Light path alignment for computed tomography of scattering material
© The Author(s) 2016
Received: 28 April 2016
Accepted: 17 June 2016
Published: 2 August 2016
We aim to estimate internal slice of the scattering material by Computed Tomography (CT). In the scattering material, the light path is disturbed and is spread. Conventional CT cannot measure the scattering material because they rely on the assumption that rays are straight and parallel. We propose light path alignment to deal with scattering rays. Each path of disturbed scattering light is approximated with a straight line. Then the light path in the object corresponding to a single incident ray is modeled as straight paths spreading from incident point. These spreading paths are aligned to be parallel, so that they can be used directly by conventional reconstruction algorithm.
Measurement of the interior of an object is a key technique in various fields such as the inspection of food for finding foreign objects or the observation inside a human body.
X-ray computed tomography (CT) is one of those techniques that has been used in the inspection of the interior. However, the conventional X-ray CT is hard to use due to its cost, invasiveness, etc. Recently, optical measurement using visible (VIS)/near infrared (NIR) light is spotlighted because the optical system is inexpensive and safe compared to the X-ray.
However, optical measurement is challenging due to the scattering property of VIS/NIR light. In the case of the conventional CT, the interior is estimated using a mathematical inverse transform based on an assumption that paths are straight and parallel. Therefore, the conventional CT is not directly applicable to optical measurement when the scattering occurred in the target material. For the scattering material, optical scattering tomography (OST) [1, 4, 6] and optical diffusion tomography (ODT) [2, 3] have been proposed. In these methods, the paths of the scattered light are simulated and the interior is estimated from them. A drawback to these methods is the high computational cost due to simulating thousands of paths.
2.1 Conventional CT method
To begin with, we describe how the interior is estimated in the conventional CT. The conventional X-ray CT has been used for decades because it provides a clear visualization of the interior of the object. The conventional CT relies on the important X-ray’s property that it passes straight through the object without reflection and refraction, but it is attenuated a certain rate depending on the material.
When the paths in the object are straight and parallel, the interior is reconstructed using the inverse Radon transform. When a single ray is cast onto the object, it travels in a straight path. Its intensity is attenuated according to the path length and the absorption rate of the material. A projection of a single ray is regarded as the integral of the absorption over the straight line. The inverse Radon transform will give a relationship between the projection data obtained from the measurement and the spatial distribution of the absorption rate. Filtered back projection (FBP)  is an implementation of the inverse Radon transform, which is superior in terms of accuracy and speed.
2.2 Model of path of scattered rays
In the case of optical measurement, the interior cannot be estimated in the same way since the light path does not follow the assumption in the parallel transmission measurement. Now, we describe the model of the paths of the scattered light that our CT method relies on.
Limiting incident ray
In order to apply the conventional CT for recovering the interior, points on the surface where the ray entered and where the ray went out must be determined first.
Approximating path of scattered light
According to this model, we propose shortest-path transmission measurement to estimate the interior of the scattering object. The left-top figure of Fig. 1 illustrates the setup of our measurement. A single ray is cast toward an arbitrary point of the object, and the object’s surface is measured using a camera. Then measurement is repeated with rotating the object. As a result, sets of spreading straight rays for various angles of the incident rays are captured.
2.3 Light path alignment
According to the light path model, the paths in the object are simplified as spreading straight rays. However, it is still insufficient to apply inverse Radon transform because they are not parallel. Here, we introduce light path alignment that converts these simplified paths to parallel paths.
A schematic illustration of light path alignment is shown in Fig. 1. The light path alignment converts paths of the scattering surface measurement shown in the first row into the parallel paths in the second row. From a single measurement, scattered light through the object is observed and it is treated as a set of spreading straight paths from the incident point. Other sets of the paths are observed by casting rays from different directions. Then we pick up the paths of the same direction from these sets and align them according to the location so as to be parallel as shown in Fig. 4 c.
We denote a sinogram of shortest-path transmission measurement by g s and one of parallel transmission measurement by g p . Light path alignment is a conversion from g s to g p .
In order to formulate a conversion between both sinograms, we discuss about the relationship between the coordinates of the sinogram and the path of a ray. In the case of parallel transmission measurement as shown in Fig. 5 a, a ray that reaches x p of the projection plane with θ=θ p is stored at g p (x p ,θ p ). In the case of shortest-path transmission measurement as shown in Fig. 5 b, “ray 1” that reaches x s of the projection plane with θ=θ s is stored at g s (x s ,θ s ). We focus on the path of this ray. If this path is extended through the surface, it can be regarded as a ray of parallel transmission measurement with a virtual projection plane X ′ and θ=θ p′. Let x p′ be a location that the extended path reaches, then the ray is stored at g p (x p′,θ p′) in the sinogram of the converted sinogram. From the discussion above, light path alignment is altered by moving g s (x s ,θ s ) to g p (x p′,θ p′) like as shown in Fig. 2. This conversion is possible when each coordinate in the sinogram g p represents a unique ray in the object.
The paths of the scattered light are converted to parallel paths by this conversion, then the optical CT for the scattering object using inverse Radon transform is possible.
2.4 Optimal placement of light source
By light path alignment, the paths of shortest-path transmission measurement are converted to parallel. However, there are some unobserved paths due to the incompleteness of the measurement. For example, “ray 2” in Fig. 5 b is observed because it reaches the surface that can be seen from the camera. In contrast, “ray 3” cannot be observed because it reaches the surface that is hidden from the camera. Therefore, it is necessary to find the optimal setup of measurement to minimize the number of unobservable paths.
In the discussion so far, the light source is placed at the opposite of the camera. However, a number of the unobserved paths changes corresponding to the angle between the light source and the camera. Assume the camera is fixed; now, we derive an optimal placement of the light source relative to the camera.
We performed the numerical simulation for the evaluation of the proposed method. We confirmed that our method estimates the interior and confirmed the improvement of reconstruction by using the optimal measurement setup.
First, we show the sinogram when the scatter occurs and the reconstructed interior from it using FBP without alignment. The projection was simulated with the assumption of the ideal shortest-path transmission model where the scattering at surface is isotropic and the path of the ray in the object is straight. We performed an experiment for θ l =0° where the coverage is not complete and for θ l =90° where the measurement setup is optimal. Figure 8 b shows the sinogram and the reconstructed interior for θ l =0°. Although the sinogram consists of two curves reflecting the absorption corresponding to two circles, they are not perfect sinusoids like in the sinogram of parallel transmission measurement in Fig. 8 a. The reconstructed interior consists of two blurred shapes. The circles were not reconstructed correctly because the light paths were not aligned. Figure 8 c shows the sinogram and the reconstructed interior for θ l =90°. In this case, the light was cast perpendicularly to the orientation of the camera from the left side; thus, traces appear mainly on the right side. The reconstructed interior is totally unclear due to the corrupted shape of the sinogram.
Next, the sinogram was converted by light path alignment. Figure 8 d shows the aligned sinogram and the reconstructed interior for θ l =0°. Two sinusoidal curves appear after the light path alignment is executed. However, the sinogram is lacking its both sides since the measurement setup is not optimal. In the reconstructed interior, the shapes of the circles are reconstructed. We can see that ellipse-shaped artifacts appeared and the shape outside the artifacts is unclear. These artifacts are considered to be caused by the incompleteness of the sinogram. Figure 8 e shows the aligned sinogram and the reconstructed interior for θ l =90°. The aligned sinogram is complete and is substantially identical with the sinogram of parallel transmission measurement. The complete observation of rays was possible because all the ray through the closer half to the camera reach to the observable surface in this case. In the reconstructed interior, we can see the interior is reconstructed correctly.
We have proposed an OST using inverse mathematical transform. In order to reconstruct the interior using the inverse Radon transform, the paths in the object have to be parallel and straight. We have proposed shortest-path transmission model that approximates the paths of the scattered light as spreading straight lines from the incident point. We have also proposed light path alignment that converts approximated paths to parallel and optimal placement of the light source.
From the numerical experiments, we have presented the interior is reconstructed correctly by our proposed method. We have also shown that the reconstruction is improved by choosing the optimal placement of the light source.
For future work, we plan to measure a real object using our method.
This work was partly supported by JSPS KAKENHI Grant Numbers JP26700013, JP15K16027.
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