Image analysis with two-dimensional continuous wavelet transform

Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform (CWT) based on the 2D Euclidean group with dilations. In this case, the wavelet transform of a 2D signal (an image) is a function of 4 parameters: two translation parameters b(x), b(y), a rotation angle theta and the usual dilation parameter a. For obvious practical reasons, two of the parameters must be fixed, either (a, theta) or (b(x), b(y)), and the WT visualized as a function of the two other ones. We discuss the general properties of the CWT and apply it, both analytically and graphically, to a number of simple geometrical objects: a line, a square, an angle, etc. For large a, the analysis detects the global shape of the objects, and smaller values of a reveal finer and finer details, in particular edges and contours. If the analyzing wavelet is oriented, like the 2D Morlet wavelet, the transform is extremely sensitive to directions: varying the angle theta uncovers the directional features of the objects, if any. The selectivity of a given wavelet is estimated from its reproducing kernel.

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