Concepts inImage deformation using moving least squares
Moving least squares
Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested. In computer graphics, the moving least squares method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a point cloud through either downsampling or upsampling.
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Affine transformation
In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") is a transformation which preserves straight lines (i.e. , all points lying on a line initially still lie on a line after transformation) and ratios of distances between points lying on a straight line (e.g. , the midpoint of a line segment remains the midpoint after transformation). It does not necessarily preserve angles or lengths.
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Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e. , sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation. The most important application is in data fitting.
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Linear function
In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; a map between two vector spaces that preserves vector addition and scalar multiplication.
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Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).
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Rigid transformation
In mathematics, a rigid transformation or a Euclidean transformation is a transformation from a Euclidean space to itself that preserves distances between every pair of points. Rigid transformations include rotations, translations, reflections, or their combination.
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Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions¿constants, one variable x, elementary operations of arithmetic (+ ¿ × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).
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