Abstract
This paper extends our work on automated discovery of geometric theorems from diagrams by taking scanned and photographed images instead of images produced with dynamic geometry software. We first adopt techniques of Hough transform and randomized detection algorithms to detect geometric objects from scanned and photographed images, then use methods of image matching to recognize labels for the detected geometric objects, and finally employ numerical-computation-based methods to mine geometric relations among the objects. Experiments with a preliminary implementation of the techniques and methods demonstrate the effectiveness and efficiency of geometric information retrieval from scanned and photographed images for the purpose of discovering geometric theorems automatically.
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Notes
- 1.
In [1] it is explained how to translate mathematical problems stated in natural languages to propositions formulated in Zermelo–Fraenkel axiomatic set theory. Combination of the method of natural language processing discussed therein with our approach of theorem discovering from images of diagrams may help increase the degree of automation for mathematical problem solving in education.
- 2.
For example, using Zhang’s technique of parallel thinning [11].
- 3.
\(|\Vert OP_4\Vert -r|\le \tau _{pc}\), where \(\tau _{pc}\) is a prespecified tolerance.
- 4.
The radius of C is measured by the number r of pixel points.
- 5.
On the one hand, this condition allows a certain degree of numerical errors; on the other hand, it is applicable for both large and small radii of circles.
- 6.
Due to numerical errors of line detection, some arcs may be recognized as line segments, in particular when the radius of the circle is large.
- 7.
\(d_{\mathrm {T}, \mathrm {B}}\) is computed by the SIFT algorithm [8], where \(\mathrm {T} \in \mathbb {T}\) and \(\mathrm {B}\) is an image block obtained in step 3(b).
- 8.
Currently, carefully photographed images of diagrams can be successfully processed. The problem of processing images of diagrams carelessly photographed is still under investigation.
- 9.
The objects recognized and the theorems discovered automatically from images of diagrams are presented on the website http://geo.cc4cm.org/data/recognizer/.
- 10.
There is a theorem about the equality of the areas of the parallelograms ABCD and EBCH implied by the image. The theorem could not be mined because area relations are not among our chosen basic geometric relations.
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Acknowledgements
The authors wish to thank the referees for their constructive comments which have helped improve the paper significantly. This work has been supported by the project SKLSDE-2015ZX-18.
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Song, D., Wang, D., Chen, X. (2015). Discovering Geometric Theorems from Scanned and Photographed Images of Diagrams. In: Botana, F., Quaresma, P. (eds) Automated Deduction in Geometry. ADG 2014. Lecture Notes in Computer Science(), vol 9201. Springer, Cham. https://doi.org/10.1007/978-3-319-21362-0_10
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