<p>This thesis shows the result from investigations of methods that use distance vectors when matching pictures. The distance vectors are available in a distance map made by the Euclidean Distance Transform. The investigated methods use the two characteristic features of the distance vector when matching pictures, length and direction. The length of the vector is used to calculate a value of how good a match is and the direction of the vector is used to predict a transformation to get a better match. The results shows that the number of calculation steps that are used during a search can be reduced compared to matching methods that only uses the distance during the matching.</p>

This thesis shows the result from investigations of methods that use distance vectors when matching pictures. The distance vectors are available in a distance map made by the Euclidean Distance Transform. The investigated methods use the two characteristic features of the distance vector when matching pictures, length and direction. The length of the vector is used to calculate a value of how good a match is and the direction of the vector is used to predict a transformation to get a better match. The results shows that the number of calculation steps that are used during a search can be reduced compared to matching methods that only uses the distance during the matching.

The ordinary cross product in R3 is a widespread tool in mathematics and other sciences. It has applications in many areas such as several variable calculus, abstract algebra, geometry, and physics. In this thesis, we investigate in which Euclidean spaces R𝑛 there exist cross products. Based on the properties of the cross product in R3, we introduce two different notions of a cross product in R𝑛. Our first definition is based on the Pythagorean property and the perpendicular property of the cross product in R3. By direct calculation, we show that there is exactly one cross product in R1, no cross product in R2, and exactly two cross products in R3. We also show that if R𝑛 has a cross product, then 𝑛 = 1, 3, or 7. Our second definition uses the following self-selected properties of the cross product in  R3: the triple property, and the nondegeneracy property, leading to the notion of a semi-crossproduct. By direct computation, we discover that R3 has exactly two semi-cross products, which coincide with its cross products, moreover, there does not exist any semi-cross product in R1 or R2. The main result of the thesis is that there are no semi-cross products in R𝑛 for 𝑛 ≥ 4. As far as we know, the results of this chapter are new.

Real-time spatial awareness is essential in areas such as robotics and autonomous navigation. However, as environments expand and become increasingly complex, maintaining both a low computational load and high mapping accuracy remains a significant challenge. This thesis addresses these challenges by proposing a novel method for real-time construction of continuous Euclidean distance fields (EDF) using Gaussian process (GP) regression, hereafter referred to as GP-EDF, tailored specifically for large indoor environments. The proposed approach focuses on leveraging the inherent structural information of indoor spaces by partitioning them into rooms and constructing a local GP-EDF model for each, reducing the computational cost tied to large matrix operations in GPs. By also exploiting the geometric regularities commonly found in indoor spaces it detects walls and represents them as line segments. This information is integrated into the models’ priors to both improve accuracy and further reduce the computational expense. Comparison with two baselines demonstrated the proposed approach’s effectiveness. It maintained low computation times despite increasing amounts of sensor data, signifying a significant improvement in scalability. Results also confirmed that the EDF quality remains high and isn’t affected by partitioning the GP-EDF into local models. The method also reduced the influence of sensor noise on the EDF’s accuracy when incorporating the line segments into the model. Additionally, the proposed room segmentation method proved to be efficient and generated accurately partitioned rooms, with a high degree of independence between them. In conclusion, the proposed approach offers a scalable, accurate and efficient solution for real-time construction of EDFs, demonstrating significant potential in aiding autonomous navigation within large indoor spaces. / Realtidsrumslig medvetenhet är avgörande inom områden som robotik och autonom navigering. Emellertid, när miljöer expanderar och blir alltmer komplexa, kvarstår det en betydande utmaning att bibehålla både en låg beräkningsbelastning och hög kartläggningsnoggrannhet. Denna avhandling bemöter dessa utmaningar genom att föreslå en ny metod för realtidskonstruktion av kontinuerliga euklidiska avståndsfält (EDF) med hjälp av regression via gaussiska processer (GP), hädanefter benämnd GP-EDF, specifikt anpassad för stora inomhusmiljöer. Den föreslagna metoden fokuserar på att utnyttja den inneboende strukturella informationen i inomhusmiljöer genom att dela upp dem i rum och konstruera en lokal GP-EDF-modell för varje rum, vilket minskar den beräkningsbelastning som är kopplad till stora matrisoperationer i GP:er. Genom att även utnyttja de geometriska regelbundenheter som vanligtvis finns i inomhusutrymmen, detekterar den väggar och representerar dem som linjesegment. Denna information integreras sedan i modellernas a priori-fördelningar, både för att förbättra noggrannheten och ytterligare minska den beräkningsmässiga kostnaden. Jämförelse med två baslinjemodeller demonstrerade den föreslagna metodens effektivitet. Den bibehöll låga beräkningstider trots ökande mängder sensordata, vilket indikerar en betydande förbättring av skalbarheten. Resultaten bekräftade även att kvaliteten på EDF:en förblir hög och påverkas inte av uppdelningen av GP-EDF:en i lokala modeller. Metoden minskade även sensorbrusets inverkan på EDF:ens noggrannhet vid integrering av linjesegment i modellen. Dessutom visade sig den föreslagna rumsegmenteringsmetoden vara effektiv och genererade korrekt uppdelade rum, med en hög grad av oberoende mellan dem. Sammanfattningsvis erbjuder den föreslagna metoden en skalbar och effektiv lösning för realtidskonstruktion av EDF:er, och visar på betydande potential att underlätta autonom navigering inom stora inomhusutrymmen.