Multi-Photo Fusion through Projective Geometry

Multi-Photo Fusion through Projective Geometry

Luigi Barazzetti (Department of Building Environmental Science and Technology, Politecnico di Milano, Italy)
DOI: 10.4018/978-1-4666-4490-8.ch016
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Abstract

Multiple images of the same object can be registered and fused after identifying a proper mathematical model. This chapter illustrates the contribution of projective geometry and automated matching techniques to several photographic applications: high dynamic range, multi-focus, and panoramic photography. The method relies on the estimation of a set of planar transformations that make the data consistent. This provides pixel correspondence and allows the photographer to obtain new digital products in an almost fully automated way.
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Image Registration Through Projective Transformations

Projective geometry allows us to deal with geometric transformations and offers more powerful methods than conventional Euclidean geometry. For instance, in Euclidean geometry any two distinct points in a plane define a line. On the other hand, any two distinct lines define a point, unless the lines are parallel. Projective geometry overcomes this limitation in a symmetric way: any two distinct points determine a unique line and any two distinct lines intersect at a point. When we consider the image formation process by means of a pinhole camera, it is quite evident that Euclidean geometry is not sufficient. Properties like lengths, angles, areas, and parallelism are not preserved during the imaging process: parallel lines may intersect.

We get to projective geometry by taking Euclidean geometry and adding an extra dimension. A point in an n-dimensional Euclidean space is represented as a point in an (n+1)-dimensional projective space. Suppose we have a point (x, y)T in the Euclidean space, the homogenous coordinates of a point can be obtained by adding an extra coordinate to the pair as (λx, λy, λ)T . We say that this 3-vector is the same point in homogeneous coordinates (for any non-zero value λ). An arbitrary homogeneous vector x = (x1, x2, x3)T represents the point x = (x1 / x3, x2 / x3)T in R2.

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