Abstract—The paper considers a layer-to-layer 3D reconstruction method for diffuse optical mammotomography which uses conic geometry for time-domain measurements. The method is based on an approximate 2D reconstruction of transverse layers of a conic scattering object by inverting an integral equation with integration along a photon average trajectory. The equation is inverted using a backprojection algorithm with an original filtration of optical projections by the Vainberg-Butterworth method. A numerical experiment shows that our method reconstructs the 3D distributions of absorbing inhomogeneities embedded in a conic object much faster than the multistep Newton-like algorithms, and reconstruction accuracy remains acceptable, at least when inhomogeneities are not in the immediate vicinity of object boundaries. I. INTRODUCTION IFFUSE optical mammotomography (DOM) is a promising medical imaging method aimed at the early detection of breast cancer [1] - [5]. In this method female breast is illuminated by near-infrared radiation in the so- called therapeutic transparency window from 700 to 900 nm. Near-infrared light from an array of sources is observed with an array of receivers and then an inverse problem, i.e. the tomographic reconstruction problem is solved to localize breast tumor. Most importantly, DOM helps visualize the spatial distributions of functional optical parameters such as blood volume and blood oxygen saturation. As a result, it becomes possible to detect structures with increased vascularity that is a characteristic feature of malignancy [6]. It does mean that DOM allows primary breast cancer to be distinguished from benign lesions. Moreover, DOM has other advantages in comparison with traditional mammographic imaging modalities. Unlike X-ray mammography and computed tomography, it poses no risk from ionizing radiation. DOM is significantly more inexpensive than magnetic resonance imaging or positron emission tomography. Last, DOM allows obtaining mammograms with greater contrast than Doppler ultrasonography or ultrasonic reflectivity tomography. Manuscript received November 2, 2009. A. B. Konovalov is with the Russian Federal Nuclear Center – Zababakhin Institute of Applied Physics, PO Box 245, Snezhinsk Chelyabinsk Region, 456770 Russia (phone: +73514654639; fax: +73514652233; e-mail: a_konov@ mail.vega-int.ru). A. S. Uglov is with the Russian Federal Nuclear Center – Zababakhin Institute of Applied Physics, PO Box 245, Snezhinsk Chelyabinsk Region, 456770 Russia (e-mail: a.s.uglov@vniitf.ru). V. V. Lyubimov is with the Institute for Laser Physics of Vavilov State Optical Institute Corporation, Birzhevaya Line 12, St.Petersburg, 199034 Russia (e-mail: vv_lyubimov@mail.ru). DOM measurements are now taken mainly in two ways: the breast is compressed between two parallel glass plates with the sources and receivers of near-infrared light [1] - [3] or the breast is put into an applicator in the form of a cup or a cone [4], [5]. In the latter case, the air gaps between the breast and the glass surface of the applicator are filled with a tissue-equivalent gel to match refractive indices. In what the first way of measurement is good is that the inverse problem of diffuse optical tomography (DOT) can be fitted using exact analytical solutions of the transport or diffusion equation. The second works better in practice because it is not necessary to compress the breast thus providing comfort for patients with no risk of distorting the optical properties of the tissue under examination. If however object geometry is complicated, it is impossible to construct and use the exact analytical solutions of the transport or diffusion equation. To reconstruct the optical inhomogeneity distributions in this case, they use the Newton-like algorithms [7] based on a multiple numerical solution of the forward problem of DOT, i.e. the problem of radiation propagation through matter. A procedure of step-by-step solution approximation requires a computing time of no less than a few tens of minutes for getting 3D mammotomograms, and is therefore inapplicable for the real-time clinical explorations. In order to fasten the process of reconstruction, we have recently developed a new method based on the concept of photon average trajectory (PAT) [8] - [12]. The method allows us to reduce the reconstruction procedure to the solution of an integral equation with integration along a generally curvilinear PAT: () () () t L g t vl f dl = < > ∫ r (1) Here () g t is the optical projection (the relative signal disturbance due to optical inhomogeneities) measured for time-gating delay t , () f r is the sought function of optical inhomogeneity distribution, t <⋅> is the operator of averaging over the spatial distribution of the photons that contribute to the measured optical signal, L is the PAT from the source point to the receiver point, l is distance along the PAT, and () vl is a factor meaning the inverse relative velocity of photons moving along the PAT over time t . We consider implementations of the PAT method with the use of algebraic reconstruction techniques in [8] - [10]. It is shown in [11], [12] that 2D reconstruction by the PAT method can Diffuse Optical Mammotomography Based on Backprojection Algorithm with Vainberg-Butterworth Filtration Alexander B. Konovalov, Alexander S. Uglov, and Vladimir V. Lyubimov D Proceedings of the 4th International Symposium on Communications, Control and Signal Processing, ISCCSP 2010, Limassol, Cyprus, 3-5 March 2010 978-1-4244-6287-2/10/$26.00 ©2010 IEEE