Transfer and Invariants of Surfaces of Revolution

Diplomarbeit von Sven Utcke

Abstract:

A number of papers in the Computer Vision and Pattern Recognitionliterature have demonstrated that invariants, or equivalently structure modulo a 3D linear transformation, are sufficient for object recognition. The final stage in the recognition process is verification, where an outline is transferred from an acquisition image of the object to the target image.

For the most part recognition based on invariants has concentrated on planar objects, though some 3D invariants have been measured from single and multiple images for polyhedra, point sets, surfaces of revolution and algebraic surfaces. The work so far on surfaces of revolution has only exploited isolated points on the outline (such as bitangents), and has not addressed transfer or verification.

This thesis, for the first time, extends the transfer and extraction of invariants to surfaces of revolution using the entire outline. Given a single view of the surface, it is possible to obtain the projection in any other given view, given a minimal number of points in the target image. In particular it is is possible to reconstruct the generating curve, and thereby a rich set of invariants.

Zusammenfassung:

Eine Reihe von Veröffentlichungen in der Bildverarbeitungs- und Mustererkennungsliteratur hat gezeigt, dass Invarianten, oder alternativ Struktur Modulo einer 3D linearen Transformation, die Erkennung von Objekten aus Bildern ermöglicht. Der letzte Schritt im Erkennungsprozess ist die Verifikation, in der eine Kontur einer exemplarischen Ansicht des Objektes in das Bild zurückprojeziert wird.

Die Invarianten-basierte Erkennung hat sich im wesentlichen auf ebene Objekte beschränkt, wenngleich einzelne 3D Invarianten für Polyeder, Punkt-Mengen, rotationssymmetrische Objekte und algebraische Oberflächen existieren. Die bisherigen Arbeiten für rotationssymmetrische Objekte haben sich auf einzelne Punkte der Kontur (z.B. Bi-Tangenten) beschränkt und das Problem der Rückprojektion und Verifikation ignoriert.

In dieser Arbeit wird zum ersten Mal die ganze Kontur eines rotationssymmetrischen Objektes in die Rückprojektion und Verifikation mit einbezogen. Basierend auf einer einzigen Ansicht ist es Möglich, Ansichten aus beliebigen anderen Blickwinkeln zu erhalten. Insbesondere ist es möglich, den Querschnitt des Objektes zu rekonstruieren, und damit eine reichhaltige Auswahl an Invarianten.

Table of Contents:

	Introduction	1
	Specification	1
	Relevant Literature	2
1.	Introduction	7
1.1	The Object Class of Interest	8
1.2	The Task	8
1.3	The Chosen Imaging Geometry	11
1.4	Contributions of this Thesis	12
1.5	Outline of this Thesis	13
2.	Distinguished features	14
2.1	Tangents	16
2.1.1	The Tangent Cone	16
2.1.2	The Outline	16
2.2	The Affine Basis	19
3.	The Weak Perspective Camera	21
3.1	The underlying Geometry	22
3.1.1	The Surface of Revolution	22
3.1.2	The Weak Perspective Camera	23
3.1.3	Recovering the Generating Function	26
3.1.4	How to calculate the viewing direction	28
3.1.5	Transfer using two arbitrary views	28
3.2	Method 1. Using the Generating Curve	30
3.2.1	Summary	30
3.2.2	The Implementation	31
3.2.2.1	Computing the Axis of Symmetry	31
3.2.2.2	The Direction of View	33
3.2.2.3	Transfer	34
3.2.3	Results	35
3.3	Method 2. Using the Outline's Envelope	37
3.3.1	The underlying Geometry	37
3.3.1.1	How to Represent the Intersection	37
3.3.1.2	The Angle	38
3.3.1.3	The Envelope	38
3.3.1.4	Summary	38
3.3.2	The Implementation	39
3.3.2.1	The Direction of View	39
3.3.2.2	Acquisition and Transfer	40
3.3.3	Results	41
3.3.3.1	The Influence of Perspective Distortions	41
3.3.3.2	The Influence of Errors in the Angles	43
3.4	Comparing the two Methods	45
3.5	Affine Extensions	46
3.5.1	Unknown Aspect Ratio	46
3.5.2	Full Affine Distortions	47
4.	The Affine Camera	49
4.1	Theoretical Background	50
4.1.1	The Affine Camera	50
4.1.2	The Surface's 3D Geometry and its Image	51
4.1.3	Acquisition --- Calculating the Conics	52
4.1.4	Transfer	53
4.1.4.1	The Tangent Cone	53
4.1.5	Summary	56
4.1.5.1	Prerequisites	56
4.1.5.2	Acquisition	56
4.1.5.3	Transfer	57
4.2	Implementation	58
4.2.1	The Common Frame	58
4.2.1.1	The Axis	58
4.2.1.2	Calculating the Offset and Scale in the y-Direction	58
4.2.2	The Acquisition	59
4.2.2.1	The tangent to the outline	59
4.2.2.2	The conic tangent to the outline point	60
4.2.3	Transfer	62
4.2.3.1	Transfer using the first representation	63
4.2.3.2	Transfer using the second representation	63
4.3	Results	63
4.4	Possible Enhancements and Open Questions	65
4.4.1	Better Features than Intersections	65
4.4.2	Unused Constraints	66
5.	The Projective Camera	68
5.1	The underlying geometry	69
5.1.1	The projective Camera	69
5.1.2	The Surface's 3D Geometry	70
5.1.2.1	The Cross-ratio	74
5.1.3	Summary	75
5.2	A possible Implementation	75
5.2.1	Acquisition	76
5.2.2	Transfer	77
5.2.3	Transfer into the Canonical Frame	78
5.2.3.1	How to pick a canonical frame	78
5.3	Results	78
6.	Conclusions	81
6.1	A Recognition System	82
6.1.1	Transfer between two Views	82
6.1.2	Transfer into a Canonical Frame	82
6.1.3	How to build a Recognition System	83
6.2	Future Work	83