A. Goshtasby. Piecewise cubic mapping functions for image registration. Pattern Recognition, 20(5):525--533, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a global analytic mapping function to the image pixel positions, or by using a set of control points which specify the displacements of some points in the initial image. From these control points, a triangulation can be obtained and new locations computed by transforming each of the triangles [7, 8]. For the second step, a number of algorithms have been developed in recent years. A simple minded approach would be to take the coordinates of four neighboring pixels, compute their deformed coordinates with the mapping function from above, and perform a bilinear fill for all pixels within the ....

....appearance of the result is quite acceptable if the deformations are small and if enough data points are provided so that changes of the transform coefficients between neighboring triangles are small. A smooth deformation can be obtained by using non linear patches within the triangles. 5 In [8], a method is described which uses cubic functions obtained from a Clough Tocher subtriangulation. A problem common to all triangulation based methods for image warping is that foldover can easily occur. The term foldover describes the occurence of overlapping deformations, i.e. several ....

A. Goshtasby. Piecewise cubic mapping functions for image registration. Pattern Recognition, 20(5):525--533, 1987.

....points inside each triangular region are closer to one of its vertices than to the vertices of any other triangle. The mapping transformation is then computed for each point in the image from interpolation of the vertices in the triangular patch to which it belongs. Later, he extended this method [Goshtasby 87] so that mapping would be continuous and smooth (C 1 ) by 31 using piecewise cubic polynomial interpolation. To match the number of constraints to the number of parameters in the cubic polynomials, Goshtasby decomposed each triangle into Clough Tocher subtriangles and assumed certain partial ....

A. Goshtasby, "Piecewise Cubic Mapping Functions for Image Registration," Pattern Recognition 20, No. 5, 1987, pp525-533.

....model. In this stage, the type and parameters of spatial transform are to be determined. In case of rigid deformations, a linear transform model, the coecients of which are calculated via leastsquare technique, is commonly used [15] When non rigid deformations are present, local mapping [11] [12], 13] radial basis functions and thin plate splines [21] 27] 3] are the most popular solutions. Resampling and transformation. Finally, the sensed image is transformed over the reference one according to the above mapping model. An appropriate resampling technique (nearest neighbor, linear ....

A. Goshtasby. Piecewise cubic mapping functions for image registration. Pattern Recognition, 20(5):525-533, 1987.

....projection images to three dimensional volume images. Nonlinear transformations, which map straight lines to curves, are also sometimes useful. The best known nonlinear transformations used in image registration are polynomial functions. Linear [53, 73, 137, 174] quadratic [197] cubic [74], and higher order polynomials [191] have been used to match images. Other nonlinear transformations that have been explored are exponential warping [199] and thin plate splines [18] Nonlinear transformations are useful for deforming an anatomical atlas to fit image data. They are also useful ....

A. Goshtasby. Piecewise cubic mapping functions for image registration. Pattern Recognition, 20:525--533, 1987.

....component is resampled with a scanline algorithm, then the other. A detailed introduction to resampling algorithms can be found in [21] The problem of geometric deformation can be treated as a scattered data interpolation problem by treating each of the components of the coordinates separately [6, 15]. Below, we give a short summary of some deformation methods which offer ease of use, stability, and smoothness. For a more detailed description, see [14] Deformations based on scattered data interpolation methods compare favorably to algorithms based on control point meshes [18] and on vector ....

....nor do they display the ghostbusting phenomena described in [1] 3. 1 Triangle Based Methods A well known approach to image deformation is to determine a triangulation of the source positions of the correspondence points and performing linear or cubic interpolation within each of the triangles [5, 6]. In the scattered data interpolation literature, these methods are commonly referred to as FEM interpolation. Typically, a Delaunay triangulation is used to avoid thin triangles. Within each triangle, the simplest mapping is a linear interpolation using barycentric coordinates [5] This gives a ....

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A. Goshtasby. Piecewise cubic mapping functions for image registration. Pattern Recognition, 20(5):525--533, 1987.

....by Lawson [30] to derive optimal triangulations in which long thin triangles with small angles are avoided. Piecewise linear approximation over the triangulation is not smooth, achieving only C 0 continuity. The most common C 1 method uses the Clough Tocher triangular interpolant [7] 2] [26]. A related technique was proposed in [39] Triangulation methods, however, are sensitive to data distribution, i.e. long thin triangles cannot always be avoided. Schumaker [44] proposed a two stage method that first generates a grid of data using any method for scattered data interpolation. The ....

A. Goshtasby, "Piecewise Cubic Mapping Functions for Image Registration," Pattern Recognition, vol. 20, no. 5, pp. 525-533, 1987.

....to use a piecewise cubic transformation function. Typically, the cubic functions are formulated such that functions that map adjacent triangles in the images to each other have the same derivative across the triangular edges. An interpolating smooth piecewise cubic function is given in [3] [6]. In piecewise cubics, when a control point is moved in the target image, the move will affect the tangents across all triangles having that point as a vertex. Therefore, not only triangles having that point as the vertex need to be resampled, the triangles adjacent to them need to be resampled ....

A. Goshtasby, Piecewise cubic mapping functions for image registration, Pattern Recognition, vol. 20, no. 5, pp. 525-533, 1987.

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