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.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Low rank subspace clustering (LRSC)

We consider the problem of fitting a union of subspaces to a collection of data points drawn from one or more subspaces and corrupted by noise and/or gross errors. We pose this problem as a non-convex optimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean and self-expressive dictionary plus a matrix of noise and/or gross errors. By self-expressive we mean a dictionary whose atoms can be expressed as linear combinations of themselves with low-rank coefficients. In the case of noisy data, our key contribution is to show that this non-convex matrix decomposition problem can be solved in closed form from the SVD of the noisy data matrix. The solution involves a novel polynomial thresholding operator on the singular values of the data matrix, which requires minimal shrinkage. For one subspace, a particular case of our framework leads to classical PCA, which requires no shrinkage. For multiple subspaces, the low-rank coefficients obtained by our framework can be used to construct a data affinity matrix from which the clustering of the data according to the subspaces can be obtained by spectral clustering. In the case of data corrupted by gross errors, we solve the problem using an alternating minimization approach, which combines our polynomial thresholding operator with the more traditional shrinkage-thresholding operator. Experiments on motion segmentation and face clustering show that our framework performs on par with state-of-the-art techniques at a reduced computational cost.

Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

On the Closure of the Tame Automorphism Group of Affine Three-Space

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of C3 is not closed, when the latter is seen as an infinite-dimensional algebraic group.

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.