# Smoothed Shock Filtering: Algorithm and Applications

## Abstract

**:**

## 1. Introduction

## 2. Smoothed Shock Filtering: Principle, Algorithm and Impact of Parameters

#### 2.1. Algorithm Description

Algorithm 1: Smoothed shock filtering |

#### 2.2. Impact of the Parameters

## 3. A Robust Approach for Image Denoising

**Definition**

**1**

## 4. Image Enhancement for Improving Classification and Segmentation

#### 4.1. Image Segmentation

#### 4.2. Image Classification

## 5. Discussion and Future Works

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**From a part of Lena image (

**a**), i.e., $\mathcal{V}\left(\mathbf{p}\right)$ around the central pixel $\mathbf{p}$, the standard histogram (

**d**) and smoothed histogram ${\widehat{f}}_{\mathbf{p}}$ from Equation (3) (

**e**) are, respectively, depicted. We also present examples of smoothed dilation (

**b**) and erosion (

**c**) obtained over the whole patch.

**Figure 2.**Scale-space representation obtained from input image (

**a**), and the result from the filter for 20 iterations. The output images after 1, 5, 10, 15, and 20 iterations are presented, with a zoomed part and a height map. At the bottom, residual images are depicted with their corresponding height maps (

**b**–

**f**).

**Figure 3.**Impact of smoothing by augmenting the Gaussian kernel W with a flower picture (

**top**) and a zoomed part of it (

**bottom**).

**Figure 4.**Impact of $\rho $ parameter on smoothed morphological operators, with two zoomed parts of the image.

**Figure 5.**Evaluation of $(\alpha ,\sigma )$-robustness for several image denoising algorithms, compared to smoothed shock filtering, by studying quality function decrease through scales of noise (

**a**) or numerically by appreciating the $(\alpha ,\sigma )$ values for each algorithm (

**b**).

**Figure 6.**Evaluation of $(\alpha ,\sigma )$-robustness, by the graphical (

**a**) and numerical (

**b**) approaches, for the smoothed shock filtering and the DnCNN network, learned with two configurations.

**Figure 9.**Segmentation obtained from one CT slice, with different methods, after 10, 20, and 30 iterations. A part of the segmentation is also presented.

**Figure 11.**Segmentation results of HCC, from top to bottom: ROI selected by an expert; smoothed shock filter applied on these images; ground-truth of tumors’ contours; and segmentations obtained with our pipeline.

Setup | Gaussian Kernel | Laplacian Operator | Number of Iterations |
---|---|---|---|

Standard, for most applications | ${\sigma}_{w}=3$ | $\rho =0.1$ | $1\le {n}_{it}\le 5$ |

Water colorization | ${\sigma}_{w}>3$ | $\rho =0.1$ | ${n}_{it}\ge 5$ |

Sharpening | – | $\rho >0.1$ | ${n}_{it}\ge 5$ |

Scale-space representation | – | – | ${n}_{it}\ge 10$ |

Original shock filtering | – | $\rho =0.5$ | – |

Smoothed median filtering | ${\sigma}_{w}>0$ | $\rho =0.0$ | – |

Median filtering | ${\sigma}_{w}=0$ | $\rho =0.0$ | – |

Method | Recall | Precision | F-Measure |
---|---|---|---|

FLIRT | 0.761 | 0.733 | 0.747 |

OriginalShock | 0.797 | 0.777 | 0.787 |

Median | 0.824 | 0.811 | 0.817 |

SmoothedShock | 0.829 | 0.811 | 0.820 |

**Table 3.**Performance of classifiers, KNN (

**a**) and Naive Bayes (

**b**) approaches, for four texture classification datasets, by considering three pre-processing filters: smoothed shock filter, Gaussian filter, and anisotropic diffusion. For each filter, we present the best Correct Classification Rate (CCR), together with the feature and the combination of iterations leading to this rate. The leftmost column is the best rate without any pre-processing.

(a) KNN | |||||||

Dataset | Original | Smoothed shock | Gaussian | Diffusion | |||

CCR (feat.) | CCR (feat.) | It. | CCR (feat.) | It. | CCR (feat.) | It. | |

Outex | 75.59 (LBPV) | 84.78 (GLDM) | $\{2,4,\cdots ,18\}$ | 83.01 (CLBP) | $\left\{9\right\}$ | 82.94 (CLBP) | $\left\{19\right\}$ |

Brodatz | 97.6 (CLBP) | 98.11 (CLBP) | $\left\{6\right\}$ | 97.20 (CLBP) | $\left\{2\right\}$ | 97.84 (CLBP) | $\{11,14\}$ |

Usptex | 83.1 (CLBP) | 88.66 (CLBP) | $\{1,8\}$ | 85.21 (CLBP) | $\left\{1\right\}$ | 88.57 (CLBP) | $\{1,3,\cdots ,17\}$ |

Vistex | 98.96 (CLBP) | 99.31 (CLBP) | $\left\{2\right\}$ | 98.96 (CLBP) | $\{2,3\}$ | 99.54 (CLBP) | $\left\{14\right\}$ |

(b) Naive Bayes | |||||||

Dataset | Original | Smoothed shock | Gaussian | Diffusion | |||

CCR (feat.) | CCR (feat.) | It. | CCR (feat.) | It. | CCR (feat.) | It. | |

Outex | 80.81 (LBP) | 86.47 (LBP) | $\{2,7,\cdots ,17\}$ | 83.01 (LBP) | $\left\{8\right\}$ | 85.15 (CLBP) | $\left\{4\right\}$ |

Brodatz | 96.6 (CLBP) | 98.02 (CLBP) | $\left\{2\right\}$ | 96.85 (CLBP) | $\left\{1\right\}$ | 97.47 (CLBP) | $\left\{15\right\}$ |

Usptex | 85.77 (CLBP) | 91.49 (CLBP) | $\{1,3,5\}$ | 86.43 (CLBP) | $\{1,3\}$ | 89.66 (CLBP) | $\{8,11,\cdots ,20\}$ |

Vistex | 97.33 (CLBP) | 98.50 (CLBP) | $\{2,3\}$ | 98.50 (CLBP) | $\{1,3,5\}$ | 97.92 (CLBP) | $\{1,8\}$ |

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**MDPI and ACS Style**

Vacavant, A.
Smoothed Shock Filtering: Algorithm and Applications. *J. Imaging* **2021**, *7*, 56.
https://doi.org/10.3390/jimaging7030056

**AMA Style**

Vacavant A.
Smoothed Shock Filtering: Algorithm and Applications. *Journal of Imaging*. 2021; 7(3):56.
https://doi.org/10.3390/jimaging7030056

**Chicago/Turabian Style**

Vacavant, Antoine.
2021. "Smoothed Shock Filtering: Algorithm and Applications" *Journal of Imaging* 7, no. 3: 56.
https://doi.org/10.3390/jimaging7030056