Homographic p-norms: Metrics of homographic image transformation

• We propose homographic p-norms, metrics of homographic image transformation.
• We show that homographic p-norms satisfy metric properties.
• For use of the metrics, we provide several homographic 2-norm expressions.
• We compare their approximation errors, and validate the efficacy of the metrics.
• The metrics can be applied to various image processing and analysis tasks.

Images often need to be aligned in a single coordinate system, and homography is one of the most efficient geometric models to align images. This paper presents homographic p-norms, scalar metrics of homographic image transformation, and to the best of our knowledge these are the most rigorous definition of scalar metrics quantifying homographic image transformations. We first define a metric between two homography matrices, and show it satisfies metric properties. Then we propose metrics of a single homography matrix for a general planar region, and ones for a usual rectangle image. For use of the proposed metrics, we provide useful homographic 2-norm expressions derived from the definition of the metrics, and compare the approximation errors of the metrics with respect to the exact metric. As a result, the discrete version of the metric obtained by pixel-wise computation is greatly close to the exact metric. The proposed metrics can be applied to evaluation of transformation magnitude, image closeness estimation, evaluation of camera pose difference, selection of image pair in stereo vision, panoramic image mosaic, and deblurring. Experimental results show the efficacy of the proposed metrics.

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