Graphics (and related) Terms

Graphics, Geometric Modeling Math and Numerical Methods Physics & Mech Eng CFD Image (Signal) Processing

Always under construction...additions, corrections, modifications are WELCOME!

NOTES

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Graphics, Geometric Modeling

Also see:
Term Meaning
Barycentric coordinate Coordinates of a point in space in terms of other points. For example the barycentric coordinates of a point along a line between two points would be (t,1-t). The barycentric coordinates of a point inside a triangle is based on the following. Draw lines from the point, P, to each of the three vertices of the triangle, A,B,C. The barycentric coordinates of P are:
(area(P,B,C)/area(A,B,C),area(P,A,C)/area(A,B,C),area(P,A,B)/area(A,B,C))
BCC - Body Centered Cubic A celullar lattice that includes a node in the middle of each cell, popular in chemistry but also used in some volumetric approaches. For example, see Body-centered cubic.
Chamfer the transition surface, between two major surfaces which meet at a convex corner, usually created by the manufacturing process.
Characteristic polyhedron The polyhedron formed by the control points for a surface patch
Continuity (parametric continuity; also referred to as C1, C2, etc.) parametric continuity: For two parametric curves ra(u) and rb(u) if ra(1) = rb(0) they are c0 continuous (the endpoints meet) if ra'(1) = rb'(0) they are c1 continuous (first derivative of the endpoints is same or tangents are the same) if ra''(1) = rb''(0) they are c2 continuous (second order continuous or change in tangents is the same at the end points)
Continuity (geometric continuity; also referred to as G1, G2, etc.) G1 continuity is maintained if the unit tangent vectors are identical (versus C1 continuity in which the tangent vectors are identical). G2 is a similar idea but a bit more complicated in the way the math works out.
Contour tree From http://www.dmu.ac.uk/~mt/research.html: "A Contour Tree is a hierarchical structure describing which plateaus are totally enclosed by other plateaus." Also see http://www.cs.ubc.ca/~hcarr/research/contourtrees.html.
Control polygon Any sequence (mesh) of points generally used to control the shape of a curve (surface) - as opposed to the term polygon used in modeling that usually refers to a sequence of points that define a closed border of a planar surface.
Convex hull The minimal bounding polygon (polyhedron) of a collection of points in 2-space (3-space) that is convex. You can also talk about the convex hull of higher order curves (surfaces) in which case the hull bounds the curves (surfaces).
CSG Constructive Solid Geometry. An object description kept as a binary tree in which leaf nodes are transformed instances of primitive objects such as sphere, cylinder, cube and internal nodes are boolean combinations of the subtrees, i.e., union, difference, intersection.
Displacement map Similar to a texture map except the surface of the object is actually displaced by the map.
Extrinsic an extrinsic property of a surface is a property of the surface which is relative to the space in which it is embedded. The height of a point is extrinsic; curvature at a point is not (it is intrinsic)
Fair (intuitive) Refers to a surface not having extraneous bumps and wiggles.
Fillet the transition surface, between two major surfaces which meet at a concave corner, usually created by the manufacturing process.
Free form deformation (FFD) A method to distort the shape of an object in which a 3D local coordinate grid is placed around an object. The vertices of the object are located within the grid by noting their coordinate definitions relative to it. The grid is then interactively distorted by the user. The object points are then remapped back into the distorted grid using tricubic interpolation thus distorting the object.
Frenet frame A.k.a., moving trihedron, moving triad. Local coordinate system defined at a point on a curve, p(u), made up of tangent vector (pu), principle normal vector, (puu-(DOT(puu,pu)/|pu)|2)*pu) and binormal (puXpu).
Gamma correction Modifying the color palette to compensate for non-linear response of a monitor so that a linear change in digital color values results in linear change in light emitted from screen.
Gaussian curvature If k1 and k2 are the principle curvatures at a point, the Gaussian curvature is k1k2.
Golden Rectangle The golden rectangle is a rectangle with sides of 1 and 1+x such that, if you chop off a 1x1 square you'd be left with a rectangle with sides x and 1 and (here's the important part) the ratio of the sides of the original rectangle (1:1+x) is equal to the sides of the rectangle left after chopping off the square (x:1). It turns out that x = sqrt(5)/2 - 1/2 (or, said another way, the sides of the original rectangle are 1 and 1/2 + sqrt(5)/2.
Hausdorff distance "two sets are within Hausdorff distance r from each other iff any point of one set is within distance r from some point of the other set" (from www.cut-the-knot.org"s page on Hausdorff distance)
Intrinsic an intrinsic property of a surface is a property of the surface which is not relative to the space in which it is embedded. Curvature at a point is an intrinsic propterty; the height of a point is not (it is extrinsic).
KD tree A binary tree which partitions k-dimensional data. At each successive node, the data is divided into two buckets which splits the data along one of the dimensions (presumably the dimension with the largest range of values).
Level set methods A numerical technique to follow the evolution of interfaces. The evolving interface is defined by an initial shape and a speed function F, defined over the points of the interface. This is used to define a higher dimensional surface in which the interface is embedded. The interface is a cut of the surface. See level sets by J.A.Sethian
Manifold A points set whose points are in one-to-one correspondence with n-tuples of real numbers, the coordinates of the points, is called an n-dimensional differentiable manifold [CEM, p.573].
"A two-dimensional surface in R^d (d>=2) is a 2-manifold if each point on it has a neighborhood that is homeomorphic to an open two-dimensional disk. In this paper we call 2-manifolds simply manifolds. A manifold surface is called compact if it is bounded and closed. For example, a sphere is a compact manifold whereas a plane is not. A manifold is called orientable if it has two distinct sides. A sphere is orientable since it has two sides, 'inside' and 'outside'. A Mobius strip or a Klein bottle is not orientable." [DTTD]
mean curvature If k1 and k2 are the principle curvatures at a point, the mean curvature is (k1+k2)/2.
Medial Axis Transform Also called Skeletonization. Reducing a region to a skeletal remnant, for example by some type of thinning operation, that reflects the main topology of the main areas of the original region reflecting the connectivity and branching structures of these areas. Distance to the surface is associated with the skeletal points so that the original regions might be reconstructed. See HIPR2's Skeletonization/Medial Axis Transform
Mip-map A.k.a. pyramidal parametrics. Method of storing filtered versions (by powers of 2 in each dimension at each stage) of a texture map.
Morse theory From http://mathworld.wolfram.com/MorseTheory.html: "A generalization of calculus of variations which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology of the manifold." Also see What is Homotopy?.
Polygon Polygon usually refers to a sequence of points that define a closed border of a planar surface in three-space (compare to control polygon). However, in some contexts, the term polygon may not necessarily imply planarity.
Principal Curvatures The maximum and minumum bending of a surface at a point. See Wolfram's Mathworld Principal Curvatures.
Quadrics, quadratic surfaces A second order algebraic surface. Quadratic surfaces are also called quadrics. See Wolfram's Mathworld Quadratic Surfaces.
Quaternions Four-tuples used to describe orientation and rotations. Avoids gimbal lock, the drawback of Euler angles. [PPRU]
Red Green Subdivision A mesh refinement approach using a regular (red) subdivision and an irregular (green) subdivision. For example, see Simplicial-based multiresolution volume datasets management: an overview
Reeb graph A Reeb graph is a skeletal representatin of the topology of an object.
R-function A real-valued function of real variables that inherit some properties of Boolean functions. They can model objects implicitly. See R-functions. See Solid Modeling with R-furnctions
Simplex (tentative) a simplex is the simplest geometric element of a particular dimension. For our purposes, it refers to a triangle or a tetrahedron.
Spherical harmonics Spherical harmonics are special functions of the spherical coordinates, theta and phi. These functions can be thought of as "standing waves on a sphere". (http://www.csd.abdn.ac.uk/~dritchie/graphics/)
Steiner points There are various definitions for Steiner points. The definition most related to graphics are points inserted into a planar straight edge graph to allow a Delaunay triangulation of the graph. See http://www-2.cs.cmu.edu/~quake/triangle.defs.html.
Superquadrics Superquadrics and their geometric properties
Tensor A tensor is a quantity that is invariant under coordinate system transformation. See, for example, it's use in the discussion of Stress at a Point (pdf). Generalization of vectors and matrices. A tensor of rank 0 is a scalar, of rank 1 a vector, rank 2 is a matrix, rank 3 a 3D rectangular array and rank k a k-dimenstional rectangular array. See Wolfram's Tensor.
Tensor product Let A=[aij] be an m x p matrix and B = [bij] an n x q matrix. Then the tensor product of A and B is the mn x pq matrix, C, which can be expressed as C = [aijB]. See [TSHORES]
Tensor product surface A surface patch that is completely defined in terms of the corner conditions it is called a tensor product surface. A tensor product patch has a rectangular topology and is expressed in a symmetric form (w.r.t u and v) as can be seen in the standard bicubic patch.
Thin plate spline A spline defined by minimizing an energy function based on curvature. See Wolfram's Thin Plate Spline.
Wavelets A mathematical technique for the hierarchical decomposition of functions - also referred to as a multiresolution technique because it represents a function as a weighted sum of base functions which differ in the resolution of features which they capture. [WCG]


Math, Numerical Methods

also see:
Term Meaning
Bijective "A one-to-one function from a set A onto a set B is called a bijection. Bijections play a fundamental role in combinatorial mathematics. A bijection from a set onto itself is usually called a permutation of that set." [IC,p.16]
Condensation Also known as "particle filters" (vision community). See Particle filters for real-time object tracking
Conjugant gradient method An iterative solver for a sparse matrix system representing a system of linear equations. See Jonathan's paper
Convex combination A linear combination of elements where the weights are 1) all greater than or equal to zero and 2) they sum to one. See PlanetMath.org. Also see convex hull, above.
Curl measure of a vector field's tendency to rotate about a point. You can think of it as the cross product between a vector of first partial derivative operators and the function.
Critical point A critical point, (x,f(x)) , of a function f is if f(x) is defined and f '(x) is either zero or undefined. The x-coordinate of the critical point is called a critical value or a critical number. (From thinkquest.org)
Divergence The sum of first partial derivatives of a function (see Laplacian operator). See Wolfram's Laplacian.
Direct Linear Transform (DLT) Least squares solution for point correspondences between, for example in camera callibration, world space points and image space points. See Camera Callibration
Dynamic programmng "Dynamic Programming is an approach developed to solve sequential, or multi-stage, decision problems; hence, the name "dynamic" programming. But, as we shall see, this approach is equally applicable for decision problems where sequential property is induced solely for computational convenience." [DYNP]
E(3) 3 dimensional Euclidean space
Edge detection See this page, for example.
Eigenvalue Value associated with an Eigenvector. It's magnitude (greater or less than one) tells whether the associated Eigenvector will diverge to infinity or collapse to zero after repeated applications of a matrix A. This is an important indicator for iterative solvers in which an error vector (expressed as a linear combination of Eigenvectors) needs to be forced to zero for a solution to be found. See Determinants, Eigen Vectors.
Eigenvector A vector which is not changed (except for a scale factor, the associated eigenvalue) by application of a matrix, Ax = kx where k (usually notated lambda) is called the eigenvalue. You might think of this as the axis of rotation if A were a 3D rotation matrix. See Determinants, Eigen Vectors.
EM-PCA First see PCA. In a nutshell, to find C matrix that relates latent variables X to observed vectors Y use Y = CX: Estimate X by X = (CTC)-1CTY, then estimate C by Cnew = YXT(XXT)-1. Do this iteratively.
Eulerian formulation v. Lagrangian formulation e.g., in fluid dynamics, whether the analysis is grid-based or particle-based.
Euler integration (tentative) Method of integrating a function by stepping in the direction of the function's derivative, i.e. f(x+dx) = f(x) + dx*f'(x), as in updating the position of a particle by it's velocity.
Factor Analysis "The main applications of factor analytic techniques are: (1) to reduce the number of variables and (2) to detect structure in the relationships between variables, that is to classify variables." From Principal Components and Factor Analysis
Finite element method (FEM)
Finite element analysis (FEA)
A numerical method in which a surface or volume is broken down into relatively small descrete elements. The relationship of one element to adjacent elements with respect to some attribute is expressed as some differential (or difference) equation. This is expressed in matrix form and (possibly non-linear) solution strategies are applied to produce a sequence to values for this attribute over time. See Internet Finite Element Resources.
Forward differencing Updating the descrete values of a variable by forming the expression which increments it from one value to the next. E.g. if p(t) = at+b, then dp = p(t+1)-p(t) = a, so initializing p = p(0) = b, we have in sequence p = b, a+b, 2a+b, 3a+b, etc. This can be done with higher order equations. For example if p(t) = at2+bt+c, then dp=p(t+1)-p(t)=2at+a+b and ddp=dp(t+1)-ddp(t)=2a so initializing p=p(0)=c and dp=dp(0)=a+b, we have (p,dp) = (c,a+b), (a+b+c,3a+b), (4a+2b+c,5a+b), (9a+3b+c,7a+b), etc.
Golden Ratio Search Search strategy similar to binary search but based on the golden ratio. (radical 5 minus 1 over 2) For the number, see Phi: That Golden Number; for the search strategy, see Golden Ratio Search
Gradient of a scalar field is a vector field where the vectors all point in the direction of higher values and the magnitude of the vectors represent the rate of change.
Halton Sequence Generator The Halton Sequence Generator is a quasi-random number sequence.
Hessian Similar to the Jacobian (really the second derivative version), it is defined as Hij = (2nd order partial of R)/((partial of ti)(partial of tj)).
Harmonic mean For a set of numbers xi, it's one over the average of one over xi. See MathWorld.
Hidden Markov Models (HHMs) See Hidden Markov Models.
Holonomic constraint (tentative) A constraint based on one level of derivative of a variable. For example a constraint based only on positional information.
Homeomorphic (tentative) A one-to-one onto mapping
Homographic A one-to-one onto mapping between two figures.
Homotopy "A continuous transformation from one function to another. A homotopy between two functions f and g from a space X to a space Y is a continuous map G from such that and , where denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping space from the first function to the second. " (From mathworld.worlfram.com) Also see What is Homotopy?
Ill-conditioned A problem in which a small change in measurements might produce a large change in the solution.
Ill-posed A problem in which there might not be a solution or there is more than on solution or the solution might not depend continuously on the data. See Ill-posed Inverse problems
Implicit Euler integration Instead of updating a function from it's current position f(x) by stepping in the direction of its tangent (f(x+dx)=f(x)+f'(x)*dx, find a point which is a step away in x whose derivative points directly away from the current point: f(x+dx)-f(x)=f'(x+dx)*dx. This take more computation per step but presumably it allows bigger steps to be taken. For example, see Implicit integration - the linear case. (Compare to Euler integration)
Independent component analysis (ICA) ICA describes how the observed data are generated by a process of linearly mixing the components. See this Tutorial
Injective "A function f is called a one-to-one function or injection if it associates different values in the range with different values in the domain." [IC, p.15]
Isomap A dimensionality reduction technique that computes the geodesic distance between two points by constructing a nearest neighbor graph from a set of points and computing distances between non-nearest neighbors from the graph. This is followed by a standard Multidimensional Scaling (MDS) dimensionality reduction technique. See this article from Science Magazine.
Jacobian A matrix, J, in which each entry i,j relates the change of variable tj to the change in variable vi so that V = J T.
Essentially it is the multidimensional version of dx/dt. T would be a vector of unknowns; V is given. If J is not square, can be solved using the pseudo inverse by multiplying both sides by the transpose of J (to form the square matrix JTJ on the right) and then multiplying both sides by the inverse (JTJ)-1 to get J+V = T where J+, the pseudo inverse of J, is (JTJ)-1JT
Kalman Filter A recursive solution to the discrete-data linear filtering problem. See The Kalman Filter
Lagrange multipliers (from [LAGR]) Lagrange multipliers provide a method of finding maxima or minima of a function subject to constraints. This method is particularly useful when more than one constraint is involved. To solve a problem with Lagrange multipliers,
  • Write down the quantity to be maximized or minimized. U se the variables to write a function such as f(x,y,z), that corresponds to this quantity. This is the objective function.
  • Write down the constraint, 0=g(x,y,z). Make sure that the objective function and constraint have the same units.
  • Form a function to differentiate:
    F(x,y,z,)=(objective function) - A*(constraint).
  • Take partials of F with respect to x, y, z, and A. Set these equal to zero.
  • Solve the system of equations that results. The values of the variables from the solution provide the desired maximum or minimum.
  • In the event that the system of equations has more than one solution, compute the value of the objective function at each solution point and take the point (s) that gives the maximum or minimum.
Also see An Introduction to Lagrange Multipliers
Laplace operator sum of the second partial derivatives of a function
Levenberg_marquardt method From Mathworld: "Levenberg-Marquardt is a popular alternative to the Gauss-Newton method of finding the minimum of a function F(x) that is a sum of squares of nonlinear functions."
Linear discriminant analysis LDA tutorial
Linear dynamical systems Lectures by Prof. Boyd at Stanford
Wikipedia's Linear dynamical system
Linear least-squares fitting a set of data points to a linear combination of any specified functions [CEM]
Linear programming
Linear optimization
optimizing a linear function bounded by linear constraints. The Simplex Method is the standard way to efficiently organize the computations.
Linear regression fitting a set of data points to a straight line model [CEM]
Lloyd's algorithm repositioning a set of points to the centroid of their respective Voronoi cell in order to more evenly distribute them. See Answers.com's Lloyd's algorithm
Lipschitz condition A condition of an upper bound on the change in a function given a change in parametric value: ||f(u2)-f(u1)|| <= L*||u2-u1|| for some finite number L in some region R of f. The Lipschitz condition is implied if the function f has fintion partial derivatives [GCTD]
LU Decomposition Decomposing a square matrix A into lower and upper triangular matrices such that A = L U. Further, the diagonal of L can be set uniformly to '1'. This decomposition can be used to solve a linear set of equations Ax=b by solving Ux = y and then Ay = b. These can be easily solved with backward and forward substitution. This decomposition can also be used to compute the inverse of a matrix by successively setting the b vector to columns of the identity matrix.
Markovian property "For a process to be Markovian, the future must depend only at the present and past should not have any effect on the future." [DYNP]
Maximum likelihood Given a model, M, that is parameterized by the set of parameters S, the probability of generating a dataset D is P(D|S). If you don't know the parameter setting but can observe data, then determine the parameters by calculating the maximum P(S|D). Compute the maximum by setting the dirivative of the equation for P(S|D) to zero. Often it is easier to deal with the log of the P(S|D), for example, when P() is a function of a Gaussian distribution and, therefore, has an exponential term in it.
Monte Carlo integration evaluates the function at a random sample of points and estimates its integral based on that random sample. For example, if the area under a curve is desired for a function y=f(x), 0 < x < 1 and 0 < f(x) < 1, then a bunch of (xi,yi) points are generated in the interval; the percentage of points for which yi < f(xi) is used as the estimation for the area under the curve.
Multidimensional Scaling "Multidimensional scaling (MDS) can be considered to be an alternative to factor analysis (see Factor Analysis). In general, the goal of the analysis is to detect meaningful underlying dimensions that allow the researcher to explain observed similarities or dissimilarities (distances) between the investigated objects." From Multidimensional Scaling
Multiresolution analysis (MRA) (tentative) analyzing a function at multiple resolutions (e.g. wavelets).
Newton's method
Newton-Raphson method
root finding by geometrically extending the tanget line at a current point Xi, until it crosses zero, then setting the next guess, Xi+1, to the abscissa of that zero-crossing. See http://uranus.eng.auth.gr/lessons/1/5.html
Particle Filters Also known as "condensation". See Particle filters for real-time object tracking
Penalty Method A method of enforcing contraints by using momentary stiff springs to re-establish violated constraints, e.g., collision.
Positive definite matrix A real symmetric matrix with postive eigenvalues. Equivalently, a symmetric matrix for which xTAx > 0 [LAA,p.250; 5] See matrix properties.
Principle component analysis (PCA) Break down data into it's principle components. That is, determine an ordered set of orthogonal vectors such that the data varies along the direction of the first vector by the greatest amount, then varies along the direction of the second vector by the next greatest amount, etc.
Steps: 1) get data, 2) subtract out mean, 3) calculate the covariance matrix, 4) get the eigenvectors and eigenvalues of the covariance matrix, 4) order the eigenvectors according to the magnitude of their associated eigenvalues.
The eigenvectors are essentially new coordinate frame in which a data point can be expressed by projecting the data point onto the eigenvectors. For data compression, choose only the top so many of the eigenvectors and project a data point onto each of these vectors.
See A tutorial on Principal Components Analysis [PDF].
Quadratic programming solving for Sj by minimizing R(Sj) subject to Ci(Sj) by taking a 2nd order Newton-Raphson step in R (irrespective of Ci), then projecting that onto the null-space of a first-order step to drive the Ci to zero.
Radial Basis Function Network Uses Gaussians or Gaussian-like functions, (monotonic, radially symmetric), hi(x), to approximate a function y = f(x) = ∑wi hi(x). The weights are learned by a training set of data and finding a least-squares fit to the data. It is a single-layer linear neural network.
See Introduction to Radial Basis Function Networks
R3 triples of real numbers
Riccati Equations See Riccati Equations
Runga Kutta (tentative) An integration method (used as opposed to the Euler method) in which a derivative half step is taken and the derivative there is used to update the value of the function, e.g. f(x+dx) = f(x) + f'(x+(dx/2)*f'(x)). This is second order Runga Kutta. Fourth order Runga Kutta is commonly used. [NR]
S3 3D vectors on the unit sphere
SE(3) Special Euclidean group - position and orientation in 3D space
Secant method a root finding method. See Secant Method.
self organizing map (SOM) "The basic Self-Organizing Map (SOM) can be visualized as a sheet-like neural-network array, the cells (or nodes) of which become specifically tuned to various input signal patterns or classes of patterns in an orderly fashion. The learning process is competitive and unsupervised..." (http://www.mlab.uiah.fi/~timo/som/thesis-som.html)
SO(3) Special orthogonal group - 3x3 matrices, columns are unit length, mutually orthogonal, determinant is 1.0. (rigid rotations)
Sigmoidal The usual definition is 'curved like the letter S'. However, dictionaries may also include 'curved like the letter C' and 'shaped like a sigma'.
Simulated Annealing A search strategy in which random perturbations from a local search are allowed but attenuated over time, simulating the physical process of annealing as liquids freeze. See Simulated Annealing.
Singular point "Some discontinuities are singular points. A singular point of a curve is any point at which the curve does not have a unique tangent. From this, it follows that the function will not be differentiable at a singular point. A singular point of a curve can occur where the curve is pointed, where it crosses itself, or where it branches in two directions." (From geocities.com/mathfair2002/)
Singular value decomposition Solution method for Ax = b. Similar to LU decomposition in that the matrix A is decomposed into more than one matrix. In this case it is decomposed into 3 matrices, A = UWV. W is a diagonal matrix. The magnitude of the diagonal values indicates relative importance of the values of A. If a value of W is 0 or close to 0, then A is singular. This can be used to reduce the dimensionality of the problem by removing 'unimportant' values of A from consideration.
Stationary point A point at which the derivative of a function vanishes, f'(x) = 0. (From mathworld.worlfram.com)
steepest descent Interative technique in finding a solution to Ax=b. You take a step in the direction of steepest descent (gradient). The magnitude of the step you take is determined by the place along the direction your stepping in which the gradient is orthogonal to that direction. [ICGM]
Stochastic sampling Taking one or more pseudo random samples of an area as representation of an area which would otherwise be integrated.
Surjective "A function f from A onto B is defined as a function from A to B such that each member of B is associated with at least one member of A. A function from A onto B is also called a surjection from A to B." [IC, p. 16]
SVM
Support Vector Machine
SVM is a particular type of Kernel machine (a large class of learning algorithms). Uses inner product of feature vectors. Use a kernel function that non-linearly map features into (hopefully) linearly separable space. From: Tutorial (pdf). Also see www.kernel-machines.org and www.suport-vector.net.
Twist vector A twist vector expresses rigid body motion. It is a 6 component vector that consists of angular velocity and linear velocity.
Underconstrained A problem is underconstrained if the constraints allow for more than one solution.
Variational calculus Variational calculus studies the variation of an integral over a path under perturbations of the path q(t). We substitute the initial path q(t) with the new path q(t) = q(t) + q(t). [MECH].
Verlet integration Calculate forces at current position, use to calculate velocity at next halfway time step, and use to update positions. Velocities leapfrog over positions. See http://www.cse.clrc.ac.uk/msi/software/Democritus/Theory/verlet.html
Viterbi search algorithm An algorithm to compute the most likely sequnce of states in a hidden Markov model given a sequence of observed outputs. It uses best possible paths to intermediate states and updates incrementally. See Viterbi Algorithm.
Wrench vector Similar to a twist vector, it is a 6 component force vector consisting of a linear component and angular component.


Physics, Mechanics, etc.

Also see:
Term Meaning
Acceleration The rate of change of velocity with respect to time.
Bernoulli's equation An equation that relates pressure, density, velocity, elevation and gravitational acceleration. See Bernoulli's Equation
Cauchy stress Cauchy stress is the force per unit area of the deformed solid, also called true stress. See Stress at a Point (pdf) or the Mechanics of Solids.
Continuum Mechanics The study of the behavior of material as a uniform continuum, neglecting it's sstructure on a smaller scale. See Introduction to Continuum Mechanics
Critical damping "A critically damped system approaches equilibrium as fast as possible without any overshoot or oscillation." From http://icva.hep.ph.ic.ac.uk/CMS/UG/Vibrations+Waves/damped/critical_damping.html. Also see: http://web.mit.edu/8.03-esg/watkins/8.03/damlim.pdf
Damping For our purposes, a force applied proportionate to and in the opposite direction of velocity. Most often encountered with springs. The damping term is added to help control the motion and preventing it from getting out of hand.
Consider
F = kx - dv
where k is the spring constant, x is the deviation from rest length, v is velocity and d is the damping constant.
Density Mass per unit volume (under specified conditions of pressure and temperature). [DICT]
Dirichlet Boundary Conditions In a fluid simultion, expressing the boundary conditions in terms of fluid velocity. See Dirichlet Boundary Conditions for example.
Elastic Collisions A material is elastic in the area in which Hooke's Law holds. The elastic limit is the where the elastic region ends. Perfectly elastic collisions are those in which no kinetic energy is lost and momentum is conserved. See Elastic Collisions
Eulerian framework Looking at a time-varying spatial function by breaking space into cells and keepting track of what happens inside of each cell. Compare to a Lagrangian framework.
Force The capacity to do work or cause physical change. [DICT]
Friction Resistance of one object moving relative to, and in contact with, another. Friction is a force linearly related to velocity, and is a function of normal force and the materials in contact.
Gravity The natural force of attraction between any two massive bodies, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. [DICT]
Green's Strain Characterizes a deformation by the change in square of infintesimal line element lengths between the deformed and undeformed surface. See (ps) for example.
Hertzian contact Contact modeling between two linear elastic objects. See Robotics and Human Augmented Systems
Hooke's Law Force is proportional to displacement as in a spring. They are related by a constant of proportionality. It is usually written as F = k*x where x is the difference between rest length (as in a spring) and current length.
Inelastic Collisions inelastic means that some energy is lost in the collision. See Inelastic Collisions
Inertia Inertia is the measure of self-resistance to acceleration. (http://www.ebtx.com/ntx/ntx13.htm)
Lagrangian framework Looking at a time-varying spatial function by tracking particles through space. Compare to a Eularian framework.
Laplace operator The summation of second order partial differentials. See The Laplace Operator.
Load  
Mass A property of matter equal to the measure of an object's resistance to changes in either the speed or direction of its motion. The mass of an object is not dependent on gravity and therefore is different from but proportional to its weight. [DICT]
Material derivative "The time rate of change of a quantity such as temperature or velocity of a material particle is known as a material derivative, and is denoted by a superimposed dot or by D/Dt." See Material Derivative
Moment The (first) moment of a quantity is quantity*distance. The second moment of a quantity is quantity*distance2.
momentum Momentum = Mass times velocity. Angular momentum = mass velocity distance (from point object is spinning or orbiting around) (http://www.astronomynotes.com/angmom/angmom.htm) Momentum (both linear and angular) is conserved in a closed system.
Normal Force The force perpencidular to the surface of contact between two bodies.
Poison's Ratio Poison's Ratio: n = e perpendicular/e parallel Ratio of longitudinal strain perpendicular/parallel to maximum compressive stress. ()
Pressure Force applied uniformly over a surface, measured as force per unit of area. [DICT]
Shear Deformation of an object which can be characterized by the relative movement of parallel surfaces in opposite directions.
Strain The axial strain is defined as the fractional change in length. Strain = (deformation of member) divided by the (original length of member). (http://physics.uwstout.edu/statics/)
Stress What is known as Axial (or Normal) Stress, often symbolized by the Greek letter sigma, is defined as the force perpendicular to the cross sectional area of the member divided by the cross sectional area. Typcially, stress is given by: Sigma = F/A. Axial stress is really the 'pressure' in a solid member. (http://physics.uwstout.edu/statics/)
Surface traction Applied distributed forces on the surface of an object.
Tension A force working to elongate an object.
Traction A force, such as adhesive friction, pulling one object towards another
viscosity Resistance to flow.
Young's Modulus The value of Young's modulus - which is a measure of the amount of force needed to produce a unit deformation - depends on the material. Young's Modulus for Steel is 30 x 106 lb/in2, for Aluminum E = 10 x 106 lb/in2, and for Brass E = 15 x 106 lb/in2. (http://physics.uwstout.edu/statics/)


CFD

From:
compressiblewhere the density changes (e.g. gas)
densitymass per unit area
diffusion a physical process in which some property of a gas is transported by the random motion of molecules. It is realted to the stress tensor and the viscosity of the gas.
dilatanttype of fluid that flow easily with a low velocity at low strain rates, but becomes more like a solid as the strain reate increases (e.g. quicksand).
hydrostaticproperty of fluid at rest
ideal fluidfluid that has no viscosity and does not flow in a turbulent manner
ideal flowwhen the friction due to viscosity is negligible
incompressiblewhere the density is fixed (e.g. water)
isotropicthe same in all directions
laminar flowpurely viscous flow (no turbulence); the fluid flows in laminas or layers
Newtoniantype of fluid where the shear stress is linearly proportional to the velocity gradient
pressurein a static fluid, the normal compressive force per unit area acting on a surface immersed in the fluid.
steady flowflow in which the velocity components and thermodynamic properties at every point in space do not change with time
surface tensionthe apparent stress in the surface layer of a liquid
thixotropictype of fluid in which stress-strain rate relationship depends on prior working or straining (e.g. printer's ink)
velocity gradientchange in velocity
visco-elasticfluids in which the shear stress is a function not only of the velocity gradient but also of ordinary strain as well.
viscositya measure of a fluid's resistance to shear when the fluid is in motion

Notation (due to text limitations):
d/dtpartial differential
rdensity (usually rho)
ppressure
uvelocity

conservation of mass: the net rate of mass flow out of the control surface, r*(du/dx+du/dy+du/dz), is equal to the time rate of decrease of mass inside the control volume, dr/dt, so:

    dr/dt = r*(du/dx + du/dy + du/dz)

for incompressible fluids (e.g. water), r doesn't change so:
    0  = (du/dx + du/dy + du/dz)

momentum equation: Ignoring the effect of viscosity results in what's called the Euler Equations. The force on the differential element will change the momentum (mass * velocity) of the stuff inside the element.


Image Processing

Also see:
Term Meaning
Arithmetic encoding A form of entropy encoding that encodes the entire message into a single number (or fraction). It's context based if the entropy of a symbol changes based upon its context (the preceeding symbol, for example). See Wikipedia's Arithemtic encoding.
Canny Edge Detector "The Canny operator works in a multi-stage process. First of all the image is smoothed by Gaussian convolution. Then a simple 2-D first derivative operator (somewhat like the Roberts Cross) is applied to the smoothed image to highlight regions of the image with high first spatial derivatives. Edges give rise to ridges in the gradient magnitude image. The algorithm then tracks along the top of these ridges and sets to zero all pixels that are not actually on the ridge top so as to give a thin line in the output, a process known as non-maximal suppression. The tracking process exhibits hysteresis controlled by two thresholds: T1 and T2 with T1 > T2. Tracking can only begin at a point on a ridge higher than T1. Tracking then continues in both directions out from that point until the height of the ridge falls below T2. This hysteresis helps to ensure that noisy edges are not broken up into multiple edge fragments." From Canned Edge Detector.
Gabor wavelets See Gabor Wavelets. I've come across them in articles talking about face recognition.
Hough transform from Amos Storkey:
The Hough Transform is a global method for finding straight lines hidden in larger amounts of other data. It is an important technique in image processing. To understand the Hough Transform, it is important to know what the Hough space is. Each point (d,T) in Hough space corresponds to a line at angle T and distance d from the origin in the original data space. The value of a function in Hough space gives the point density along a line in the data space. The Hough Transform utilises this by the following method. For each point in the original space consider all the lines which go through that point at a particular discrete set of angles, chosen a priori. For each angle T, calculate the distance to the line through the point at that angle and discretise that distance using an a priori chosen discretisation, giving value d.
Hu moments See the definition of moments (including central moments) below if you are not familiar with it.
Hu moments are based on scale invariant central moments.
If the central moments are cpq, the scale invariant central moments are:npq = cpq/c00((p+q)/2)+1
Some Hu moments are computed from these. Some Hu moments are scale and translation invariance, some are reflectance invariance, some provide reflectin discrimination, etc. Some of the Hu moments are: n20+n02; (n20-n02)2+4n112; (n30-3n12)2+(3n21-n03)2
See the Appendix in Real-time Motion Template Gradients using Intel CVLib by Jim Davis and Gary Bradski.
KL transform Karhunen-Loee (KL) transformation is also known as the principal component transformation, the eigenvector transformation or the Hotelling transformation. See Image Processing - the KL transform
LIC
Line Integrals
See Line Integrals.
LIC
Linear Integral Convolution
"Basically this plug-in will smear a image at each pixel along a line which direction and length is calculated from a vector field. This vector field is calculated as the direction or gradient at each pixel of the hue, saturation or brightness of another image." From Line Integral Convolution.
LIC
Linear Predictive Code - LPC
Estimating a value based on a linear function of previous values. Used in speech synthsis. See Text-to-speech technology
Moments For a descretized image, f(x,y), the two-dimensional moment of order p+q is:
mpq = ∑∑xpyqf(x,y).
m00 is the mass of an object.
(m10/m00, m01/m00) is the center of mass of an object.
Central moments are computed relative to the center of mass of an object.
Morphological operators Binary image operators; the operators can also be defined for greyscale images. The fundamental ones are dilation and erosion. Both use structuring element, also called a kernel, that is positioned over each pixel's neighborhood. In Dilation the pixel is set to one if the intersection of the image neighborhood and the kernel is non-empty. In Erosion, a pixel is set to '1' when the intersection with the kernel and the neighborhood is equal to the neighborhood. Closure is defined by Dilation followed by Erosion. Opening is defined by Erosion followed by Dilation. See Morphology by HIPR. Also see Wolfram's Binary Morphological Operators
Noise Fluctuations and additions to a signal from influences outside the system being studied. See Wikipedia's noise. Also see Wikipedia's the colors of noise
Optical flow Assuming intensities of points do not change (much) in short periods of time, optical flow attempts to for displacement vectors of points between successive images of a sequence. See
Roberts Cross Operator 2D spatial gradient measure on an image. It consists of a pair of 2x2 convolution masks. One mask is columms of: +1, 0; 0, -1. The other is the first rotated by 90 degrees: 0, -1; +1, 0. See Roberts Cross Edge Detector
Sobel Edge Detector 2D spatial gradient measure on an image. It consists of a pair of 3x3 convolution masks. One mask has columns of: -1,-2,-1; 0,0,0; and +1,+2,+1. The other mask has rows of: -1,-2,-1; 0,0,0; and +1,+2,+1. See Sogel Edge Detector
"Two 3x3 convolution masks are applied to each pixel, one color at a time - one with a horizontal trend and one with a vertical trend. The result of the each convolution is treated a vector representing the edge through the current pixel. If the magnitude of the sum of these two orthogonal vectors is greater than some user-specified threshold, the pixel is marked in black as an edge. Otherwise, the pixel is set to white." (From Sobel Edge Detection Filter)
Also see this demo.
Stationary signal Is a signal that repeats with the same periodicity.
Watershed algorithm Region-partitioning algorithm inspired by geographical construction of catch basins - where basins meet, watershed lines are constructed. See "The Watershed Transform: Definitions, Algorithms and Parallelization Strategies" by Roerdink and Meijster.
Zernike moments See Complex Zernike moments


Bibliography

  1. [CEM] VNR Concise Enclyclopedia of Mathematics, Van Nostrand Reinhold.
  2. [NR] Numerical Recipes
  3. [LAA] Linear Algebra and Its Applications, Academic Press.
  4. [WCG] Wavelets for Computer Graphics: Theory and Applications, Morgan Kaufmann
  5. [ICGM] Shewchuk, J.R., An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, CMU, School of Computer Science, August 4 1994.
  6. [PCA] Principle component analysis
  7. [NLOM] Non-Linear Optimization Models
  8. [DTTD] Tamal K. Dey, Kokichi Sugihara, and Chandrijit L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design, 9:457-470, 1992.
  9. [IC] Kenneth P. Bogart. Introductory Combinatorics. 2nd ed. San Diego: Harcourt Brace Jovanovich, Inc., 1990.
  10. [GCTD] Brian Von Herzen, Alan Barr, Harold Zatz, "Geometric Collisions for Time-Dependent Parametric Surfaces," SIGGRAPH 90, pp. 39-48.
  11. [LAGR] http://math.math.ucdavis.edu/~hom/calculus/Lagrangef/overview.html
  12. [MECH] http://alamos.math.arizona.edu/~rychlik/557-dir/mechanics/node3.html#action-integral
  13. [TSHORES] http://www.math.unl.edu/~tshores/Book/page_handler?page=114
  14. [DICT] http://www.dictionary.com/
  15. [PPRU] F. Sebastian Grassia, "Practical Parameterization of Rotations Using the Exponential Map", Journal of Graphical Tools, 3(3), pp. 29-48, 1998.
  16. [DYNP] http://benli.bcc.bilkent.edu.tr/~omer/research/dynprog.html
  17. [CAIS] Vassar's Concepts and Applications of Inferential Statistics
General references
  1. CERN's Rudy Bock's The Data Analysis BriefBook
  2. Computer Vision Online's Vision Geometry and Mathematics
  3. Liuz Velho and Paulo Carvalho's Mathematical Optimization in Graphics and Vision
  4. Wikipedia's Mathematics and Descrete Mathematics
  5. Matrix Properties.
  6. First step in numerical analysis
  7. CRC Concise Encyclopedia of Mathematics
  8. Key Concepts from Materials and Sturctures
  9. Computer Vision Online's Vision Geometry and Mathematics
  10. Key Concepts from Materials and Sturctures
  11. Wolfram Research Mathworld
  12. A variational formulation of the fast marching eikonal solver
  13. Not Positive Definite - Causes and Cures
  14. cut-the-knot.org
  15. thinkquest.org
  16. Geocities Mathematics Fair
  17. PlanetMath.org
  18. math.about.com


rick parent: parent at domain cse.ohio-state.edu
with help getting started from Scott King and Lawson Wade