# Hyperanalytic Wavelet-Based Robust Edge Detection

^{*}

## Abstract

**:**

## 1. Introduction

_{x}and G

_{y}, can be computed using convolution masks h

_{x}and h

_{y},

- (a)
- To have a low error rate,
- (b)
- The edge points should be well localized,
- (c)
- To circumvent the possibility to have multiple responses for a single edge.

- (1)
- To reduce the sensitivity of the edge detector to noise, a Gaussian filter is applied first;
- (2)
- To compute the gradient magnitude and direction, the derivative operators are used;
- (3)
- To get edges with a one-pixel width, non-maximum suppression is used;
- (4)
- To eliminate weak edges, the threshold with hysteresis is applied.

_{a}(x,y) is a two-dimensional Gaussian distribution and ∇

^{2}is the Laplacian. Other multiscale approaches to edge detection use morphological detectors. One method [5] applies the same morphological edge detector at various scales. A different multiscale edge detection method is based on the combination of the Canny’s detector with the scale multiplication. A truly comprehensive multiscale edge detector is the Wavelets Transform [6]. Mallat has proved the equivalence of a multiscale Canny’s edge detector with the finding of the local maxima of a wavelet transform using a Gaussian smoothing function.

#### 1.1. Motivation

- (A)
- The wavelet transforms (WT) are sparse representations of images and can be implemented with fast algorithms [6];
- (B)
- There is an inter-scale dependency between the wavelet detail coefficients from two consecutive scales [8]. In the following, the parent coefficients will be indexed by 1 and the child coefficients will be indexed by 2;
- (C)
- The probability density functions (pdf) of wavelet detail coefficients or parent-children pairs of wavelet detail coefficients are invariant to input image transformations [9];
- (D)
- The WT decorrelates the noise component of the input image [6];
- (E)
- (F)
- The CoWT have good directional selectivity and low redundancy.

#### 1.2. Contribution

_{1}in Figure 1). The noise component extracted by the first stage serves as pilot for the second stage, which is another wavelet-based denoising system. A second HWT (W

_{2}in Figure 1) is used in the second stage. By applying the two-dimensional DWT (2DDWT) to the pilot, we estimate separately, based on (2), the standard deviation σ

_{n}of each W

_{2}detail sub-band,

_{2}domain uses this feature to estimate the standard deviation of the input image’s noiseless component σ using,

_{y}represents the standard deviation of the acquired image’s WT pixels. The main advantage of this two stages denoising approach is that it avoids the limitations generally imposed on MAP filters’ denoising scheme as the low Signal-to-Noise Ratio (SNR) or the whiteness of the noise.

#### 1.3. Related Work

- 1.
- Detecting the step and linear edges from images corrupted by mixed noise (exponential and impulse) without smoothing. The authors of [15] substituted the Gaussian smoothing filter with a statistical classification technique;
- 2.
- Detecting thin-line edges, as a series of outliers using the Dixon’s r-test;
- 3.
- Suppressing the spurious edge elements and connecting the isolated missing edge elements.

## 2. Materials and Methods

**u**into a dictionary

**D**(whose columns are l elementary atoms) in the form:

_{k}represents synthesis coefficients. Among the most popular dictionaries, we can find WTs and Discrete Cosine Transform (DCT). We will denote in the following a wavelet dictionary

**D**=

**W**and the corresponding synthesis coefficients

**a**=

**w**. Hence, we will write the decomposition of the signal

**u**into a wavelet basis as:

**U**=

**Ww**.

**y**represents the coefficients of the WT of the acquired image and λ is a constant called regularization parameter.

^{p}norm,

#### 2.1. Hyperanalytic Wavelet Transform

- Since wavelets are bandpass functions, the wavelet coefficients tend to oscillate around singularities;
- The wavelet coefficients oscillation pattern around singularities is significantly perturbed even by small shift of the signal;
- The wide spacing of the wavelet coefficient samples (the calculation of the wavelet coefficients involves interleaved sampling operations in discrete time and high-pass filtering), resulting in a substantial aliasing. The inverse DWT transformation (IDWT) cancels this aliasing, of course, if the wavelet coefficients are not changed. Any wavelet coefficient processing operation, as seen in thresholding; filtering; or quantization, leading to artifacts in the reconstructed signal.

_{a}$=$ψ $+j$ H{ψ}. Further, it is equivalent to the computation of the DWT of the analytic signal associated with the input signal with the aid of the real MW ψ. Hence, the operation of analytic signal generation commutes with the computation of DWT associated with the real MW. We will name this property, using an expression taken from electronics, reduction at the input. The CDWT is a quasi-shift invariant as well [7].

_{x}denotes the Hilbert transform computed on rows and H

_{y}the Hilbert transform computed on columns.

_{x}) and the 1D Hilbert transform from the third branch is computed across the columns of the input image (H

_{y}). Finally, the fourth branch processes the result obtained after the computation of the two 1D Hilbert transforms of the input image (H

_{y}(H

_{x})). The scheme for the HWT implementation proposed in [7] contains two blocks. The first one computes the hypercomplex input image and the four 2D DWTs, whose output sequences are: d

_{1}, d

_{2}, d

_{3}and d

_{4}. The second block improves the directional selectivity of HWT. Through the linear combinations of detail coefficients belonging to each sub-band of each of the four 2D-DWTs, described in Equation (16), we can obtain the HWT directional selectivity’s enhancement:

#### 2.2. Global MAP Filters Applied in Wavelet Domain

**w**from a single noisy measurement,

**y = w + n**, where

**n**is noise. The MAP filters apply a Bayesian approach to estimate the a posteriori (or ‘posterior’) distribution: the probability distribution of

**w**, given the measurement

**y**, i.e., $p\left(w|y\right)$, contains all the information that we can infer about a given

**y**. In cases where the posterior distribution of

**w**is unimodal (having a unique maximum in

**W**) then the value

**W**represents the MAP estimate of

**w**. We can compute the posterior distribution $p\left(w|y\right)$ with the aid of Bayes’ theorem,

**w**given the data $\mathrm{Li}\left(w|y\right)$. The inference

**w = W**implies that the noise can be expressed as

**n = y**−

**W**. Hence, the likelihood is $\mathrm{Li}\left(w|y\right)=p\left(y|w=\mathit{W}\right)={p}_{n}\left(y-w\right)$, where p

**is the noise’s pdf. The MAP estimation of**

_{n}**w**is given by (10). The solution of (10) is a marginal MAP filter [27], if

**w**and

**n**are scalars. When

**w**and

**n**are vectors, the solution of (10) is a global MAP filter [8].

**x**=

**s**+

**n**

_{i}, such a denoising method supposes the following steps:

- 1.
- Computation of the WT of the observation: $\mathrm{WT}\left\{x\right\}=\mathrm{WT}\left\{s\right\}+\mathrm{WT}\left\{{n}_{i}\right\}$ and separation of approximation and detail coefficients;
- 2.
- Non-linear filtering of detail coefficients and restructuration of WT by the concatenation of approximation coefficients with the new detail coefficients;
- 3.
- Computation of the inverse WT (IWT).

**y**= WT{

**x**}, the WT of the noiseless component of the input image by

**w**= WT{

**s**} and the WT of the noise by

**n**= WT{

**n**

_{i}}.

^{W}p

_{n}and

^{W}p

_{w}are Gaussians,

^{st}p

_{n}and a Laplacian prior

^{st}p

_{w}:

#### Bishrink Filter

_{1}, y

_{2}) as a circle with the center in the origin of the plane and with the ray:

_{2}. The corresponding coefficients at the next decomposition level represent the parent wavelet coefficients y

_{1}. The sub-images of parent wavelet coefficients are extended by doubling the lines and columns to obtain the same size as the sub-images of child wavelet coefficients, in order to compute in each pixel the magnitude of vector

**y**: $\sqrt{{y}_{1}^{2}+{y}_{2}^{2}}$, required by (26) [8]. The next step supposes the computation of the local noiseless component’s standard deviations in each sub-band, obtaining a sub-band adaptive bishrink filter, or locally in each sub-band, using moving windows of size 7 × 7, obtaining a local adaptive bishrink filter. For each position of the window, a local estimation of the noisy child wavelet coefficients variances, ${}_{}{}^{l}\widehat{\mathsf{\sigma}}{}_{{y}_{1}}^{2},$ is realized, and next is applied relation (3) using the noise standard deviation already estimated to obtain the local estimation of noiseless components variances:

#### 2.3. Multiplicative Noise

**x**=

**sn**

_{i}.

**x**) = log(

**s**) + log(

**n**

_{i}),

**s**) represents the noiseless component of the input image and that log(

**n**

_{i}) represents the additive noise, we can apply the HWT-marginal ASTF or HWT-Bishrink denoising associations, which are additive noise-denoising kernels. Due to the decorrelation effect of WTs, the statistical model of the HWT of log(

**n**i) is Gaussian in case of the HWT-marginal ASTF association and bivariate Gaussian in case of the HWT-Bishrink association. These priors were considered for log(

**n**i) in the majority of references about wavelet-based despeckling already cited, as seen for example in [45] or [46]. Based on experiments, in different papers, as for example [9], was proved that the statistical model of the wavelet detail coefficients of noiseless images is characterized by a heavy tail distribution, such as a Laplacian, and that this model is invariant to different transformations, such as the logarithm. Therefore, the statistical model of HWT of log(

**s**) is Laplacian in case of HWT-marginal ASTF association and bivariate symmetric Laplacian in the case of HWT-Bishrink association.

#### 2.4. Proposed Denoising Method

_{1}). To compute this HWT, we have used the Daubechies MW with two Vanishing Moments (VM). Supposing the Additive White Gaussian Noise (AWGN) model for the input image, we obtain the image after the computation of W

_{1}:

_{1}, at the first decomposition level (${D}_{1}^{3}$), are retained, to estimate the four noise standard deviations, ${\widehat{\sigma}}_{n}$, using (2). Next is realized the estimation of local noiseless components standard deviations in each sub-band and at any decomposition level, using moving windows 7 × 7 pixels in size. For each window position, the local variances of noisy child wavelet coefficients are estimated and after is applied the relation (3) using the noise standard deviations already estimated, to obtain the local estimation of noiseless components standard deviations ${}_{}{}^{l}\widehat{\sigma}$. The following step is to apply the marginal ASTF, whose input–output relation is:

_{1}is the threshold computed with the following relation:

_{1}, is obtained after the computation of the four inverse 2D DWTs. By computing the difference of images x and s

_{1}:

_{2}) with an improved variant of bishrink filter.

_{2}in Figure 1) is applied to the pilot $\widehat{n}$, in order to compute the standard deviations of the wavelet detail, ${\widehat{\mathsf{\sigma}}}_{{n}_{2}}$, of each W

_{2}detail sub-band. Next, W

_{2}is applied to the image x. Both HWTs denoted as W

_{2}, are computed using the Daubechies 9/7 biorthogonal MW. The HWT of x is used for the bishrink filtering. Next, we separate the child y

_{1}and parent y

_{2}wavelet coefficients for any sub-band and pair of successive decomposition levels. We extend the parent sub-images by resizing with factor 2, obtaining sub-images of same size as the corresponding child sub-images. Next, we estimate the local standard deviation of the noiseless component of the input image, ${}_{}{}^{l}\widehat{\mathsf{\sigma}}{}_{2}$, for each sub-band and decomposition level with the aid of constant array elliptic estimation windows. The principal axes of those ellipses are oriented following the preferential orientations of the sub-bands: ±atan (1/2), ±π/4 and ±atan (2). The authors of [55] introduced the elliptic directional windows in conjunction with the 2D DWT. The authors of [7] generalized this idea. Another source of bias for the noiseless component estimation is the small size of the estimation window [56]. Despite the fact that the mean of detail coefficients equals zero for the ensemble of the entire image, the mean of the detail coefficients in the estimation window could be different of zero. Therefore, we compute first the mean in the estimation window and next we compute the local variances. Extracting the square roots of variances, we obtain the sub-images containing the estimation of local standard deviations of the image y, ${\widehat{\mathsf{\sigma}}}_{y}$ in each sub-band and decomposition level.

#### 2.5. Performance Measures

_{p}represents the number of pixels of the considered image, s

_{q}are the pixels of the noiseless component and ${\widehat{s}}_{q}$ are the pixels of the denoising result.

## 3. Results

_{0}= 7. We have implemented all algorithms in MATLAB using the WaveLab Toolbox [57].

#### 3.1. Images Affected by Synthesized Noise

_{n}= 15.

_{n}= 30.

_{n}= 30. We can remark the same behavior of the proposed edge detection method concerning the robustness against noise as in the case of images Lena and Boat.

#### 3.2. Real Remote Sensing Images

## 4. Discussion

#### 4.1. Images Affected by Synthesized Speckle

#### 4.2. Real Remote Sensing Images

#### 4.3. Comparison with Modern Despeckling Methods

^{2}loss; the L

^{2}difference between the gradient of the two images and the Kullback–Leibler divergence between the distributions of the two images. The three terms are designed to emphasize the spatial details, the identification of strong scatterers, and the speckle statistics, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 2.**Comparison of results obtained — ${\mathsf{\sigma}}_{n}=30$. First line: original image and its edge map obtained with Canny’s detector; second line: noisy image and its edge map obtained by applying Canny’s detector; third line: denoised image and the result of the proposed method.

**Figure 4.**Comparison of Canny’s detector applied directly to noisy image (

**left**) and proposed edge detection method (

**right**)—${\mathsf{\sigma}}_{n}=30$.

**Figure 5.**Comparison of Canny’s detector applied directly on noisy image (

**left**) and proposed edge detection method (

**right**) for—.${\mathsf{\sigma}}_{n}=30$.

**Figure 6.**Results of different despeckling methods applied to a test image [40]—left column and the corresponding edge maps obtained with the Canny’s edge detector (right column). First line: the test image; second line: Lee despeckling method; third line: Kuan despeckling method; fourth line: MBD algorithm; fifth line: Frost despeckling method and last line: proposed despeckling method.

**Figure 7.**Results of the proposed edge detection method applied to an aerial SAR image. Left column, up to down: original image; first stage denoising result; final denoising result. Right column, up to down: edge maps of corresponding images obtained applying the Canny’s detector.

**Figure 9.**(

**Left**): Region of a Sentinel-1 Stripmap GRDH SAR image of the Agulhas current ocean registered the 11-03-2015; (

**Right**): Result obtained applying the Canny’s edge detection method.

**Figure 10.**Results of the proposed edge detection method applied to a SONAR image. We are grateful to GESMA for making this image available to us. Left column: original image (up); first stage’s denoising result (middle); final denoising result (down). Right column: edge maps of corresponding images obtained applying the Canny’s detector.

**Table 1.**Image Lena. A comparison of the results obtained for the proposed method and for Canny’s edge detector applied directly on the noisy image, for different values of noise standard deviation.

Proposed Method | Canny’s Method | ||||
---|---|---|---|---|---|

First Step | Final Result | Directly on Noisy Image | |||

${\mathsf{\sigma}}_{n}$ | Input PSNR | Output PSNR | Output SSIM | Edges’ MSE | Edges’ MSE |

10 | 28.13 | 35.19 | 0.9989 | 0.04 | 0.07 |

15 | 24.59 | 33.41 | 0.9983 | 0.06 | 0.09 |

20 | 22.10 | 32.06 | 0.9977 | 0.07 | 0.14 |

25 | 20.21 | 31.06 | 0.9971 | 0.07 | 0.2 |

30 | 18.61 | 30.20 | 0.9964 | 0.08 | 0.21 |

**Table 2.**Image Boat. A comparison of the results obtained for the proposed method and for Canny’s edge detector directly applied to the noisy image, for different values of the noise standard deviation.

Proposed Method | Canny’s Method | ||||
---|---|---|---|---|---|

First Step | Final Result | Directly on Noisy Image | |||

${\mathsf{\sigma}}_{n}$ | Input PSNR | Output PSNR | Output SSIM | Edges’ MSE/no. of missed pixels | Edges’ MSE/no. of false edge pixels |

10 | 28.13 | 33.11 | 0.9981 | 0.05/1122 | 0.06/1218 |

15 | 24.59 | 31.20 | 0.9970 | 0.07/2828 | 0.09/5247 |

20 | 22.10 | 29.86 | 0.9959 | 0.09/5443 | 0.14/13,538 |

25 | 20.21 | 28.82 | 0.9948 | 0.1/6893 | 0.18/17,130 |

30 | 18.61 | 28.08 | 0.9935 | 0.1/6978 | 0.20/26,644 |

**Table 3.**Image Barbara. A comparison of the results obtained for the proposed method and for Canny’s edge detector directly applied to the noisy image, for different values of the noise standard deviation.

Proposed Method | Canny’s Method | ||||
---|---|---|---|---|---|

First Step | Final Result | Directly on Noisy Image | |||

${\mathsf{\sigma}}_{n}$ | Input PSNR | Output PSNR | Output SSIM | Edges’ MSE/no. of missed pixels | Edges’ MSE/no. of false edge pixels |

10 | 28.13 | 33.23 | 0.9987 | 0.05/516 | 0.06/3672 |

15 | 24.59 | 31.31 | 0.9978 | 0.06/523 | 0.09/7995 |

20 | 22.10 | 29.41 | 0.9968 | 0.07/2052 | 0.14/16,922 |

25 | 20.21 | 28.21 | 0.9956 | 0.08/3235 | 0.17/25,815 |

30 | 18.61 | 27.06 | 0.9943 | 0.09/4335 | 0.2/29,272 |

NL | Noisy | Result in [48] | HWT - | HWT - | |||
---|---|---|---|---|---|---|---|

Marginal ASTF | Bishrink | ||||||

D4 | B9/7 | D4 | B9/7 | D4 | B9/7 | ||

1 | 12.1 | 26.0 | 26.2 | 25.4 | 25.6 | 25.7 | 26.2 |

4 | 17.8 | 29.3 | 29.6 | 29.9 | 30.0 | 29.9 | 30.4 |

16 | 23.7 | 32.9 | 33.1 | 33.2 | 32.9 | 33.0 | 33.3 |

NL | Noisy | SA-WB MMAE | MAP-S | PPB | SAR-BM3D | H-BM3D | Prop. |
---|---|---|---|---|---|---|---|

1 | 12.1 | 25.0 | 26.3 | 26.7 | 27.9 | 26.4 | 26.4 |

4 | 17.8 | 29.0 | 29.8 | 29.8 | 29.6 | 31.2 | 30.6 |

16 | 23.7 | 32.4 | 33.2 | 32.7 | 34.1 | 34.5 | 33.5 |

**Table 6.**Comparison of components performance in the two-stage denoising system proposed in terms of ENL.

Method | Parameters | |
---|---|---|

ENL | Noise Rejection | |

Input image | 2 | Unavailable |

First stage (HWT-marginal ASTF) | 3.4 | Worst result |

Entire System | 7.61 | Best result |

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**MDPI and ACS Style**

Isar, A.; Nafornita, C.; Magu, G.
Hyperanalytic Wavelet-Based Robust Edge Detection. *Remote Sens.* **2021**, *13*, 2888.
https://doi.org/10.3390/rs13152888

**AMA Style**

Isar A, Nafornita C, Magu G.
Hyperanalytic Wavelet-Based Robust Edge Detection. *Remote Sensing*. 2021; 13(15):2888.
https://doi.org/10.3390/rs13152888

**Chicago/Turabian Style**

Isar, Alexandru, Corina Nafornita, and Georgiana Magu.
2021. "Hyperanalytic Wavelet-Based Robust Edge Detection" *Remote Sensing* 13, no. 15: 2888.
https://doi.org/10.3390/rs13152888