Monogenic signal

The monogenic signal is a generalization of the analytical signal to more than one dimension based on the Riesz transformation. The monogenic signal is used in image processing . With it, images can be broken down into local amplitude and local phase .

Definitions

Monogenic signal

Let d be a natural number and a function. Then the monogenic signal is defined by ${\displaystyle f\in L^{2}(\mathbb {R} ^{d})}$ ${\displaystyle f_{M}}$

${\displaystyle f_{M}:=(f,R_{1}f,\ldots ,R_{d}f),}$

where denotes the j-th component of the Riesz transform. ${\displaystyle R_{j},j=1,\ldots ,d}$

Riesz transformation

Let d be a natural number. The jth component, the Riesz transform, is defined by ${\displaystyle j=1,\ldots ,d,}$ ${\displaystyle R_{j}}$

${\displaystyle R_{j}f(x):=c_{d}\lim _{\epsilon \to 0}\int _{0<\epsilon \leq \left|y\right|}{\frac {y_{j}}{\left|y\right|^{d+1}}}f(x-y)\,\mathrm {d} y}$

With

${\displaystyle c_{d}={\frac {\Gamma \left({\frac {d+1}{2}}\right)}{\pi ^{(d+1)/2}}},}$

where denotes the gamma function . ${\displaystyle \Gamma }$

The Riesz transformation is defined as a d-dimensional vector of the j components ${\displaystyle R_{j}}$

${\displaystyle Rf(x):=(R_{1}f(x),\ldots ,R_{d}f(x)).}$

Relationship with the analytic signal and the Hilbert transformation

For the Riesz transformation is the Hilbert transformation and the monogenic signal corresponds in this case to the analytic signal if the vector of the monogenic signal is understood as a complex number, i.e. H. ${\displaystyle d=1}$ ${\displaystyle {\mathcal {H}}}$

${\displaystyle \left(f(x),R_{1}f(x)\right)=f(x)+iR_{1}f(x)=f(x)+i{\mathcal {H}}f(x)}$

Decomposition in phase and amplitude

The monogenic signal allows a multi-dimensional signal to be broken down into local amplitude and local phase . The local amplitude in this case is defined by ${\displaystyle A}$

${\displaystyle A(x)={\sqrt {f^{2}(x)+(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},}$

the local phase angle through ${\displaystyle \alpha }$

${\displaystyle \alpha (x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|,}$

the local phase direction by ${\displaystyle u}$

${\displaystyle u(x)={\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}}$

and the local phase by ${\displaystyle \phi }$

${\displaystyle \phi (x)=\alpha (x)u(x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|{\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}.}$

example

Zone lens test image

local amplitude

local orientation (local phase direction shown as angle, white = , black = 0) ${\displaystyle \pi }$

local phase angle

Application in image analysis

If the function is understood as a two- or three-dimensional image, the monogenic signal has the following possible applications: ${\displaystyle f}$

• The local phase can be understood as a kind of optical flow of an image. The local phase direction indicates a flow direction, the local phase angle a flow strength.
• Using a multi-scale analysis , the monogenic signal can be used to extract structures from images independent of brightness and illuminance.

literature

• M. Felsberg, G. Sommer: The monogenic signal . In: IEEE Transactions on Signal Processing . tape 49 , no. 12 , 2001, p. 3136-3144 .
• S. Held, M. Storath, P. Massopust, B. Forster: Steerable Wavelet Frames Based on the Riesz Transform . In: IEEE Transactions on Image Processing . tape 19 , no. 3 , 2010, p. 653-667 .

software

The following software packages implement the monogenic signal on a multi-scale basis

• Monogenic Wavelet Toolbox for ImageJ, Technical University of Munich
• MonogenicJ: A ImageJ plugin for wavelet-based monogenic analysis of images EPF Lausanne