# A Systematic Multichimera Transform for Color Image Representation

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## Abstract

**:**

## 1. Introduction

- A simple image transform is proposed based on recursively finding the similarity between a precomputed codebook and image blocks.
- A simple set of 2D functions are derived to build a codebook independently of image contents and dimensions. The size of the codebook is relatively proportional to the image block size.
- This transform supports different image block sizes, which eventually leads to obtaining different compression ratios.
- The matching process between image and codebook is directly conducted without any complex transformations or huge mathematical calculations.

## 2. Theoretical Background

#### 2.1. Advantages of Mathematical Representation Instead of Data

**Simplified processing:**Image processing fields such as filtering, edge detection, resizing, and color conversion require working on each pixel (point in the image). As a result, a huge number of mathematical operations are required. If data are converted into some mathematical representations, this reduces the required number of operations.**Image analysis and classification:**Image analysis and classification depend on similarity between image details. This similarity is sensitive to many factors, such as resizing, rotation, noise contamination, and color changes, while the similarity between mathematical functions could be simpler, faster, and more stable in comparison with the similarity between images. For example, to remove image background, a full analytical process is required to first capture the background and then scan all points to remove the background. If the image is represented as a set of functions, the detection and removal of the background could be simpler because the modification of some coefficients in the function is expected to remove the image background. Our previous work in [25] applied the multichimera transform to image analysis and reconstruction.**Image security:**converting an image block into a mathematical representation provides autocoding for the image because an intruder cannot restore the image without having a particular function library.

#### 2.2. Properties of the Proposed Mathematical Representation

**Efficiency of 2D image representation:**the first condition to satisfy a successful transformation; output values of the mathematical expression should be able to give a three-dimensional surface similar in topology to the original image points with small error.**Simple form functions:**Mathematical expressions or functions should be as simple and commonly used as possible; but this requirement might conflict with the first condition. To solve this conflict, an optimization process could be implemented with some skill and experience to determine the type of function depending on the basis of the transformation process, and taking into account the rest of the conditions.**Suitability for fractals:**An important goal of fractal geometry is to describe images in terms of transformations that in some way keep images unaltered. One of the most common properties of fractal geometry is the complex form that results from a simple process.**Approval for digital:**Logical operations and functions are the simplest mathematical form with the fastest implementation, and are more consistent with the computer language. The set of the 2D binary functions that are employed to represent the image in the proposed transform satisfies this condition (see Section 3.1). These functions would make the conversion process more flexible.

## 3. Systematic Multichimera Transform (SMCT)

#### 3.1. Codebook Establishment

**1D function generation:**a set of 1D binary functions called fractal half functions are generated by firstly choosing a mother (basis) function ($f1$) with all its elements set to ones. This represents the first level. The functions in the second level are generated by dividing the mother function into two halves using binary step functions. A similar procedure is applied to each function in the previous level to generate the functions in the other intermediate levels. Lastly, the function at the last level is generated by setting all its elements to zeros. Figure 2 indicates the generation of the 1D functions for any N. There are N functions separated over Z levels where $Z=\left(lo{g}_{2}N\right)+1$. Except for the last level, there are ${2}^{Z-1}$ functions in each level where Z is the level number. For example, if we consider the value of $N=8$, a set of eight binary 1D functions is derived as follows.

**2D functions generation:**1D functions generated in the previous step are used to construct a set of 2D functions. Each of these 2D functions represents a particular pattern that eventually forms the final codebook. The 2D functions are produced by implementing the outer product between each pair of 1D functions as demonstrated below.

#### 3.2. Image Encoding

**Step 1:**Resize the color image to turn its size into a multiple of N, and then split the new color image into three separate images: red (${I}_{R}$), green (${I}_{G}$), and blue (${I}_{B}$).

**Step 2:**Separate each image (${I}_{R}$, ${I}_{G}$, and ${I}_{B}$) into nonoverlapping blocks of size $N\times N$ as

**Step 3:**From each block in ${I}_{R}$, ${I}_{G}$, and ${I}_{B}$, estimate three coefficient vectors $(l{x}^{c},l{y}^{c},{m}^{c})$ using

**Steps 4–9**.

**Step 4:**Set $B={B}_{p,s}^{c}$ where ${B}_{p,s}^{c}$ denotes the current image block, and $(p,s)$ is the index of the image block over the vertical and horizontal directions, respectively.

**Step 5:**Repeat

**Steps 6–9**for K times.

**Step 6:**Find the maximal value of the block (B), and then multiply it by a constant (T) to estimate the value of m at index k as

**Step 7:**Compare the image block (B) with each block in the codebook (CB) by multiplying (CB) by ${m}_{k}^{c}$ and then calculating the mean absolute error (MAE) between them using

**Step 8:**Find the index ($x,y$) of the CB block with the minimal MAE (i.e., the best match) to estimate the value of $L{X}^{c}$ and $L{Y}^{c}$ such that $l{x}^{c}$ stores the x values and $l{y}^{c}$ stores the y values as

**Step 9:**Update the values of the image block (B) as

**Step 10:**Split the elements of the coefficient vectors $(l{x}^{c},l{y}^{c},{m}^{c})$ estimated for all image blocks into ${I}_{R}$, ${I}_{G}$, and ${I}_{B}$, and rearrange them into separate matrices to denote the coefficients collected in a particular color channel and in a certain iteration (k), as shown in Figure 4. Each cell in this figure refers to a matrix of size ($P\times S$) pixels. More precisely, matrix $L{X}_{1}^{R}$ contains the values of $lx$ coefficients collected over all blocks in ${I}_{R}$ and at $k=1$. The remaining matrices in Figure 4 are similarly interpreted.

**Steps 1**and

**2**represent the preprocessing stage.

**Steps 3–9**represent the matching stage, and

**Step 10**represents the postprocessing stage. Figure 5 shows the representation of the Lenna image in the SMCT domain. In this representation, we set N = 8, T = 0.25, and K = 4. According to these settings, the algorithm was implemented four times. The representation in Figure 5b consists of three horizontal parts that belong to the representation of the red, green, and blue channels.

#### 3.3. Image Decoding

**Step 1:**Split the image vertically into three identical sections that correspond to the encoded parts of red, green, and blue channels.

**Step 2:**Split each section horizontally into K identical subsections that correspond to the three coefficient matrices that are estimated in each iteration k.

**Step 3:**Split each subsection vertically into three separate units to access the entries of each coefficient matrix $(L{X}_{k}^{c},L{Y}_{k}^{c},{M}_{k}^{c})$ individually.

**Step 4:**To estimate the dimension of the original image, multiply the size of any coefficient matrix by N as

**Step 5:**Set a new empty matrix (A) of size $(Z,V,3)$ and split it into blocks of size $N\times N$.

**Step 6:**Loop over the elements of the $L{X}_{k}^{c}$, $L{Y}_{k}^{c}$, and ${M}_{k}^{c}$ in three color channels and estimate the values of the corresponding blocks in A by multiplying the normalization factor in the M matrix by the codebook block in the index specified by the elements in the $LX$ and $LY$ matrices as

## 4. Experiments and Results

#### 4.1. Length of Coefficient Vector (K) versus Quality and $CR$

#### 4.2. Influence of Parameter T versus Quality

#### 4.3. Parameter K versus Parameter N

#### 4.4. Comparison with Standard Transforms

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMCT | Systematic multichimera transform |

SSIM | Structural similarity index |

PSNR | Peak signal-to-noise ratio |

DWT | Discrete wavelet transform |

DCT | Discrete cosine transform |

WHT | Walsh–Hadamard transform |

KLT | Karhunen–Loeve transform |

JPEG | Joint Photographic Experts Group |

PCA | Principal component analysis |

MAE | Mean absolute error |

CR | Compression ratio |

CB | Codebook |

MSE | Mean square error |

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**Figure 8.**Visual comparison of different transforms. (

**left**) Results of applying DWT; (

**center**) results of applying DCT; (

**right**) results of our proposed transform.

DWT | DCT | SMCT |
---|---|---|

Elements of its (Haar) basis functions consist of −1 and +1 only. | Elements of its basis functions are between −1 and +1. | Elements of its basis functions consist of 0 and 1. |

The energy of its elements is fixed. | The energy of its elements is fixed. | The energy of its elements is decreasing. |

Orthogonal | Orthogonal | Nonorthogonal |

Symmetrical | Symmetrical | Symmetrical |

Image | Quality Metric | K Parameter | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

PSNR | 14.81 | 25.48 | 26.99 | 27.57 | 27.86 | 28 | 28.07 | 28.1 | |

Lenna | SSIM | 0.535 | 0.946 | 0.959 | 0.964 | 0.966 | 0.967 | 0.968 | 0.968 |

CR | 75.42 | 34.1 | 28.57 | 27.21 | 26.55 | 26.24 | 26.04 | 25.94 | |

PSNR | 14.53 | 19.46 | 20.28 | 20.71 | 20.97 | 21.1 | 21.16 | 21.18 | |

Baboon | SSIM | 0.375 | 0.644 | 0.69 | 0.72 | 0.735 | 0.743 | 0.747 | 0.748 |

CR | 77.16 | 21.29 | 17.45 | 16.1 | 15.45 | 15.13 | 14.98 | 14.92 | |

PSNR | 16.03 | 24.38 | 25.36 | 25.7 | 25.83 | 25.89 | 25.91 | 25.92 | |

Peppers | SSIM | 0.736 | 0.939 | 0.95 | 0.955 | 0.957 | 0.957 | 0.958 | 0.958 |

CR | 59.15 | 32.07 | 27.41 | 26.23 | 25.67 | 25.39 | 25.28 | 25.20 | |

PSNR | 14.72 | 27.34 | 29.32 | 29.84 | 30.02 | 30.11 | 30.14 | 30.15 | |

House | SSIM | 0.508 | 0.928 | 0.952 | 0.96 | 0.963 | 0.964 | 0.965 | 0.966 |

CR | 105.41 | 40.85 | 34.63 | 32.94 | 31.97 | 31.53 | 31.14 | 30.96 | |

PSNR | 11.38 | 23.37 | 24.68 | 25.1 | 25.27 | 25.35 | 25.38 | 25.39 | |

Airplane | SSIM | 0.585 | 0.793 | 0.842 | 0.86 | 0.869 | 0.873 | 0.876 | 0.877 |

CR | 105.26 | 40.37 | 30.32 | 28.88 | 28.02 | 27.59 | 27.41 | 27.27 | |

PSNR | 14.12 | 21.93 | 23.09 | 23.59 | 23.81 | 23.92 | 23.96 | 23.97 | |

Lake | SSIM | 0.523 | 0.83 | 0.864 | 0.878 | 0.884 | 0.887 | 0.888 | 0.888 |

CR | 66.62 | 25.99 | 21.74 | 20.63 | 20.09 | 19.88 | 19.78 | 19.76 |

T | Quality Metric | K Parameter | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

0.15 | PSNR | 14.79 | 25.37 | 26.86 | 27.46 | 27.79 | 27.95 | 28.04 | 28.08 |

SSIM | 0.534 | 0.945 | 0.958 | 0.963 | 0.965 | 0.967 | 0.968 | 0.968 | |

0.25 | PSNR | 14.81 | 25.48 | 26.99 | 27.57 | 27.86 | 28.00 | 28.07 | 28.10 |

SSIM | 0.535 | 0.946 | 0.959 | 0.964 | 0.966 | 0.967 | 0.968 | 0.968 | |

0.35 | PSNR | 14.82 | 25.50 | 27.00 | 27.55 | 27.82 | 27.95 | 28.00 | 28.02 |

SSIM | 0.535 | 0.947 | 0.959 | 0.964 | 0.966 | 0.967 | 0.967 | 0.967 | |

0.45 | PSNR | 14.82 | 25.42 | 26.92 | 27.48 | 27.74 | 27.86 | 27.91 | 27.92 |

SSIM | 0.535 | 0.947 | 0.959 | 0.964 | 0.965 | 0.966 | 0.967 | 0.967 | |

0.55 | PSNR | 14.79 | 25.14 | 26.69 | 27.30 | 27.58 | 27.70 | 27.75 | 27.77 |

SSIM | 0.532 | 0.945 | 0.958 | 0.963 | 0.965 | 0.965 | 0.966 | 0.966 | |

0.65 | PSNR | 14.72 | 24.54 | 26.21 | 26.99 | 27.33 | 27.48 | 27.55 | 27.57 |

SSIM | 0.528 | 0.941 | 0.956 | 0.961 | 0.963 | 0.964 | 0.964 | 0.965 | |

0.75 | PSNR | 14.53 | 23.55 | 25.38 | 26.32 | 26.82 | 27.06 | 27.18 | 27.24 |

SSIM | 0.518 | 0.932 | 0.950 | 0.957 | 0.961 | 0.962 | 0.963 | 0.963 | |

0.85 | PSNR | 14.23 | 22.21 | 24.09 | 25.27 | 25.95 | 26.35 | 26.56 | 26.67 |

SSIM | 0.502 | 0.917 | 0.940 | 0.951 | 0.956 | 0.958 | 0.960 | 0.960 | |

0.95 | PSNR | 13.73 | 20.48 | 22.35 | 23.64 | 24.52 | 25.09 | 25.45 | 25.65 |

SSIM | 0.476 | 0.891 | 0.922 | 0.937 | 0.946 | 0.951 | 0.954 | 0.955 |

Image | Quality Metric | Transform | ||
---|---|---|---|---|

DWT | DCT | SMTC | ||

Lenna | PSNR | 26.77 | 27.2 | 27.57 |

SSIM | 0.954 | 0.955 | 0.964 | |

Baboon | PSNR | 20.28 | 20.41 | 20.71 |

SSIM | 0.655 | 0.657 | 0.720 | |

Peppers | PSNR | 25.39 | 26.18 | 25.7 |

SSIM | 0.946 | 0.953 | 0.955 | |

House | PSNR | 28.41 | 29.07 | 29.84 |

SSIM | 0.937 | 0.937 | 0.960 | |

Airplane | PSNR | 24.78 | 25.31 | 25.10 |

SSIM | 0.836 | 0.773 | 0.860 | |

Lake | PSNR | 23.00 | 23.61 | 23.59 |

SSIM | 0.849 | 0.852 | 0.878 | |

Average | PSNR | 24.77 | 25.29 | 25.41 |

SSIM | 0.862 | 0.854 | 0.889 |

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**MDPI and ACS Style**

Abdulsattar, F.S.; Zaghar, D.; Khalaf, W. A Systematic Multichimera Transform for Color Image Representation. *Symmetry* **2022**, *14*, 516.
https://doi.org/10.3390/sym14030516

**AMA Style**

Abdulsattar FS, Zaghar D, Khalaf W. A Systematic Multichimera Transform for Color Image Representation. *Symmetry*. 2022; 14(3):516.
https://doi.org/10.3390/sym14030516

**Chicago/Turabian Style**

Abdulsattar, Fatimah Shamsulddin, Dhafer Zaghar, and Walaa Khalaf. 2022. "A Systematic Multichimera Transform for Color Image Representation" *Symmetry* 14, no. 3: 516.
https://doi.org/10.3390/sym14030516