# A Model of Pixel and Superpixel Clustering for Object Detection

^{1}

^{2}

^{*}

## Abstract

**:**

_{0}= 1, 2,..., etc., colors. For the selected hierarchy of pixel clusters, the objects-of-interest are detected as the pixel clusters of optimal approximations, or as their parts, or unions. The paper develops the known idea in cluster analysis of the joint application of Ward’s and K-means methods. At the same time, it is proposed to modernize each of these methods and supplement them with a third method of splitting/merging pixel clusters. This is useful for cluster analysis of big data described by a convex dependence of the optimal approximation error on the cluster number and also for adjustable object detection in digital image processing, using the optimal hierarchical pixel clustering, which is treated as an alternative to the modern informally defined “semantic” segmentation.

## 1. Introduction

- We propose a mathematical model for detecting objects in an image, assuming that the sequence of approximation errors of optimal approximations in a successively increasing number of colors is convex.
- We formalize the concepts of images, objects, and superpixels, distinguishing them from each other by structure to unify object detection without referring to specific examples of images and objects.
- To verify the object detection model, the criterion of minimum standard deviation $\sigma $ or approximation error $E~{\sigma}^{2}$ is used instead of using “ground truth” samples that do not take into account image ambiguity.
- We offer three modernized versions of the classical methods for real-life minimizing approximation errors and customizable object detection by hierarchical image approximations.

## 2. Pixel Clustering vs. Image Segmentation

## 3. Definition of Superpixels

## 4. Approach to Superpixels

## 5. Model of Hierarchical Approaching of Optimal Approximations

- is described by a convex sequence of approximation errors, similar to the hierarchy of optimal approximations;
- is a binary hierarchy, where each pixel cluster either coincides with an indivisible superpixel or is divided into two.

- all pixels of the feature map are assigned initial zero values, and the current markup value is assumed to be equal to one;
- clusters $i$ of the hierarchy of image approximations are scanned in order from smaller to larger heterogeneity $H\left(i\right)$;
- from the number of clusters $i$ with heterogeneity values $H\left(i\right)$ not lower than the established threshold ${H}_{threshold}$, $H\left(i\right)\ge {H}_{threshold}$: (a) the next cluster $j$ is selected, marked with zero values on the object map; (b) pixels of cluster $j$ on the object map are assigned the current markup value; (c) the current markup value is incremented by one.

## 6. Recursive Ward’s Method

- at $\eta =2$, $C$ falls off like ${N}^{2}\to {N}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\to {N}^{\raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$15$}\right.}\to {N}^{\raisebox{1ex}{$256$}\!\left/ \!\raisebox{-1ex}{$255$}\right.}\to \dots \to {N}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\left(t-1\right)$}\right.}....\to N$, where $t={2}^{2i},\text{}i=1,2,\dots $;
- at $\eta =3$, $C$ falls off like ${N}^{3}\to {N}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\to {N}^{\raisebox{1ex}{$81$}\!\left/ \!\raisebox{-1ex}{$65$}\right.}\to {N}^{\raisebox{1ex}{$6561$}\!\left/ \!\raisebox{-1ex}{$6305$}\right.}\to {N}^{\raisebox{1ex}{$43,046,721$}\!\left/ \!\raisebox{-1ex}{$42,981,185$}\right.}\dots \to N$.

- to calculate the hierarchy of approximations for each cluster of optimal approximation by the original Ward method, as for an independent image;
- to rebuild the hierarchy of the image approximations without modifying the calculated pixel clusters, reordering the merging of clusters so that the resulting hierarchy of approximations is described by a convex sequence of approximation errors ${E}_{g}$, where $g\le {g}_{0}$;
- to complete the hierarchy to the full one, iteratively enlarging ${g}_{0}$ clusters of the optimal image approximation using Ward’s original method.

## 7. CI Method for Improving Structured Approximations

- From ${g}_{0}$ clusters, such cluster $i\cup j$ is selected, the division of which into two is accompanied by the maximum drop $\mathrm{max}H$ in the approximation error $E$.
- From ${g}_{0}$ × ${g}_{0}$ of cluster pairs, that pair of clusters ${i}^{\prime}$, ${j}^{\prime}$ is selected, the merging of which is accompanied by a minimum increment $\mathrm{min}\Delta {E}_{merge}$ of $E$ approximation error.
- At $\mathrm{max}H\le \mathrm{min}\Delta {E}_{merge}$, the processing ends. If $\mathrm{max}H>\mathrm{min}\Delta {E}_{merge}$, cluster $i\cup j$ is divided into two clusters $i$ and $j$, and a pair of clusters is merged with a minimum increase in $E$—either a pair of clusters ${i}^{\prime}$, ${j}^{\prime}$, or a new pair of clusters generated by the appearance of clusters $i$ and $j$. Next, the processing is resumed.

## 8. K-Meanless Method for Improving Structured Approximations

## 9. Discussion of the System of Methods for E Minimization

- The CI method, i.e., by means of counter operations of splitting one of the clusters in two with the subsequent merging of a pair of other clusters;
- The K-meanless method, i.e., reclassification operation, partially reassigns the pixels from one to another cluster.

## 10. Dynamic Table of Ordered Image Approximations

^{2}= 232,833,270,784 image approximations.

## 11. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Optimal (on the left), really optimized two-segment approximation (central), and iteratively segmented (on the right) image approximation with the standard deviations σ =30.64564, σ = 31.60341 and σ = 50.33156.

**Figure 2.**Structural representations of image, objects, and superpixels. The plots illustrate the standard deviation σ of approximations from the image depending on cluster number g. The upper purple curve describes the hierarchical segmentation of the image. The black solid curve describes the superpixel hierarchy. The lower gray curve describes the sequence of optimal image approximations, and the red curve describes the object binary hierarchy of approximations obtained by the hierarchical Otsu method [9,10,26].

**Figure 3.**The number $s$ of superpixels as a function of the number $g$ of optimal image partitions involved in dividing $N$ pixels into superpixels due to the accumulation of boundaries between clusters. The dotted curve shows the case of hierarchical optimal approximations.

**Figure 4.**Optimal approximations of a standard image in 1–8 tones, ordered from left to right and top to bottom.

**Figure 5.**Hierarchy of superpixels generated by eight optimal approximations, ordered from left to right and top to bottom.

**Figure 6.**Model of hierarchical approaching of optimal approximations. The limiting lower gray curve, assumed to be predominantly convex, describes the optimal image approximations. The bold red convex curve tangent to the gray curve at g = g

_{0}describes the target hierarchy of image approximations. Another thin red convex curve tangent to the gray curve describes the target hierarchy of approximations at the maximum value g

_{0}= g

_{1}. The rest upper thin red convex curve describes the hierarchy of approximations produced by Ward’s original method. The dotted black curve describes the hierarchy of superpixel approximations or enlarged pixels intended to initialize Ward’s pixel clustering.

**Figure 8.**Hierarchical approaching of optimal approximations according to original Ward’s method. The limiting lower gray curve, assumed to be predominantly convex, describes the minimal approximation errors ${E}_{g}$ of the optimal image approximations depending on the number $g$ of colors. The red solid convex curve describes approaching the image by the hierarchy of approximations for Ward’s pixel clustering. The dashed line shows the upper limit for both error sequences.

**Figure 9.**The CI method for minimizing the approximation error with the same number of clusters in the image approximation. The limiting lower gray curve, which is assumed to be predominantly convex, describes the optimal image approximations. The upper dotted curve describes the generation of some approximation of the image, specifically, by hierarchical segmentation. The intermediate convex red curve describes the resulting hierarchy of approximations obtained by the CI method in combination with Ward’s method.

**Figure 10.**K-meanless method of minimizing the approximation error for a constant number of clusters in image approximation. The limiting lower gray curve, treated as predominantly convex, describes the optimal image approximations. The red convex curve tangent to the gray curve describes the hierarchy of approximations obtained using the K-meanless method.

**Figure 11.**Parameterized approaching of an image by a hierarchical sequence of approximations in g

_{0}= 1, 2,..., N colors. The lower gray convex curve describes E

_{g}sequence of optimal image approximations. The remaining red convex curves describe the hierarchies E

_{g}sequences of image approximations each containing at least one optimal approximation, in g

_{0}= 1, 2,..., N colors.

**Figure 13.**Dynamic Table of N × N image approximations, ordered along the columns and the main diagonal. Diagonal elements are highlighted in red. Dynamic Table demonstrates the hierarchies of image approximations arranged in columns.

**Figure 14.**Customizable selection of desired object hierarchy. The input composite image from [46] is displayed in the top line. The middle line shows the result of object detection according to [46], where the surfer’s body is poorly detected. The bottom line shows the proposed method results using g

_{0}= 3, g = 14.

g | σ | s | σ_{s} |
---|---|---|---|

1 | 55.8832 | 1 | 55.8832 |

2 | 30.6456 | 2 | 30.6456 |

3 | 21.2174 | 4 | 18.1409 |

4 | 14.9645 | 7 | 11.3062 |

5 | 11.6976 | 11 | 8.71359 |

6 | 10.0398 | 16 | 6.85761 |

7 | 8.46072 | 18 | 5.67555 |

8 | 7.51121 | 24 | 5.27755 |

9 | 6.81359 | 27 | 4.62883 |

10 | 6.14397 | 30 | 3.84431 |

11 | 5.57864 | 33 | 3.71779 |

12 | 5.11403 | 36 | 3.18455 |

13 | 4.75689 | 39 | 3.09882 |

14 | 4.42306 | 41 | 2.91261 |

15 | 4.17825 | 43 | 2.61455 |

16 | 3.92460 | 46 | 2.27893 |

17 | 3.70326 | 50 | 2.10761 |

18 | 3.50441 | 55 | 1.95239 |

19 | 3.32383 | 60 | 1.68547 |

20 | 3.15658 | 63 | 1.64135 |

^{1}A complete table of standard deviations of optimal approximations for the Lena image is published in [26].

**Table 2.**The 15 × 15 fragment of Dynamic Table of (1774 × 272)

^{2}= 232,833,270,784 σ values denoting image approximations

^{1}.

g_{0} | 1, 2 | 3 | 4 | 5, 12 | 6 | 7, 9 | 8 | 10, 11 | 13, 14, 15 | |
---|---|---|---|---|---|---|---|---|---|---|

g | ||||||||||

1 | 82.7624 | 82.7624 | 82.76244 | 82.7624 | 82.7624 | 82.7624 | 82.76244 | 82.7624 | 82.76244 | |

2 | 41.9853 | 45.6371 | 44.61491 | 48.8133 | 47.1703 | 47.1703 | 47.5201 | 46.2158 | 49.70901 | |

3 | 34.1905 | 30.5178 | 32.17524 | 31.9416 | 32.6948 | 31.0402 | 31.40646 | 30.7935 | 32.2961 | |

4 | 27.4511 | 27.4203 | 26.25674 | 27.99 | 28.3736 | 26.4501 | 27.7935 | 27.1017 | 27.13618 | |

5 | 24.1529 | 24.3494 | 23.98987 | 23.4904 | 23.5732 | 23.8579 | 24.41134 | 24.5259 | 23.81052 | |

6 | 21.5788 | 21.9211 | 22.23782 | 21.1801 | 20.922 | 21.2423 | 21.8049 | 21.9532 | 21.05243 | |

7 | 20.2058 | 19.8155 | 20.65018 | 19.9869 | 19.769 | 19.6739 | 19.9542 | 20.1168 | 20.00413 | |

8 | 18.744 | 18.5328 | 19.08621 | 18.9059 | 18.7525 | 18.443 | 18.39776 | 18.644 | 18.93809 | |

9 | 17.5868 | 17.6728 | 17.99386 | 17.7844 | 17.8988 | 17.297 | 17.46976 | 17.3337 | 17.94591 | |

10 | 16.6735 | 16.8386 | 17.05136 | 16.726 | 17.0394 | 16.7309 | 16.80458 | 16.514 | 16.92667 | |

11 | 15.8308 | 16.1807 | 16.34836 | 15.8691 | 16.3098 | 16.1497 | 16.1853 | 15.788 | 15.95407 | |

12 | 15.3293 | 15.553 | 15.86705 | 15.2115 | 15.7082 | 15.6135 | 15.59814 | 15.2757 | 15.2257 | |

13 | 14.834 | 15.0178 | 15.37627 | 14.7058 | 15.0876 | 15.1058 | 15.10836 | 14.7567 | 14.67922 | |

14 | 14.3752 | 14.5941 | 14.91152 | 14.2989 | 14.5841 | 14.603 | 14.62753 | 14.2415 | 14.20433 | |

15 | 13.9324 | 14.1667 | 14.49436 | 13.9219 | 14.0773 | 14.0968 | 14.15316 | 13.7995 | 13.79826 |

^{1}The diagonal elements are highlighted in bold and are listed in the first table row.

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**MDPI and ACS Style**

Nenashev, V.A.; Khanykov, I.G.; Kharinov, M.V.
A Model of Pixel and Superpixel Clustering for Object Detection. *J. Imaging* **2022**, *8*, 274.
https://doi.org/10.3390/jimaging8100274

**AMA Style**

Nenashev VA, Khanykov IG, Kharinov MV.
A Model of Pixel and Superpixel Clustering for Object Detection. *Journal of Imaging*. 2022; 8(10):274.
https://doi.org/10.3390/jimaging8100274

**Chicago/Turabian Style**

Nenashev, Vadim A., Igor G. Khanykov, and Mikhail V. Kharinov.
2022. "A Model of Pixel and Superpixel Clustering for Object Detection" *Journal of Imaging* 8, no. 10: 274.
https://doi.org/10.3390/jimaging8100274