Detecting Directionality in Random Fields Using the Monogenic Signal

Abstract

Detecting and analyzing directional structures in images is important in many applications since one-dimensional patterns often correspond to important features such as object contours or trajectories. Classifying a structure as directional or non-directional requires a measure to quantify the degree of directionality and a threshold, which needs to be chosen based on the statistics of the image. In order to do this, we model the image as a random field. So far, little research has been performed on analyzing directionality in random fields. In this paper, we propose a measure to quantify the degree of directionality based on the random monogenic signal, which enables a unique decomposition of a 2D signal into local amplitude, local orientation, and local phase. We investigate the secondorder statistical properties of the monogenic signal for isotropic, anisotropic, and unidirectional random fields. We analyze our measure of directionality for finite-size sample images, and determine a threshold to distinguish between unidirectional and non-unidirectional random fields, which allows the automatic classification of images.

Anisotropic Gaussian wave models

“…A big demand for anisotropic models is nowadays observed, in particular by practitioners in geostatistics, offshore engineering, heterogeneous material or medical imaging (see for instance Richard and Biermé, 2010, Klatt, 2016, Allard et al, 2016, but also for more theoretical studies dedicated to image synthesis and analysis, optics, cosmology or arithmetic (Biermé et al, 2015, Olhede et al, 2014, De Angelis et al, 2016, Ade et al, 2016, Kurlberg and Wigman, 2017.…”

Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling

“…It is straightforward to prove the following isometry property by . By , we have the contractive property for each variable, namely, In monogenic signal analysis, the Riesz transform is the tool for defining local features such as the local amplitude, orientation, and instantaneous phase . Riesz transform is a singular integral, and when d =1, it reduces to be the Hilbert transform .…”

“…jj R k f jj L 2 ¼ jj f jj L 2 (3) by (2). By (3), we have the contractive property for each variable, namely, jjR k f jj L 2 ≤ jj f jj L 2 : In monogenic signal analysis, the Riesz transform is the tool for defining local features such as the local amplitude, orientation, and instantaneous phase.…”

“…By (3), we have the contractive property for each variable, namely, jjR k f jj L 2 ≤ jj f jj L 2 : In monogenic signal analysis, the Riesz transform is the tool for defining local features such as the local amplitude, orientation, and instantaneous phase. [1][2][3][4][5] Riesz transform is a singular integral, and when d=1, it reduces to be the Hilbert transform. 6 Various methods have been established to recover the Hilbert transform.…”

Color monogenic wavelet transform for multichannel image denoising

“…Then, in Held et al (2010), the new framework of monogenic wavelet transform which is based on the hyper-complex monogenic by using Riesz transforms is proposed. The MWT has been successfully applied to various areas such as medical image processing (Alessandrini et al 2013), pattern recognition (Dong et al 2014), edge detection (Olhede et al 2014) and texture classification . However, MWT fails to handle color images.…”