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This research is conducted in the summer of 2015 and is possible by the support of various agency, in particular, by the grant of Prof. Angulo Nieves and the New York City Research Initiative.The purpose of this research is to reveal the mathematics and applications of the computer animation techniques of warping and morphing. A warp is a twist or distortion in the form of an object in an image while a morph is the smooth and gradual transformation of an object in one image into the object in another image. Linear algebra makes these computer animation techniques possible; the first phase of this research delves into how those mathematical processes translate into image warps and morphs. Image morphs and morphs were identified as affine transformations of original images. The second part of this study requires the analysis and application of image warping and morphing techniques in an array of fields. The team utilized the computer software, Abrosoft Fantamorph and Morpheus in order to create a series of warps and morphs. This shows an example of the use of technology in undergraduate research. The final phase of this research was to identify what uses NASA can have for these computer animation techniques and what further research can be done to expand our knowledge of warps and morphs. By identifying the mechanics of warps and morph, we can discover how they can assist scientists and organization, such as NASA, to create depictions of objects, ideas, places, and events. Ultimately, studying morphing and warping techniques allows us to find better ways to represent visual data - whether it is images of the ozone hole or maps of the ever-changing weather in a region. The limitations that were found during the study can be used to conduct further research about warps and morphs - such as distorting images using quadratics or varying the rate at which each part of a transformation happens.

The Mathematics and Applications behind Image Warping and Morphing