1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

Complex variables and applications

Inside-outside determination is a basic building block for higher-level geometry processing operations. Generalized winding numbers provide a robust answer for triangle meshes, regardless of defects such as self-intersections, holes or degeneracies. In this paper, we further generalize the winding number to point clouds. Previous methods for evaluating the winding number are slow for completely disconnected surfaces, such as triangle soups or-in the extreme case- point clouds. We propose a tree-based algorithm to reduce the asymptotic complexity of generalized winding number computation, while closely approximating the exact value. Armed with a fast evaluation, we demonstrate the winding number in a variety of new applications: voxelization, signing distances, generating 3D printer paths, defect-tolerant mesh booleans and point set surfaces.

Fast winding numbers for soups and clouds

Diffusion curve primitives are a compact and powerful representation for vector images. While several vector image authoring tools leverage these representations, automatically and accurately vectorizing arbitrary raster images using diffusion curves remains a difficult problem. We automatically generate sparse diffusion curve vectorizations of raster images by fitting curves in the Laplacian domain. Our approach is fast, combines Laplacian and bilaplacian diffusion curve representations, and generates a hierarchical representation that accurately reconstructs both vector art and natural images. The key idea of our method is to trace curves in the Laplacian domain, which captures both sharp and smooth image features, across scales, more robustly than previous image- and gradient-domain fitting strategies. The sparse set of curves generated by our method accurately reconstructs images and often closely matches tediously hand-authored curve data. Also, our hierarchical curves are readily usable in all existing editing frameworks. We validate our method on a broad class of images, including natural images, synthesized images with turbulent multi-scale details, and traditional vector-art, as well as illustrating simple multi-scale abstraction and color editing results.

/pdf/hierarchical-diffusion-curves-for-accurate-automatic-image-edgxg7rzt6.pdf

Hierarchical diffusion curves for accurate automatic image vectorization

We propose a new algorithm for random-access evaluation of diffusion curve images (DCIs) using the fast multipole method. Unlike all previous methods, our algorithm achieves real-time performance for rasterization and texture-mapping DCIs of up to millions of curves. After precomputation, computing the color at a single pixel takes nearly constant time. We also incorporate Gaussian radial basis functions into our fast multipole representation using the fast Gauss transform. The fast multipole representation is not only a data structure for fast color evaluation, but also a framework for vector graphics analogues of bitmap editing operations. We exhibit this capability by devising new tools for fast diffusion curve Poisson cloning and composition with masks.

/pdf/fast-multipole-representation-of-diffusion-curves-and-points-23ymabhqve.pdf

Fast multipole representation of diffusion curves and points

Vectorization turns photographs into vector art. Manual vectorization, where the artist traces over the image by hand, requires skill and time. On the other hand, automatic approaches allow users to generate a result by setting a few global parameters. However, global settings often leave too much detail/complexity in some parts of the image while missing important details in others. We propose interactive vectorization tools that offer more local control than automatic systems, but are more powerful and high-level than simple curve editing. Our system enables novices to vectorize images significantly faster than even experts with state-of-the-art tools.

Interactive Vectorization

There is currently significant interest in freeform, curve-based authoring of graphic images. In particular, "diffusion curves" facilitate graphic image creation by allowing an image designer to specify naturalistic images by drawing curves and setting colour values along either side of those curves. Recently, extensions to diffusion curves based on the biharmonic equation have been proposed which provide smooth interpolation through specified colour values and allow image designers to specify colour gradient constraints at curves. We present a Boundary Element Method (BEM) for rendering diffusion curve images with smooth interpolation and gradient constraints, which generates a solved boundary element image representation. The diffusion curve image can be evaluated from the solved representation using a novel and efficient line-by-line approach. We also describe "curve-aware" upsampling, in which a full resolution diffusion curve image can be upsampled from a lower resolution image using formula evaluated orrections near curves. The BEM solved image representation is compact. It therefore offers advantages in scenarios where solved image representations are transmitted to devices for rendering and where PDE solving at the device is undesirable due to time or processing constraints.

http://biblio.telecom-paristech.fr/cgi-bin/download.cgi?id=14268

Peter Ilbery

Papers

Biharmonic diffusion curve images from boundary elements