the photomixing concept. This concept transfers an optical beat signal to the terahertz frequency domain using a PCA. One requirement of this approach is the source of the optical beat signal, which will be discussed in the next chapter.