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Mismatched boundary conditions such as air-sample interface introduce artifacts that obscure internal information in the reconstructed laser computed tomographic (CT) images. Here, we demonstrate enhancement of target structure in the laser CT images by correcting the projection data using the experimentally determined angle dependence of sample surface attenuation. The images reconstructed with the corrected projection data are shown to have improved image contrast. Our proposed correction to laser CT reconstruction is effective for visualizing internal structure with small variations in the attenuation coefficients that would otherwise be masked by the dominant surface attenuation.
©1998 Optical Society of America
1. Introduction
Recently, optical tomography techniques are widely studied and are currently expected for various applications. The advantages of using optical tomography over conventional imaging modalities have been well documented in areas such as fluid dynamics, flame analysis, atmospheric monitoring etc. [1–8]. The imaged medium or targets in the above mentioned examples were transparent or semi-transparent and image reconstruction techniques were similar to those used in X-ray CT. In recent years, however, the advantages of using light for biomedical imaging have been realized and optical tomographic studies are extended into this promising area of research.
The heterogenous and highly scattering nature of biological tissues present unique problems for transillumination laser CT. Biological tissues, in general, are layered structures with building blocks ranging from a few nanometers to 100’s of microns that are potential light scattering centers. Large water content, circulating body fluids and presence of absorbing pigments further compound our understanding of photon migration in tissues. Although various experimental and theoretical methods have been proposed to determine the optical characteristic of tissues [9], there still remain uncertainties regarding actual light transport in tissues. Lack of suitable image reconstruction algorithms for laser CT of biological tissues are a direct manifestation of these uncertainties.
One of the simplest method for tomographic image reconstructions is to use the filtered back projection (FBP) method that has been established for X-ray CT. Modified version of the FBP have also been proposed for laser CT of biological tissues [10,11]. Backprojection image reconstruction method is mathematically based on Radon transform. According to the theory of Radon transform, a cross section of an object can be reconstructed using the ray integral values of the attenuation coefficient. The ray integral can be considered equivalent to the Beer’s law in optical imaging and laser CT images describe the distribution of the total attenuation coefficient. Theoretically, total attenuation coefficient depends on the scattering coefficient and the absorption coefficient at a given wavelength. However, in a practical imaging system, the total attenuation also depends on surface effects. The incident light is strongly affected by refractive index mismatch at boundaries such as the air-tissue interface. Other than refractive index mismatches, the surface effects also include light reflections, wave front distortions and speckle effects etc. If the attenuation due to surface effects is larger than the attenuation of both the scattering and the absorption events, the information on the target structure is poor and artifacts result in the laser CT images reconstructed by backprojection method. Further, the attenuation is also not unique for the same part of a tissue target at different incident angles. Therefore, the surface effects would depend on both the boundary conditions and the incident angle in laser CT images.
In this paper, we propose a method for the correction of surface effects in the laser CT images. Experimentally determined attenuation due to different incident angles at the sample surfaces were used to correct the surface effects in the reconstructed laser CT images. Improved laser CT image contrast is demonstrated on correction of the surface effects.
2. Theory
In CT scans using a single beam, projection data are acquired by translating and rotating the beam. The coordinates as seen in Fig. 1, (r, s) are related to the coordinates (x, y) by
Let rn ( -N/2 ≤ n ≤ N/2 ) be a distance from the origin making an angle φm ( 1 ≤ m ≤ M ) with the x-axis, where N and M are the data size of one projection and the number of projections, respectively. The on-axis photons into the sample, Iin(φm,rn) ,is represented as
where I 0 is the incident power, α is a constant dependent on the boundary conditions and the incident angle. In translational scanning, change in the incident beam position brings about a change in α. In rotational scanning, change in angle φm, brings about a change in the incident angle at the same position of the sample surface. Then, remaining photons are attenuated by absorption and scattering in the sample. Let’s assume that transmitted photons follow Beer’s law. The transmitted photons, It(φm,rn) can be represented as
Fig. 1 Incident angle dependence in laser CT scan. I0 is incident intensity, Iin(φm,rn) is light intensity launched into sample, It(φm ,rn ) is transmitted intensity, θ is incident angle and R is the sample radius.
where μt is the total attenuation coefficient. The total attenuation is then represented as
In Eq. (4), the first term depends on both the incident angle and the boundary conditions, and the second term depends on the optical path length. As CT reconstruction methods such as FBP consider the ray integral values of attenuation coefficient, the existence of other attenuation factors cause blurring or artifacts in the reconstructed CT images. Therefore, projection data need to be corrected for the surface effects attenuation.
3. Experimental setup and results
The experiments were carried out with the coherent detection imaging (CDI) system that is based on the optical heterodyne detection technique. The CDI method has unique properties such as high directionality, selectivity and sensitivity [12] for the detection of the directional, forward scattered, coherence-retaining photons, emerging from a biological tissue. As the CDI method detects on-axis or near-axis photons, FBP method was reported to be used for laser CT image reconstruction of biological tissues in vivo and in vitro [13–16].
Fig. 2 Schematic diagram of the coherent detection imaging system. L1 and L2 are collimated lenses. M1, M2, and M3 are mirrors. BS and PBS are beam splitter and polarizing beam splitter, respectively. AOM1 and AOM2 are acousto-optic modulators.
A schematic of the measurement system is shown in Fig. 2. Continuous wave and single frequency He-Ne laser (Spectra-Physics, model 117A) operating at 632.8 nm was used as the light source. Collimated output (ϕ 0.8 mm) of the laser is split into the signal beam and the local oscillator beam. Acousto-optic modulators are used to modulate the signal beam and the local oscillator beam to 40 MHz and 40.1 MHz, respectively. The signal beam after passing through the sample is mixed with the local oscillator and impinges on a silicon photodiode generating a signal at the intermediate frequency (i.f.) of 100 kHz. The i.f. signal is then fed to a FFT analyzer that is interfaced to a personal computer. Sample is mounted on a translational-rotational stepping motor stages that are controlled by a personal computer. Dynamic range of the measurement system is ~140 dB and optical power to the order of 10-17 W could be detected.
The sample, 30 mm in diameter, was made from clear acrylic. At the near center of the acrylic block, a hole, 12 mm in diameter, was made. Outer surface of the sample was ground by sand paper to make the surface rough, and the hole was filled with 20 ml/l of Intralipid-10% (Kabi Pharmacia AB, Sweden) (Fig. 4 (a)). The measured scattering coefficient of the Intralipid-10% solution is 0.412 [ml-1·l·cm-1]. This value is consistent with earlier reports using the optical heterodyne detection method [17,18] and the direct collimated detection method [19].
Surface effects of the sample were estimated using a 5 mm thin acrylic plate with the same composition as that of the sample. The surfaces of the acrylic plate were also ground in the similar fashion to that of the sample. To determine the attenuation dependence on the incident angle, the acrylic plate was rotated by 180° at 1° interval and the averaged amplitude of the i.f. signal was recorded. The variations in the attenuation at the surface are approximated by a gaussian function. The observed data and the approximated function are displayed in Fig. 3. The approximated function, f(θ), is represented as
where θ [radian] is the incident angle. As the attenuation due to the acrylic plate at an incident angle of 0° to the surface normal is negligible, the first term in Eq. (4) can be rewritten as
In laser CT scans of the sample, the translational and rotational steps were set to 500 μm and 6° (30 projections), respectively. To correct the surface effects in laser CT images reconstructed by the FBP method, it is necessary to subtract the angle dependence function, f(θ), from the total attenuation coefficient as represented in Eq. (4). The corrected attenuation coefficient can now be represented as
Fig. 3 Attenuation due to changes in the incident angle of a 5 mm thick acrylic plate with ground surfaces.
Fig. 4 (a) Schematic representation of the sample. The acrylic sample was 30 mm in diameter with a 12 mm in diameter hole filled with Intralipid-10% 20 ml/l. The outer surface of the sample was ground by sand paper to make the surface rough.
(b) Representative acquired projection and its corrected form using Eq. (7).
(c) Laser CT image without surface effects correction.
(d) Laser CT image with surface effects correction.
(e) Profiles through the laser CT image before and after correction.
where R is the sample radius. In Fig. 4 (b), a representative acquired projection data and its corrected form using Eq. (7) are displayed. Each of the 30 projections was corrected in a similar manner and the image reconstruction was carried out by the FBP method.
Fig. 4 (c) and Fig. 4 (d) are the laser CT images reconstructed with the acquired projections and the corrected projections, respectively. As seen in Fig. 4 (d), the ring artifact at the surface disappears and the image contrast of target structure with the intralipid solution is enhanced. The laser CT image profiles at the position as shown in Fig. 4 (a) are displayed in Fig. 4 (e). Here again, it can be seen that the image reconstructed with the corrected projections displays significant decreased surface effects.
Image contrast of the laser CT image in Fig. 4 (c) and Fig. 4 (d), C, can be defined as follow,
where Itarget is a value of averaged attenuation in the region of the intralipid solution. Imax and Imin are maximum and minimum attenuation value in laser CT image, respectively. The value of C is improved from 0.36 to 0.95 by correction of surface effects.
4. Conclusion
We propose a simple correction method for the surface effects in laser CT and demonstrate reduction of image artifacts and improvement of image contrast. Our model can also be applied to other laser CT techniques that involve algebraic or iterative reconstruction methods. As attenuation in biological samples is mainly due to scattering and as scattering coefficients are a factor of 100 or larger than absorption coefficients, it is imperative that large attenuation at the surface be corrected to visualize small changes in the internal scattering or absorption of tissues. Although the correction of surface effects is an effective method for biological applications, attenuation due to surface effects are present in most samples due to mismatched boundaries such as the air-sample interface and our proposed method can be applied to other fields such as fluid dynamics, flame analysis, atmospheric monitoring etc., where laser CT techniques are in use.
Footnotes
| * | Y. Watanabe is also at Biophotonics Information Laboratories. watanabe@seitai.yamagata-rit.go.jp |
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