The Geometry of Multiple Images: The Laws that Govern the Formation of Multiple Images of a Scene and Some of Their Applications

Citation

Faugeras, O. and Luong, Q.T. (2002), "The Geometry of Multiple Images: The Laws that Govern the Formation of Multiple Images of a Scene and Some of Their Applications", Industrial Robot, Vol. 29 No. 3, pp. 287-288. https://doi.org/10.1108/ir.2002.29.3.287.2

The Geometry of Multiple Images describes the geometric laws that relate different views of a scene. It is a book about mathematics and vision, containing definitions, lemmas, propositions and theorems. Unlike other texts, this book considers affine and Euclidean geometries as special cases of projective geometry.

The book comprises 11 chapters and starts with “A tour into multiple image geometry”. Chapter 2 discusses Projective, affine and Euclidean geometries, while Exterior and double or Grassman‐Cayley algebras are addressed in Chapter 3. Topics covered in this chapter include: Plüker relations; the meet operator; and duality and the Hodge operator.

The case of perspective projection; the affine and Euclidean models: the case of parallel projection; and departures from the pinhole model: nonlinear distortion, are addressed in Chapter 4, “One camera”.

Chapters 5 and 6 discuss Two views: the fundamental matrix, and Estimating the fundamental matrix respectively. Subjects addressed include: configurations with no parallax; ambiguity and the critical surface; and computing the uncertainty of the fundamental matrix.

Chapter 7 addresses Stratification of the binocular stereo and applications, while Chapters 8 and 9 discuss Three views: the trifocal geometry, and Determining the trifocal tensor. These two chapters present topics including: the geometry of three views from the viewpoint of two; constraints that characterise the trifocal tensors; and a note about the “change of view” operation.

Chapter 10, Stratification of n ≥ 3 views and applications, discusses canonical representation of n views; projective stratum; affine and Euclidean strata; and stereo rings. Chapter 11, the final chapter of the book, addresses Self‐calibration of a moving camera: from affine or projective calibration to full Euclidean calibration.

This is a well written book that is a superb reference text for those researching or working with machine vision and computer graphics. Its coverage plunges into the mathematical depths of vision, making it suitable for those designing machine vision algorithms but potentially daunting to the casual reader.