# Unified Mathematical Formulation of Monogenic Phase Congruency

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## Abstract

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## 1. Introduction

## 2. Phase Congruency

#### 2.1. Local Energy and Phase Congruency

#### 2.2. Approximation of Fourier Components

#### 2.3. Calculation of Phase Congruency

#### 2.4. Phase Noise Compensation

#### 2.5. Importance of the Frequency Component Distribution

#### 2.6. Phase Congruency Sensitivity

- (i)
- $W\left(x\right)$ is a weighting function, according to the frequency distribution, as it appears in Equation (21).
- (ii)
- Phase congruency quantification function ( $\lfloor 1-\alpha \left|\delta \left(x\right)\right|\rfloor $):This term corresponds to the simple quantization of the PC, i.e., it measures the phase congruency with values between zero and one without making any adjustment. According to this expression, only small values of average phase deviation, $\delta \left(x\right)$, are taken into account, while $\alpha $ limits its range, serving as an adjustment parameter of the PC sensitivity.
- (iii)
- Noise compensation ( $\u230aE\left(x\right)-T\u230b/(E\left(x\right)+\epsilon )$): This factor is used to attenuate the phase congruency by acting as a threshold below which the term becomes zero when $E\left(x\right)\le T$. This prevents false edges from being detected in high-noise regions. Thus, the threshold T is estimated according to the image noise level. To avoid division by zero in Equation (25), a tiny constant $\epsilon $ is added, as in Equation (13).

#### 2.7. Expansion of Phase Congruency to Two Dimensions Using Monogenic Filters

## 3. General Mathematical Formulation of Phase Congruency

## 4. Practical Considerations for the Use of the PC

- Define the size of the smallest features to be detected to find the value of ${\lambda}_{{}_{min}}$ given in pixels.
- Define the scale factor m.
- Define the largest desired scale size to determine how many are necessary to achieve the maximum scale. To this end, the limitation given by the image dimensions must be considered, according to Equation (43).$$N\le \frac{ln({\lambda}_{{}_{max}}/{\lambda}_{{}_{min})}}{ln\left(m\right)}+1.$$
- Adjust ${\sigma}_{o}$ depending on the closeness between the edges to be detected, making it smaller as the edges are closer.

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PC | Phase Congruency |

MPC | Monogenic Phase Congruency |

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**Figure 1.**Results obtained using phase congruency and the Canny edge detector on a synthetic image. It can be seen how the phase congruency allows one to obtain precise edges in ridges and less abrupt changes of the grey level. (

**a**) Original image. (

**b**) Image obtained with Canny’s method. (

**c**) Image obtained with PC. (

**d**) Synthetic image with Gaussian noise of $\sigma =10$. (

**e**) Image obtained with Canny’s method. (

**f**) Image obtained with PC.

**Figure 3.**Graphical representation of the phase congruency of a signal, its local energy and the amplitude sum of the Fourier components. (

**a**) Polar diagram. (

**b**) Triangular phase congruency inequality, where ${\sum}_{n}{A}_{n}\left(x\right)$ is always greater than or equal to the energy $E\left(x\right)$ where the concept of average phase deviation $\varphi \left(x\right)$ is introduced [13].

**Figure 4.**Frequency spectra of the filters used to obtain the components ${A}_{n}\left(x\right)$ [5].

**Figure 5.**Graphical representation of the use of wavelet and monogenic filter banks to obtain the frequency components for the calculation of phase congruency. (

**a**) Wavelet filters, in which six filters are used to calculate each of the four scales. The spectrum of one of the directional filters is shown in dark color. (

**b**) Monogenic filters, where only one filter is used for each of the four scales.

**Figure 7.**Phase congruency obtained with two quantization functions on the baboon image. (

**a**) Baboon image. The first, shown in (

**b**–

**d**), is the absolute value, proposed by Kovesi. The second, shown in (

**e**–

**g**), is the exponential function. The results of the global parameters suggested by default by Kovesi are shown in (

**b**,

**e**). The results achieved by setting the scale factor to 1.5 in (

**c**,

**f**) and by making ${\lambda}_{min}=8$ in (

**d**,

**g**).

**Figure 8.**Detail of the edge detection in the yellow box of the ship image. (

**a**) Ship image with yellow box. Columns (

**b**–

**d**) show the PC results. The absolute value proposed by Kovesi is used as the quantization function in the first row and the exponential function in the second one. In Column (

**b**), the default parameters are used. Column (

**c**) shows the results obtained using the scaling factor $m=1.5$. Column (

**d**) presents the results obtained by making ${\lambda}_{min}=8$.

**Figure 9.**Edge detection profile in the cameraman’s image. (

**a**) Cameraman’s image with yellow line indicating the profile plotted in (

**b**). (

**b**) Horizontal profile of the different results obtained with the phase congruency of the images in the second and third rows. The solid and dashed lines indicate the quantization function used, exponential or absolute value respectively. Red shows the response when the default parameters are used, shown in (

**c**,

**f**); green when the scale factor is set to $m=1.5$, depicted in (

**d**,

**g**); and blue when ${\lambda}_{min}=8$, illustrated in (

**e**,

**h**). The second row shows the results using the absolute value function, and the third row shows the results obtained with the exponential.

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**MDPI and ACS Style**

Forero, M.G.; Jacanamejoy, C.A.
Unified Mathematical Formulation of Monogenic Phase Congruency. *Mathematics* **2021**, *9*, 3080.
https://doi.org/10.3390/math9233080

**AMA Style**

Forero MG, Jacanamejoy CA.
Unified Mathematical Formulation of Monogenic Phase Congruency. *Mathematics*. 2021; 9(23):3080.
https://doi.org/10.3390/math9233080

**Chicago/Turabian Style**

Forero, Manuel G., and Carlos A. Jacanamejoy.
2021. "Unified Mathematical Formulation of Monogenic Phase Congruency" *Mathematics* 9, no. 23: 3080.
https://doi.org/10.3390/math9233080