Local versus global in quasiconformal mapping for medical imaging. November 20, 2008 conformal same form or shape mapping is an important technique used in complex analysis and has many applications in di erent physical situations. Solving the steady state heat equation using conformal mappings. Quasiconformal maps were first introduced in higher. Comparative analysis of quasiconformal deformations in shape. A quasiconformal mapping of the open nball induces a quasiconformal mapping on the boundary. Intuitively, quasi conformal maps are orientationpreserving homeomorphisms with bounded conformality distortions. When m is bounded, f is called a quasi conformal mapping. Pdf local versus global in quasiconformal mapping for. A similar fact for regularly oscillating functions holds.
Recently, an efficient quasistatic technique has been developed to calculate the conductor loss by using the conformal mapping technique 21, 22, which is based on quasitem approximation. Quasiconformal hybrid multimodality image registration. Under a quasiconformal map, an infinitesimal circle is mapped to an infinitesimal ellipse with bounded eccentricity. Two further properties of quasi conformal maps that we will find useful are i they can be composed to produce further quasi conformal mappings. Planar morphometry, shear and optimal quasiconformal mappings. Quasiconformal mappings in nspace and the rigidity of. Using qclr, landmarkaligned diffeomorphisms between images or surfaces can be obtained, even with a large geometric difference or a large number of landmark constraints. For instance, the robertsonwalker spacetimes are quasieinstein manifolds. A note on extremal quasiconformal mappings springerlink. Hence the plane theory of conformal mappings does not directly generalize to the higher dimensions. Intuitively, quasiconformal maps are orientationpreserving homeomorphisms with bounded conformality distortions. We describe how circle packing can be used to obtain quasi conformal mappings of surfaces and demonstrate the advantages of this approach by producing quasi conformal at maps with data from the visible man and from an mri volume of. Quasiconformal approaches for two and threedimensional. An overview 47 where, z is defined in the complex zplane xy plane, shown in fig.
Quasiconformal transformation optics qcto enabled modified. Worked examples conformal mappings and bilinear transformations example 1 suppose we wish to. A construction of anisotropic meshes based on quasiconformal. Deformation similarity measurement in quasiconformal shape space. In this paper, we first propose the quasiconformal landmark registration qclr algorithm to obtain diffeomorphic 11 and onto registrations between images or surfaces. The angular distortion of a quasiconformal map can be represented by beltrami. The most essential property of a quasiconformal mapping t is that the moduli of a quadrilateral fi in d and its fimage satisfy the relation 1 k1 mod fi mod ffi k mod fi.
If w is a quasi conformal harmonic mapping between two riemann surfaces with compact and smooth boundaries and approximate analytic metrics, then w is bilipschitz continuous with respect to internal metrics. Very recently, quasiconformal mapping transformation optics has also been utilized to atten the traditional luneburg lens, and satisfactory results are reported 32. Quasiconformal mappings appeared in explicit form in the works of h. Efficient quasiconformal map for simulation of diffusion at. Discrete conformal mapping a conformal mapping of a riemannian surface to another one is a continuous onetoone function. If is differentiable at the point, then the linear mapping transforms a ball of the tangent space into an ellipsoid the ratio of the major to the minor semiaxis of which is. Planar morphometry, shear and optimal quasiconformal. The eccentricity and the orientation of the ellipse can be represented by a complex. This result extends some recent results where, instead of an elliptic differential operator, only the laplace operator is considered. Quasiconformal maps form a branch of complex analysis. Under a quasi conformal map, an infinitesimal circle is mapped to an infinitesimal ellipse with bounded eccentricity. Conformal maps preserve both angles and the shapes of infinitesimally small. Quasinearly subharmonic functions and conformal mappings 247 now we use the identity.
Supine and prone colon registration using quasiconformal mapping. By the measurable riemann mapping theorem the new dilatation admits a quasiconformal solution of 3, f f. Tooth morphometry using quasiconformal theory deepai. Although there are some numerical quasi conformal mapping tech. Preprint 1 computing quasiconformal maps on riemann. The following example shows how badly behaved a homeomorphism can. Quasi static analysis, in particular the use of conformal mapping, is well established as a useful technique for the calculation of propagation constants of quasi tem transmission lines. Introduction the concept of quasilocal energy is very important for the study of many aspects of the physics of spacetime. An extremal quasiconformal mapping f of a domain d is said to be of nonlandslide type if the set e f. Besides the need for computing quasiconformal maps, the applications of quasiconformal maps is also of great interest in applications such as shape deformation, surface registration, texture mapping, brain mapping, mesh editing, and etc. More on the conformal mapping of quasilocal masses. Design of metamaterialcoated arrays through quasiconformal transformation optics ephrem t. Siam journal on imaging sciences society for industrial.
If w is a quasiconformal harmonic mapping between two riemann surfaces with compact and smooth boundaries and approximate analytic metrics, then w is bilipschitz continuous with respect to internal metrics. An orientationpreserving mapping is called quasi conformal or a mapping with bounded distortion if. The application of conformal mapping to the calculation of the complex internal impedance of conductors is less widely. Solving the steady state heat equation using conformal. We show that the jacobian jf of a quasiconformal mapping. Its application ranges from the study of black hole physics and spacetime thermodynamics, and all the way to cosmology. This result supports the above claim that the class of. Active control of sound waves via threedimensional quasi. Quasistatic conductor loss calculations in transmission. We also deploy our method to compare a variety of drosophila wing shapes and show that our approach allows us to recover aspects of phylogeny as marked by. The theory of quasiconformal mappings in higher dimensions, i.
S 2 isotopic to frelative to the boundary, we have kf kh. As a generaliza tion of conformal maps, quasiconformal maps are of great interest in. One of the important concepts of riemannian geometry is conformal mapping. Thompson, and shingtung yau ieee transactions on medical imaging, vol. Quasiconformal geometry 3 the following theorem is due to rademacherstepanov see v1 for a proof. The method is based on extension of classical results of. This step adjusts the conformal mapping to be quasi conformal. Quasiconformal maps and the mapping theorem matrix editions.
Suppose that d is a simply connected domain in c that is. Quasiconformal hybrid multimodality image registration 811 registration result can improve the image fusion quality and place a signi. The formal definition of quasiconformal maps on the complex plane is given below. Quasiconformal geometry of monotone mappings page 3 of 18 theorem 20 implies that s is bilipschitz if and only if. Suppose that d is a simply connected domain in c that is not equal to. Quasiconformal mapping encyclopedia of mathematics. Lavrentev, one of the founders of the theory of quasiconformal mappings. Introduction to quasiconformal mappings in nspace stony brook. Fortunately, this ratio can be determined accurately to 1 part in 10 5 using the following simple expression 6. To make our choice optimal, we impose the condition that the. Tooth morphometry using quasiconformal theory sciencedirect. We describe how circle packing can be used to obtain quasiconformal mappings of surfaces and demonstrate the advantages of this approach by producing quasiconformal at maps with data from the visible man and from an mri volume of. Worked examples conformal mappings and bilinear transfor. If f is a harmonic quasi conformal mapping between two jordan domains d and.
One challenge in development of the quasi conformal geometry is to compute effective numerical quasi conformal mappings. Recent connections and applications of planar quasiconformal. For certain reasons3, we usually allow, in the above definition, the pres. A novel approach based on the shape space concept is pro. Also quasieinstein manifold can be taken as a model of the perfect. The theory of quasiconformal mappings divides traditionally into two branches, the mappings in the plane and the case of higher dimensions. Conformal mapping of carotid vessel wall and plaque. One challenge in development of the quasiconformal geometry is to compute effective numerical quasiconformal mappings. The registration of these two scans is desirable so that the user can clarify situations or confirm polyp findings. Conformal mappings of quasieinstein manifolds admitting. Gehring in c rings and quasiconformal mappings in space 55, trans. Deformation similarity measurement in quasiconformal. This conformal mapping describes the transformation from the circular cavity unit disk.
B d is an ooweight in b if and only if d is a john domain. The study is based on a concept, introduced by the authors in their earlier papers 17, 18 and developed here in 2, of asymptotic homogeneity of a quasiconformal mapping at a given point, which is equivalent to a special kind of differentiability going back to belinskii 6, p. Local versus global in quasi conformal mapping for medical imaging. There are some solutions that extrude or rotate 2d conformal mapping to 3d, but these methods are essentially 2d and cannot control sound wave propagation in 3d space. Created as part of a mcnair research project with the purpose of presenting the findings of my research on the conformal mapping solution of the steady state heat. Compared to the conformal mapping results in figure 8, it is clear that the registration achieves greater accuracy when using the internal features. The goal in these applications is to nd mappings with desired properties that can be expressed. It is uniquely extremal if the inequality 4 is strict when h6f. In doing so, we are motivated by the fact that, in recent years it has become common amongst the imaging community, to consider images as higher dimensional manifolds embedded in rn or rn see, e.
Quasiconformal mappings have appeared as an attractive subset of the general transformation optics method because they only require isotropic, dielectriconly materials. This technique was applied to a coplanar waveguide cpw and symmetric twin conductors, and calculated resistance agreed well to ex. Design of metamaterialcoated arrays through quasi conformal transformation optics ephrem t. Aoo condition for the jacobian of a quasiconformal mapping. Local versus global in quasiconformal mapping for medical. In virtual colonoscopy, ct scans are typically acquired with the patient in both supine facing up and prone facing down positions. The importance of quasiconformal mappings in complex analysis was realized by ahlfors and teich muller in the 1930s. Comparative analysis of quasi conformal deformations in shape space 493 a b fig. This problem can be reduced to that of finding univalent holomorphic maps f, g of the unit disk and its complement into the extended complex plane, both admitting. Computational quasi conformal geometry has been developed, which extends the quasi conformal theory into a discrete setting. They show rigorously that the quasiconformal parameterization exists uniquely, and is invariant to similarity transformations, independent of resolution, and orientationpreserving.
Recently, quasi conformal geometry has attracted attention from different fields, such as applied mathematics, computer vision and medical imaging. The proposed quasipolygonal conformal mapping is given by %. Genus zero surface conformal mapping and its application to brain surface mapping pdf xianfeng gu, yalin wang, tony f. This metric is a symmetric form and utilizes the local changes of curvature to measure the similarity between deformations. Such mappings are called quasiconformal reflections.
Two further properties of quasiconformal maps that we will find useful are i they can be composed to produce further quasiconformal mappings. Figure 11 shows the registration results for 3 segments based on such harmonic maps. As the 2d carotid template lies on an lshaped nonconvex domain, onetoone correspondence of the mapping operation was not guaranteed. Computational quasiconformal geometry and its applications on. Design of metamaterialcoated arrays through quasiconformal. Comparative analysis of quasiconformal deformations in. Preprint 1 computing quasiconformal maps on riemann surfaces. On quasiconformal selfmappings of the unit disc and. Quasinearly subharmonic functions and conformal mappings vesna koji. Given any two conformal metrics their ratio is a nonnegative function.
Supine and prone colon registration using quasiconformal. Note that 10 provides a single approximation for all times with a claimed accuracy better than 0. To address this issue, an iterative beltrami differential chopping and smoothing procedure. An orientation preserving homeomorphism f c is called quasiconformal, if it is kquasiconformal for some k. A construction of anisotropic meshes based on quasi.
Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable riemann mapping theorem, proved by lars ahlfors and lipman bers. A quasiconformal mapping of the open nball induces a homeomorphism on the boundary. Cortical surface flattening using least square conformal. Chapter 7 quasiconformal maps and beltrami equation. The general problem of constructing a quasiconformal mapping from one simplyconnected domain onto another satisfying 2 was posed and solved by m. Quasi conformal transformation optics qcto enabled modified luneburg lens design using broadband anti reflective layer.
I found the sub ject difficult to learn, mainly because i had a hard time appreciating how smooth the. The constant kmeasures how near a mapping is to being conformal, i. I article pdf available in proceedings of the national academy of sciences 409. Quasistatic analysis, in particular the use of conformal mapping, is well established as a useful technique for the calculation of propagation constants of quasitem transmission lines. Pdf planar morphometry, shear and optimal quasiconformal.
It is different from the classical metricbased or high dimensional embedding mesh adaptation methods. Qr to a rectangle r taking the z i to the corners of r and taking the arc of. We also make a comparison with freesurfer 4 and caret 3 about performance and distortion. The theorem generalizes the riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Conformal mapping and its applications suman ganguli1 1department of physics, university of tennessee, knoxville, tn 37996 dated. The formal definition of quasi conformal maps on the complex plane is given below. The category of maps that one usually considers in the higherdimensional theory are the quasiregular mappings, or, if injective, quasiconformal. In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths more formally, let and be open subsets of. In this work, we make use of the transformation optics based on quasiconformal mapping to design a directive and planar focusing. Transformation optical design is generally complicated by the requirement for highly anisotropic and inhomogeneous constituent materials. In mathematics, conformal welding sewing or gluing is a process in geometric function theory for producing a riemann surface by joining together two riemann surfaces, each with a disk removed, along their boundary circles. It is a deep result in geometric function theory, it establishes strong regularity properties for quasiconformal homeomorphisms.
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