Multiple-scattering Optical Tomography with Layered Material Toru Tamaki, Bingzhi Yuan, Bisser Raytchev, Kazufumi Kaneda Graduate School of Engineering, Hiroshima University, Japan tamaki@hiroshima-u.ac.jp Yasuhiro Mukaigawa ISIR, Osaka University, Japan mukaigaw@am.sanken.osaka-u.ac.jp Abstract—In this paper, we propose an optical tomography for optically dense media in which multiple scattering is dominant. We model a material by many layers of voxels, and light scattering as a distribution from a voxel in one layer to other voxels in the next layer. Then we write attenuation of light along a light path by an inner product of vectors, and formulate the scattering tomography as an inequality con- straint minimization problem solved by interior point methods. We show experimental results with numerical simulation for evaluating the proposed method. Keywords-scattering of light, optical tomography, scattering tomography, inverse scattering, inequality constraint optimiza- tion I. I NTRODUCTION Tomography, an inverse problem to see inside materials by observing output of a structured input, is an important issue in physics, medical imaging, computer vision and related research field [1]–[7]. A well known application widely used for medical purpose is Computed Tomography (CT) with X-ray [8]. The property of penetrating a body enable us to observe the X-ray emitted from a source at the opposite side of the body and reconstruct a 3D attenuation map. Diagnoses based on CT is very popular, however an unnecessary exposure of X-ray must be minimal because of the damage to a human body. Furthermore, different aspects from different modalities such as MRI, PET, and SPECT [9] are required for medical diagnosis. Alternative technologies have therefore been developed over the years. We focus on optical tomography, a modality of infrared light, in particular scattering tomography [4], [7]. Infrared light is safe to human body; devices can be small comprised of LEDs and chips, and less expensive than X-ray devices. A disadvantage is that light is scattered and diffused as well as attenuated inside a body, while X-ray is assumed not to be scattered. This problem makes the situation much more difficult. If no scattering, we would use a linear transform for this inverse problem like as the Radon transform for CT. Due to the scattering, a light path is no longer straight but rather complicated inside a body, and the observed light is not a sharp impulse but a blurred distribution of light. There are two main approaches to the scattering tomog- raphy. One is Diffuse Optical Tomography (DOT) [3] which assumes that the light is extremely scattered inside the media. In other words, at each point of the media the light scattering is modeled by isotropic diffusion. This process can be modeled by partial differential diffusion equations and solved by Finite Element Method. DOT has an advantage of the ability to trace how the light goes at each small time step (e.g. 10 pico seconds), however the diffusion assumption is often not adequate because infrared light of particular wavelength can penetrate human body with relatively small amount of scattering. The other approach is a tomography with single-scattering assumption [4]. If the media is optically thin and the scattering is quite low, then we can assume that the scattering event happens only once for each light path, i.e, single scattering is the dominant event. This assumption is opposite to that of DOT in which the dominant event is multiple scattering because the media is optically thick and scattering event happens so many times as the light travels through the media. Of course this single- scattering assumption is too strict to a variety of materials including human body. Recently Ishii et al. [7] have proposed a scattering tomog- raphy for moderately scattered media, i.e., multiple forward scattering is dominant but diffusion is not. They assumed that the light is attenuated inside the material when the path of the light passes thought a hidden object, which absorbs any light completely. To measure the attenuation due to the hidden object, they use a reference media in which there is no hidden object in order to compare the reference with the observed media. Obviously the use of reference is not practical in real situations, and hidden objects do not well model the distribution of attenuation inside real materials. In this paper, we propose a multiple-scattering optical to- mography without any reference to estimate the distribution of attenuation at each voxel in the material. As like Ishii et al. [7], we assume that multiple forward scattering events take place inside the media. We model the propagation of light though the media with thin layers: at each layer of the media, light is scattered from one layer to the next layer. This is a reasonable assumption because a layered model has been used for realistic human faces [10] and for anatomical description of human skin [11]. Also in the limit of infinitesimal thickness of layers, it can model continuous materials. In this paper we focus on 2D cases where the material is in 2D space divided into grid, however we can stack 2D materials to extend the method to 3D materials as described in [4], [7].