In this paper we discuss the feasibility of using singular points in a scale space representation (referred to as top points) for image matching purposes. These points are easily extracted from the scale space of an image and they form a compact description of the image. The image matching problem thus becomes a point cloud matching problem. This is related to the transportation problem known from linear optimization and we solve it by using an earth movers distance algorithm. To match points in scale space a distance measure is needed as Euclidean distance no longer applies. In this article we suggest a metric that can be used in scale space and show that it indeed performs better than a Euclidean distance measure. To distinguish between stable and unstable top points we derive a stability norm based on the total variation norm which only depends on the second order derivatives at the top point. To further improve matching results we show that other features at the top points can also increase the accuracy of matching.

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Abstract. We consider the use of so-called top-points for object retrieval. These points are based on scale-space and catastrophe theory, and are invariant under gray value scaling and offset as well as scale-Euclidean transformations. The differential properties and noise characteristics of these points are mathematically well understood. It is possible to retrieve the exact location of a top-point from any coarse estimation through a closed-form vector equation which only depends on local derivatives in the estimated point. All these properties make top-points highly suitable as anchor points for invariant matching schemes. In a set of examples we show the excellent performance of top-points in an object retrieval task.

Abstract. Local feature matching is an essential component of many image retrieval algorithms. Euclidean and Mahalanobis distances are mostly used in order to compare two feature vectors. The first distance does not give satisfactory results in many cases and is inappropriate in the typical case where the components of the feature vector are incommensurable, whereas the second one requires training data. In this paper a stability based similarity measure (SBSM) is introduced for feature vectors that are composed of arbitrary algebraic combinations of image derivatives. Feature matching based on SBSM is shown to outperform algorithms based on Euclidean and Mahalanobis distances, and does not require any training.