La corrispondenza fra ideali e varietà algebriche, affini e proiettive

L’obiettivo di questa tesi è costruire una corrispondenza tra oggetti algebrici, gli ideali, e oggetti geometrici, le varietà algebriche, e studiarne il comportamento nel caso affine e proiettivo. Nel caso affine, lavorando in campi algebricamente chiusi, si descrive una corrispondenza biunivoca tra ideali radicali e varietà affini non vuote. Ciò permette di riformulare ogni affermazione sulle varietà affini in un’affermazione sugli ideali radicali, e viceversa; in particolare si descrivono le relazioni tra le operazioni su ideali e su varietà: alla somma degli ideali corrisponde l’intersezione di varietà, a prodotto e intersezione di ideali corrisponde l’unione di varietà, al quoziente di ideali corrisponde la chiusura di Zariski della differenza insiemistica delle varietà. Inoltre ad ogni ideale primo corrisponde una varietà irriducibile e agli ideali massimali corrispondono i punti dello spazio. Nel caso proiettivo invece, si considerano ideali omogenei e varietà proiettive, definite da polinomi omogenei. Restringendosi a campi algebricamente chiusi, si ha una corrispondenza biunivoca tra varietà proiettive non vuote e ideali radicali omogenei contenuti in un particolare ideale, (x_0,…,x_n). Con queste restrizioni la corrispondenza tra le operazioni algebriche e geometriche è la stessa studiata nel caso affine. Infine si introduce la chiusura proiettiva di una varietà affine, che è la più piccola varietà proiettiva che contiene una varietà affine data.

Milano. Un progetto per il riassetto urbano dell'area piazza d'armi. Urbano e rurale nell'architettura della citta

Il disegno del verde acquisisce un ruolo imprescindibile nell’intera area soggetta a studio. Se in prossimità delle zone edificate di progetto esso assume una dialettica propria, derivata dallo studio dei tracciati urbani sopra citati, in adiacenza di Via Carlo Tosi, prospiciente all’ospedale San Carlo, va ad assumere un carattere affine alla griglia ortogonale sopra descritta, rispondendo non più alle giaciture del costruito esistente ma all’inclinazione dei campi risalenti al 1878, epoca a cui corrisponde la carta di Manovra.

Varietà algebriche affini

In questa tesi introdurremo la topologia di Zariski sullo spazio affine n-dimensionale. Ne mostreremo alcune proprietà e arriveremo a dimostrare che ogni insieme algebrico è esprimibile come unione finita di varietà.

Absence of somatostatin SST(2) receptor internalization in vivo after intravenous SOM230 application in the AR42J animal tumor model

Among clinically relevant somatostatin functions, agonist-induced somatostatin receptor subtype 2 (sst(2)) internalization is a potent mechanism for tumor targeting with sst(2) affine radioligands such as octreotide. Since, as opposed to octreotide, the second generation multi-somatostatin analog SOM230 (pasireotide) exhibits strong functional selectivity, it appeared of interest to evaluate its ability to affect sst(2) internalization in vivo. Rats bearing AR42J tumors endogenously expressing somatostatin sst(2) receptors were injected intravenously with SOM230 or with the [Tyr(3), Thr(8)]-octreotide (TATE) analog; they were euthanized at various time points; tumors and pancreas were analyzed by immunohistochemistry for the cellular localization of somatostatin sst(2) receptors. SOM230-induced sst(2) internalization was also evaluated in vitro by immunofluorescence microscopy in AR42J cells. At difference to the efficient in vivo sst(2) internalization triggered by intravenous [Tyr(3), Thr(8)]-octreotide, intravenous SOM230 did not elicit sst(2) internalization: immunohistochemically stained sst(2) in AR42J tumor cells and pancreatic cells were detectable at the cell surface at 2.5min, 10min, 1h, 6h, or 24h after SOM230 injection while sst(2) were found intracellularly after [Tyr(3), Thr(8)]-octreotide injection. The inability of stimulating sst(2) internalization by SOM230 was confirmed in vitro in AR42J cells by immunofluorescence microscopy. Furthermore, SOM230 was unable to antagonize agonist-induced sst(2) internalization, neither in vivo, nor in vitro. Therefore, SOM230 does not induce sst(2) internalization in vivo or in vitro in AR42J cells and pancreas, at difference to octreotide derivatives with comparable sst(2) binding affinities. These characteristics may point towards different tumor targeting but also to different desensitization properties of clinically applied SOM230.

One-Step (18)F-Labeling of Carbohydrate-Conjugated Octreotate-Derivatives Containing a Silicon-Fluoride-Acceptor (SiFA): In Vitro and in Vivo Evaluation as Tumor Imaging Agents for Positron Emission Tomography (PET)

The synthesis, radiolabeling, and initial evaluation of new silicon-fluoride acceptor (SiFA) derivatized octreotate derivatives is reported. So far, the main drawback of the SiFA technology for the synthesis of PET-radiotracers is the high lipophilicity of the resulting radiopharmaceutical. Consequently, we synthesized new SiFA-octreotate analogues derivatized with Fmoc-NH-PEG-COOH, Fmoc-Asn(Ac?AcNH-?-Glc)-OH, and SiFA-aldehyde (SIFA-A). The substances could be labeled in high yields (38 ± 4%) and specific activities between 29 and 56 GBq/?mol in short synthesis times of less than 30 min (e.o.b.). The in vitro evaluation of the synthesized conjugates displayed a sst2 receptor affinity (IC?? = 3.3 ± 0.3 nM) comparable to that of somatostatin-28. As a measure of lipophilicity of the conjugates, the log P(ow) was determined and found to be 0.96 for SiFA-Asn(AcNH-?-Glc)-PEG-Tyr³-octreotate and 1.23 for SiFA-Asn(AcNH-?-Glc)-Tyr³-octreotate, which is considerably lower than for SiFA-Tyr³-octreotate (log P(ow) = 1.59). The initial in vivo evaluation of [¹?F]SiFA-Asn(AcNH-?-Glc)-PEG-Tyr³-octreotate revealed a significant uptake of radiotracer in the tumor tissue of AR42J tumor-bearing nude mice of 7.7% ID/g tissue weight. These results show that the high lipophilicity of the SiFA moiety can be compensated by applying hydrophilic moieties. Using this approach, a tumor-affine SiFA-containing peptide could successfully be used for receptor imaging for the first time in this proof of concept study.

Femur Specific Polyaffine Model to Regularize the Log-Domain Demons Registration

Osteoarticular allograft transplantation is a popular treatment method in wide surgical resections with large defects. For this reason hospitals are building bone data banks. Performing the optimal allograft selection on bone banks is crucial to the surgical outcome and patient recovery. However, current approaches are very time consuming hindering an efficient selection. We present an automatic method based on registration of femur bones to overcome this limitation. We introduce a new regularization term for the log-domain demons algorithm. This term replaces the standard Gaussian smoothing with a femur specific polyaffine model. The polyaffine femur model is constructed with two affine (femoral head and condyles) and one rigid (shaft) transformation. Our main contribution in this paper is to show that the demons algorithm can be improved in specific cases with an appropriate model. We are not trying to find the most optimal polyaffine model of the femur, but the simplest model with a minimal number of parameters. There is no need to optimize for different number of regions, boundaries and choice of weights, since this fine tuning will be done automatically by a final demons relaxation step with Gaussian smoothing. The newly developed synthesis approach provides a clear anatomically motivated modeling contribution through the specific three component transformation model, and clearly shows a performance improvement (in terms of anatomical meaningful correspondences) on 146 CT images of femurs compared to a standard multiresolution demons. In addition, this simple model improves the robustness of the demons while preserving its accuracy. The ground truth are manual measurements performed by medical experts.

Geometry-Aware Multiscale Image Registration via OBBTree-Based Polyaffine Log-Demons

Non-linear image registration is an important tool in many areas of image analysis. For instance, in morphometric studies of a population of brains, free-form deformations between images are analyzed to describe the structural anatomical variability. Such a simple deformation model is justified by the absence of an easy expressible prior about the shape changes. Applying the same algorithms used in brain imaging to orthopedic images might not be optimal due to the difference in the underlying prior on the inter-subject deformations. In particular, using an un-informed deformation prior often leads to local minima far from the expected solution. To improve robustness and promote anatomically meaningful deformations, we propose a locally affine and geometry-aware registration algorithm that automatically adapts to the data. We build upon the log-domain demons algorithm and introduce a new type of OBBTree-based regularization in the registration with a natural multiscale structure. The regularization model is composed of a hierarchy of locally affine transformations via their logarithms. Experiments on mandibles show improved accuracy and robustness when used to initialize the demons, and even similar performance by direct comparison to the demons, with a significantly lower degree of freedom. This closes the gap between polyaffine and non-rigid registration and opens new ways to statistically analyze the registration results.

Simultaneous Multiscale Polyaffine Registration by Incorporating Deformation Statistics

Locally affine (polyaffine) image registration methods capture intersubject non-linear deformations with a low number of parameters, while providing an intuitive interpretation for clinicians. Considering the mandible bone, anatomical shape differences can be found at different scales, e.g. left or right side, teeth, etc. Classically, sequential coarse to fine registration are used to handle multiscale deformations, instead we propose a simultaneous optimization of all scales. To avoid local minima we incorporate a prior on the polyaffine transformations. This kind of groupwise registration approach is natural in a polyaffine context, if we assume one configuration of regions that describes an entire group of images, with varying transformations for each region. In this paper, we reformulate polyaffine deformations in a generative statistical model, which enables us to incorporate deformation statistics as a prior in a Bayesian setting. We find optimal transformations by optimizing the maximum a posteriori probability. We assume that the polyaffine transformations follow a normal distribution with mean and concentration matrix. Parameters of the prior are estimated from an initial coarse to fine registration. Knowing the region structure, we develop a blockwise pseudoinverse to obtain the concentration matrix. To our knowledge, we are the first to introduce simultaneous multiscale optimization through groupwise polyaffine registration. We show results on 42 mandible CT images.

Accurate and robust reconstruction of proximal femur from sparse intraoperative data and dense point distribution model for surgical navigation

Constructing a 3D surface model from sparse-point data is a nontrivial task. Here, we report an accurate and robust approach for reconstructing a surface model of the proximal femur from sparse-point data and a dense-point distribution model (DPDM). The problem is formulated as a three-stage optimal estimation process. The first stage, affine registration, is to iteratively estimate a scale and a rigid transformation between the mean surface model of the DPDM and the sparse input points. The estimation results of the first stage are used to establish point correspondences for the second stage, statistical instantiation, which stably instantiates a surface model from the DPDM using a statistical approach. This surface model is then fed to the third stage, kernel-based deformation, which further refines the surface model. Handling outliers is achieved by consistently employing the least trimmed squares (LTS) approach with a roughly estimated outlier rate in all three stages. If an optimal value of the outlier rate is preferred, we propose a hypothesis testing procedure to automatically estimate it. We present here our validations using four experiments, which include 1 leave-one-out experiment, 2 experiment on evaluating the present approach for handling pathology, 3 experiment on evaluating the present approach for handling outliers, and 4 experiment on reconstructing surface models of seven dry cadaver femurs using clinically relevant data without noise and with noise added. Our validation results demonstrate the robust performance of the present approach in handling outliers, pathology, and noise. An average 95-percentile error of 1.7-2.3 mm was found when the present approach was used to reconstruct surface models of the cadaver femurs from sparse-point data with noise added.

Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes.

Fitting 3D morphable models using implicit representations

We consider the problem of approximating the 3D scan of a real object through an affine combination of examples. Common approaches depend either on the explicit estimation of point-to-point correspondences or on 2-dimensional projections of the target mesh; both present drawbacks. We follow an approach similar to [IF03] by representing the target via an implicit function, whose values at the vertices of the approximation are used to define a robust cost function. The problem is approached in two steps, by approximating first a coarse implicit representation of the whole target, and then finer, local ones; the local approximations are then merged together with a Poisson-based method. We report the results of applying our method on a subset of 3D scans from the Face Recognition Grand Challenge v.1.0.

An Empirical Study of Non-Rigid Surface Feature Matching of Human from 3D Video

This paper presents an empirical study of affine invariant feature detectors to perform matching on video sequences of people with non-rigid surface deformation. Recent advances in feature detection and wide baseline matching have focused on static scenes. Video frames of human movement capture highly non-rigid deformation such as loose hair, cloth creases, skin stretching and free flowing clothing. This study evaluates the performance of six widely used feature detectors for sparse temporal correspondence on single view and multiple view video sequences. Quantitative evaluation is performed of both the number of features detected and their temporal matching against and without ground truth correspondence. Recall-accuracy analysis of feature matching is reported for temporal correspondence on single view and multiple view sequences of people with variation in clothing and movement. This analysis identifies that existing feature detection and matching algorithms are unreliable for fast movement with common clothing.

Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in RnRn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive HαHα measure for some fixed α>mα>m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.