A strong-type Furstenberg-S\'{a}rk\"{o}zy theorem for sets of positive measure

For every $\beta\in(0,\infty)$, $\beta\neq 1$ we prove that a positive measure subset $A$ of the unit square contains a point $(x_0,y_0)$ such that $A$ nontrivially intersects curves $y-y_0 = a (x-x_0)^\beta$ for a whole interval $I\subseteq(0,\infty)$ of parameters $a\in I$. A classical Nikodym set counterexample prevents one to take $\beta=1$, which is the case of straight lines. Moreover, for a planar set $A$ of positive density we show that the interval $I$ can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter S\'{a}rk\"{o}zy-type theorem by Kuca, Orponen, and Sahlsten.Comment: 12 pages, 3 figure

Framework for 4D medical data compression

U ovom radu predložen je novi programski okvir za kompresiju četvero-dimenzionalnih (4D) medicinskih podataka. Arhitektura ovog programskog okvira temelji se na različitim procedurama i algoritmima koji detektiraju vremenske i prostorne zalihosti u ulaznim 4D medicinskim podacima. Pokret kroz vrijeme analizira se pomoću vektora pomaka koji predstavljaju ulazne parametre za neuronske mreže koje se koriste za procjenu pokreta. Kombinacijom segmentacije, pronalaženja odgovarajućih blokova i predikcijom vektora pomaka, zajedno s ekspertnim znanjem moguće je optimirati performanse sustava. Frekvencijska svojstva se analiziraju proširenjem wavelet transformacije na tri dimenzije. Za mirne volumetrijske objekte, moguće je konstruirati različite wavelet pakete s različitim filtrima koji omogućavaju širok raspon analiza frekvencijskih zalihosti. Kombinacijom uklanjanja vremenskih i prostornih zalihosti moguće je postići vrlo visoke omjere kompresije.This work presents a novel framework for four-dimensional (4D) medical data compression architecture. This framework is based on different procedures and algorithms that detect time and spatial (frequency) redundancy in recorded 4D medical data. Motion in time is analyzed through the motion fields that produce input parameters for the neural network used for motion estimation. Combination of segmentation, block matching and motion field prediction along with expert knowledge are incorporated to achieve better performance. Frequency analysis is done through an extension of one dimensional wavelet transformation to three dimensions. For still volume objects different wavelet packets with different filter banks can be constructed, providing a wide range of frequency analysis. With combination of removing temporal and spatial redundancies, very high compression ratio is achieved