Abstract Blur is an image degradation that makes object recognition challenging. Restoration approaches solve this problem via image deblurring, deep learning methods rely on the augmentation of training sets. Invariants with respect to blur offer an alternative way of describing and recognising blurred images without any deblurring and data augmentation. In this paper, we present an original theory of blur invariants. Unlike all previous attempts, the new theory requires no prior knowledge of the blur type. The invariants are constructed in the Fourier domain by means of orthogonal projection operators and moment expansion is used for efficient and stable computation. Applying a general substitution rule, combined invariants to blur and spatial transformations are easy to construct and use. Experimental comparison to Convolutional Neural Networks shows the advantages of the proposed theory.

Vector fields are a special kind of multidimensional data, which are in a certain sense similar to digital color images, but are distinct from them in several aspects. In each pixel, the field is assigned to a vector that shows the direction and the magnitude of the quantity, which has been measured. To detect the patterns of interest in the field, special matching methods must be developed. In this paper, we propose a method for the description and matching of vector field patterns under an unknown affine transformation of the field. Unlike digital images, transformations of vector fields act not only on the spatial coordinates but also on the field values, which makes the detection different from the image case. To measure the similarity between the template and the field patch, we propose original invariants with respect to total affine transformation. They are designed from the vector field moments. It is demonstrated by experiments on real data from fluid mechanics that they perform significantly better than potential competitors.

The paper presents a new theory of combined moment invariants to Gaussian blur and spatial affine transformation. The blur kernel may be arbitrary oriented, scaled and elongated. No prior information about the kernel parameters and about the underlaying affine transform is required. The main idea, expressed by the Substitution Theorem, is to substitute pure blur invariants into traditional affine moment invariants. Potential applications of the new descriptors are in blur-invariant image recognition and in robust template matching.

Vector field (VF) is a special kind of multidimensional data. In each pixel, VF contains information about the direction and the magnitude of the measured quantity. To detect the patterns of interest in the field, special matching methods must be developed. We propose a method for the description and matching of VF patterns under an unknown affine transformation of the field. Unlike digital images, transformations of VFs act not only on the spatial coordinates but also on the field values, which makes the detection different. To measure the similarity between the template and the field patch, we propose original invariants with respect to affine transformation designed from moments. Their performance is demonstrated by experiments on real data from fluid mechanics.

We propose a method for the recognition of vector field patterns under an unknown rotation. The rotation is modeled as a total transformation, which is applied on both spatial coordinates and field values. The invariants are constructed from orthogonal Gaussian-Hermite moments. Their numerical stability and recognition power are shown to be better than those of the invariants published so far.