Shape from texture based on the ridge of continuous wavelet transform

We propose a new shape from texture method based on the ridge of continuous wavelet transform. This method determines the orientations of a planar surface in a direct way under the perspective projection model. The variations of the image projected from a planar surface can be accurately characterized by the ridge of the continuous wavelet transform. The ridge of the 1-D signal and 2-D image are represented as a ridge curve and ridge plane, respectively. Ridges represent the energy concentration in the time-frequency plane where the energy is a local maxima. We show that the ridge of the projected image is a parabolic plane with a rotation angle equal to the tilt angle of the planar surface. The ridge is then rotated with the angle such that the slant effect appears in the X-axis and plays no role along the Y-axis. As a result, the rotated ridge plane can be regarded as the plane composed of many 1-D ridge curves. The slant angle of the 2-D image is thus obtained from the derived slant angle of the 1-D signal. A voting method and a curve fitting method are developed to obtain the slant angle of the 1-D signal. Several synthetic and real-world images have demonstrated the robustness and accuracy of our method.