Browsing by Subject "convolution"

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  • Rossi, Eino; Shmerkin, Pablo (2020)
    The L-q dimensions, for 1 <q <infinity, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the Lq dimension improve under convolution? This can be seen as a variant of the well-known L-p-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the L-q dimension. We also study the case q = infinity, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the L-q norms of convolutions due to the second author.
  • Enwald, Joel (Helsingin yliopisto, 2020)
    Mammography is used as an early detection system for breast cancer, which is one of the most common types of cancer, regardless of one’s sex. Mammography uses specialised X-ray machines to look into the breast tissue for possible tumours. Due to the machine’s set-up as well as to reduce the radiation patients are exposed to, the number of X-ray measurements collected is very restricted. Reconstructing the tissue from this limited information is referred to as limited angle tomography. This is a complex mathematical problem and ordinarily leads to poor reconstruction results. The aim of this work is to investigate how well a neural network whose structure utilizes pre-existing models and known geometry of the problem performs at this task. In this preliminary work, we demonstrate the results on simulated two-dimensional phantoms and discuss the extension of the results to 3-dimensional patient data.