Adaptive estimation of a function from its Exponential Radon Transform in presence of noise

Single Photon Emission Computed Tomography (SPECT) imaging is a valu able diagnostic tool that is frequently used to detect the presence of tumors ins ide a patient's body. The idea behind SPECT imaging can be described very briefly in the following m anner: A small amount of radioactive tracer attached to some nutrient is injecte d in the patient's body. After a brief interlude (ranging from a few minutes to a few hours), a SPECT scanner is used to measure the radioactive emissions from the body in a range o f directions by moving the scanner around the body. Along each line, the data represent s the intensity of emissions from a point along that line. This data can be mathematically interpret ed as an attenuated Radon transform. From the attenuated Radon transform data, one then tries to image the inside of the patient's body to locate the presence of tumors. If on e makes the simplifying assumption that the attenuation is constant, then the attenuat ed Radon transform reduces to the case of what is known as the exponential Radon transform. We point the interested reader to [19] and [31] for a more detailed overview. In the setting of the current article, our focus of investigation is t he estimation of a function from its stochastic (i.e.