Projective Geometry and Computer Vision. Mathematics Research Paper

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Projective Geometry and Computer Vision
Projective geometry is now been accepted as one of the most reliable and efficient forms to display or formulate geometric concepts. The projective plan and associated concepts are further improved by the implication of advanced computer vision technologies. The computer vision is an automatic representation of the analysis and interpretation of the images. The projective geometry represents the images as the sum of “n” vectors in “n” dimensional space and predicts the interpretation of the image in “n+1” plans (Bădescu 42). The common assumption is the fact that two vectors ‘a’ and ‘b’ are considered equal if they representing the same line in the 2D plane. The ease of considering the space in form points is useful. The computer vision is therefore, the most appropriate technique to interoperate the data or information represented by an image. Today, the advance computer visions in coordination with projective geometry is used in many medical, industrial, and robotic applications. Like many other computer based applications the field of computer vision has a close relation to mathematics. The computer vision can be applied to any mathematical modelling process and drive the useful information from the images (Paragios and Zabih). In this research essay, we are going to apply the computer vision and projective analysis to develop a generic C++ code to analyze “vector or cross products” and “scalar products” (Bădescu 42).
The vector product generally refers to the vector obtained as result of multiplication between two or more vectors, while the scalar product is known as the “scalar or numeric” value obtained as the product of two vectors. The mathematical representation of both concepts are represented below:
Dot product: a * b = xaxb + yayb = |a||b|cos α.To explain its meaning, lets draw dashed line l perpendicular to the vector a.Cosine properties apply that:- if a * b > 0, then b points to the same side of l as a (as in the picture);- if a * b = 0, then b is collinear with l, so it's perpendicular to a;- if a * b < 0, then b points to the opposite side of l from a.To sum it up, dot product sign indicates whether two vectors point in about the same direction.Now let's look at the absolute value. |b||cos α| = |OP| — the projection of b onto a.So, the dot product's absolute value is the projection length of b onto a times the length of a.Nothing special, just remember that we can find projection length and angle cosine from it.
2D cross product: a ^ b = xayb - yaxb = |a||b|sin α.This name is not canon, but I didn't find a better one.Again, for clarity we draw a line, but this time dashed line d through the...