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The Photogrammetric Reconstruction Task Solved by Bundle Adjustment Directly with Pixel Bundles

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Abstract

At present the reconstruction task of photogrammetry is performed mostly in several steps: one feature-based bundle solution followed by many dense image matchings and fusion of intermediate results. To increase the performance of reconstruction to a maximum a different definition is proposed, starting from the approach of strict digital image inversion and the correspondence condition of image grey values with the unknown function of object surface intensity (or brightness). This leads again to a bundle solution (similar to the classical solution), however, directly with the image grey values as measurements so that each pixel represents a ray and each image a high-density bundle of rays (pixel bundle). All images of a block are forced to generate optimally one function of object surface intensity together with the object surface and the orientation of the images (WROBEL, Proceedings SPIE 804, pp 325–334. In German language: Bildmessung und Luftbildwesen 1987). The diverse tasks of modern life need different methods of 3D reconstructions in diverging performance. Here, we concentrate on a high-fidelity approach of both the functional and stochastic model of image evaluation. The fundamental derivations are presented, main properties (e.g. its statistical potential) are discussed and highlighted in comparison to deficiencies of existing approaches. Topographical reconstructions demonstrate the validity of the results with respect to the surface and the standard deviations of its heights.

Zusammenfassung

Die Rekonstruktionsaufgabe der Photogrammetrie gelöst durch Bündelausgleichung direkt mit Pixelbündeln . Die Rekonstruktionsaufgabe der Photogrammetrie wird gegenwärtig meist schrittweise gelöst mit einer merkmalsbasierten Bündellösung gefolgt von vielen dichten Bildzuordnungen plus Fusion zum Endresultat. Um die Leistungen der Rekonstruktion zum Maximum zu erhöhen, wird für sie eine andere Definition gegeben. Sie startet vom strengen Ansatz der digitalen Bildinversion und der Korrespondenzbedingung von Bildgrauwerten mit der unbekannten Funktion der Objekthelligkeit. Dies führt wieder zu einer Bündellösung (ähnlich der klassischen), jedoch direkt mit den Bildgrauwerten als Messwerten, so dass jedes Pixel einen Abbildungsstrahl repräsentiert und jedes Bild ein Strahlenbündel sehr hoher Dichte (Pixelbündel). Alle Bilder eines Blocks werden der Bedingung unterworfen, eine einzige Funktion der Objekthelligkeit optimal zu erzeugen gemeinsam mit der Objektoberfläche und der Orientierung der Bilder (WROBEL, Proceedings SPIE 804, pp 325–334. In German language: Bildmessung und Luftbildwesen 1987). Die Aufgaben des modernen Lebens benötigen Methoden für 3D Rekonstruktionen in sehr unterschiedlicher Leistung. Wir konzentrieren uns hier auf einen Ansatz in hoher Treue für das stochastische wie auch für das funktionale Modell der Bildauswertung. In diesem Beitrag werden die fundamentalen Ableitungen des Ansatzes aufgezeigt, die Haupteigenschaften (z.B. ihr hohes statistisches Potential) diskutiert und hervorgehoben im Vergleich zu Unzulänglichkeiten von existierenden Ansätzen. Topographische Rekonstruktionen demonstrieren die Gültigkeit der Ergebnisse gleichermaßen für die Oberfläche und die Standardabweichungen ihrer Höhen.

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Notes

  1. The right-hand side of the mapping equation, a mix of homogeneous \((\mathbf K )\) and inhomogeneous terms \(({{\varvec{R}}}, {{\varvec{X}}}, {{\varvec{X}}}_o)\), can be numerically computed by taking the results of the bundle solution producing the homogeneous \(3\times 1\) vector \((\mathbf{v }_\mathbf x {+}{} \mathbf x ) {=} \mathbf x ^*\). This vector \((x^*w^*, y^*w^*, w^*)^T\) is simply decomposed into the elements of \(\mathbf x \) and \(\mathbf v _\mathbf x \): \((x^*w^*/w^*, y^*w^*/w^*)^T {=} (x^*, y^*)^T\), from which with the given measurement data x and y we finally obtain \((x^*{-}x, y^*-y)^T {=} (v_x, v_y)^T\).

  2. Lets have a look at the remarkable structure of the equations of an image (see (3) to (6)): each equation contains the six (in general non-zero) coefficients of the orientation parameters of that image (altogether a huge number of equations!). In the example of Table 1 we have only 64 equations with non-zero coefficients for an arbitrary S-facet. So, we can expect that the orientation parameters can be better estimated than the other parameters.

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Acknowledgements

I am grateful to Martin Schlüter and Johannes Schneider for preparation of the figures and Karl-Rudolf Koch for discussions about the statistical approach. Thanks also to Jochen Meidow and to the team of editors of this journal for hints to improve the clarity of this paper.

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  1. Institute of Geodesy, Remote Sensing and Image Analysis, Darmstadt University of Technology, Franziska-Braun-Strasse 7, D-64287, Darmstadt, Germany

    Bernhard P. Wrobel

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Wrobel, B.P. The Photogrammetric Reconstruction Task Solved by Bundle Adjustment Directly with Pixel Bundles. PFG 85, 377–388 (2017). https://doi.org/10.1007/s41064-017-0037-9

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