Abstract
This article presents a theory for constructing hierarchical networks in such a way that the networks are guaranteed to be provably scale covariant. We first present a general sufficiency argument for obtaining scale covariance, which holds for a wide class of networks defined from linear and nonlinear differential expressions expressed in terms of scale-normalized scale-space derivatives. Then, we present a more detailed development of one example of such a network constructed from a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives, we couple such primitive computations in cascade over combinatorial expansions over image orientations. Scale-space properties of the computational primitives are analysed, and we give explicit proofs of how the resulting representation allows for scale and rotation covariance. A prototype application to texture analysis is developed, and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets.
Highlights
We have presented a theory for defining handcrafted or structured hierarchical networks by combining linear and nonlinear scale-space operations in cascade
After presenting a general sufficiency condition to construct networks based on continuous scale-space operations that guarantee provable scale covariance, we have in more detail developed one specific example of such a network constructed by applying quasi quadrature responses of first- and second-order directional Gaussian derivatives in cascade
The present work is intended as initial work in this direction, where we propose the family of quasi quadrature networks as a new baseline for handcrafted networks with associated provable covariance properties under scaling and rotation transformations
Summary
The recent progress with deep learning architectures [1,2,3,4,5,6,7,8,9,10] has demonstrated that hierarchical feature representations over multiple layers have higher potential compared to approaches based on single layers of receptive fields. Compared to earlier approaches of related types [38,39,40,41,42], our quasi quadrature model has the conceptual advantage that it is expressed in terms of scale-space theory in addition to well reproducing properties of complex cells as reported by [34,43,44,45] Thereby, this functional model of complex cells allows for a conceptually easy integration with transformation properties, truly provable scale covariance, or a generalization to affine covariance provided that the receptive field responses are computed in terms of affine Gaussian derivatives as opposed to regular Gaussian derivatives.
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