Maximizing Image Information Using Multi-Chimera Transform Applied on Face Biometric Modality

Abstract

With the development of mobile technology, the usage of media data has increased dramatically. Therefore, data reduction represents a research field to maintain valuable information. In this paper, a new scheme called Multi Chimera Transform (MCT) based on data reduction with high information preservation, which aims to improve the reconstructed data by producing three parameters from each $16\times 16$ block of data, is proposed. MCT is a 2D transform that depends on constructing a codebook of 256 picked blocks from some selected images which have a low similarity. The proposed transformation was applied on solid and soft biometric modalities of AR database, giving high information preservation with small resulted file size. The proposed method produced outstanding performance compared with KLT and WT in terms of SSIM and PSNR. The highest SSIM was $0.87$ for the proposed scheme MCT of the full image of AR database, while the existed method KLT and WT had $0.81$ and $0.68$, respectively. In addition, the highest PSNR was $27.23$ dB for the proposed scheme on warp facial image of AR database, while the existed methods KLT and WT had $24.70$ dB and $21.79$ dB, respectively.

1. Introduction

Feature detection is considered a major step of image processing and computer vision for image compression, pattern recognition, image classification, and image registration. These features have to be invariant to specific spatial transformations such as rotation, translation, and scaling [1]. The concept of feature detection has been used in different fields and different applications such as medical images [2], remote sensing [3], face recognition [4], human shape recognition [5], indexing and image retrieval [6]. Feature detection builds the feature representation for any image and searches for the corresponding similarity between the pixels for the same scene depending on the salient features that are extracted from the image to increase the valuable information with reduced data [7].

Many transformations are used in image processing and data analysis, e.g., Karhunen-Loeve Transform (KLT), Wavelet Transform (WT), Discrete Cosine Transform (DCT), Discrete Fourier Transform (DFT), and Chimera Transform (CT), which reduce the amount of the raw data or extract low level features to be used for further processing and detection [8,9,10,11,12]. KLT is considered a representation of a stochastic process for an infinite linear combination of orthogonal functions; it creates the most precise illustration (with the assumption that the error has the same power in all dimension) of the original data. This transform spreads an input data in terms of a set of basis, which is called “eigenvalues”, where it uses only few KLT coefficients to represent the most amount of the data [8]. Therefore, KLT can be used for features dimensionality reduction, while it could not cope with clusterization tasks [13]. WT decomposes the image sub-blocks into different levels of resolution, by taking only the low frequency band to introduce a new feature set; by doing this, the wavelet coefficient size is reduced, and a compressed size of feature vector is produced that can be applied for any classification tool [14]. However, the number of computations increased dramatically when a multi-level WT is applied on the data. DCT demonstrates a finite sequence of data points as a sum of cosine functions oscillating at different frequencies; as the frequency increases, the coefficient’s value decreases, and vice versa. Applying DCT for the input image and taking only some coefficients, since the DC component and low frequency components carrying the maximum information, this compressed vector can represent the original image, and any classification model can be applied to this reduced size vector [15]. Fractal coding is an effective tool for describing image textures. This transformation extracts the DC component of each image block, and then calculates the contrast coefficients recursively to find the best geometric shape of the block. For each range block, the corresponding best matching of domain block can be estimated from the pool of domain block using self-similarity technique. On the fractal decoding side, the contrast coefficients and DC component are recursively applied to reconstruct the image (blocks), and consequently, the original image is constructed [16,17].

Hence, most image processing fields try to achieve maximum information with minimum data using different schemes. Lossless image compression techniques aim for maximizing the information rather than minimizing the data, while lossy image compression approaches trying to optimize these two factors [18]. Image data reduction techniques such as PCA, KLT, and feature detection target are considered a preprocessing step that is used to minimize the data rather than maximize the information [19].

Some other factors are used to measure the efficiency of the transform, such as complexity and execution time; however, other metrics need to be calculated such as the content of minimum data metric, which is the ratio between the input ($8\times 8$ or $16\times 16$ blocks) and the output (a small number of integer values) of the process, and maximum information metric, which represents Peak Signal to Noise Ratio (PSNR) and/or Structure Similarity Index Measure (SSIM).

In this article, a scheme is proposed to obtain the maximum information from only a small amount of image data using similarity and statistical operations. This scheme is called Multi-Chimera Transform (MCT), which is used to represent each input data of a $16\times 16$ block using 3 integer values (output) with a valuable information contents (PSNR and SSIM). To show the effectiveness of the proposed scheme (MCT), we applied it to biometric database for extracting useful information from facial image. The method was compared with KLT, which is considered as a powerful transform for information extraction, and WT, which is the most used transform for information compression.

Kukharev et al. presented a two-step approach that utilized as a first step Karhunen-Loeve Transform (KLT) because it gives a good approximation of input data but with large number of eigenvalues and as the second step Linear Discriminant Analysis (LDA) to reduce the data dimensionality. To show the useful advantage of their method, they applied it on some image data and proved the method has a high compression level and optimal representation in feature-space for image recognition and browsing large graphical databases [13]. In [20], Haar-like features are used as a detection technique, whereas the recognition stage of the system is based on eigenimages and principle component analysis. Hafed and Levine confirmed the effectiveness and robustness of using the discrete cosine transform for feature extraction and showed that the KLT is more computationally expensive in comparison with DCT regarding training [15]. Hui and Zak arranged the magnitudes coefficients of the Fourier transform, which is applied to the patterns of the data base, in a vector in descending order and selected the first P largest coefficients for use in further processing instead of the original patterns. P is defined as a design parameter, and the vector of P descending coefficients is referred to as the signature vector of the corresponding pattern [11]. Aznaveh et al. decreased the large amount of memory required to save the face database by applying KLT (also called Hotelling transform) to each block of the image database separately [21].

This work proposes a robust scheme for maintaining valuable information with data reduction, which can be applied on different applications such as object recognition, and biometric verification system. However, for evaluation purposes, our study focuses on face biometric modality with the existence of occlusion using the proposed data reduction transform MCT. The following four contributions are made:

• We suggest a novel scheme for image transform, which can be used for data reduction with maximum information.
• A large-scale evaluation of the proposed scheme using 5913 facial images was performed. The warp faces of AR had (2600) images, the full image of AR had (3315) images, and the full image from websites had (16) images.
• Implement a proposed scheme, which is invariant for subject diversities (142 persons), light conditions (right side, left side, and both sides), facial expression (neutral, smile, anger, scream), and facial occlusions (prescription eyeglasses, sunglasses, scarf, and hat) within different sessions (two sessions 14 days apart).
• The proposed scheme successfully maintained the information for solid biometric such as face and ocular biometric modalities, and it could be applied to preserve the information of soft biometric modalities such as eyebrow, eyeglasses, sunglasses, scraf, and hat.

The remained of this paper is organized as follows. Section 2 focuses on the problem of the data reduction. Section 3 introduces the suggested scheme. Section 4 represents the study results and discussions. Section 5 states the conclusion and the future work.

2. Problem Statement of Image Data Reduction

A new strategy to transfer an image from spatial domain to a new innovator (Chimera) domain is called Multi-Chimera Transform (MCT). In general, traditional transforms such as DCT and WT, transform the image data from spatial domain to its domain using a set of orthogonal functions and the inverse of this process. These are shown in Equations (1) and (2), respectively.

$$c_i=\Psi \left(i\right)\underset{i=1}{\overset{n}{⊛}}D$$

(1)

$$d_i=\Psi {\left(i\right)}^{-1}\underset{i=1}{\overset{n}{⊛}}C$$

(2)

where, $D=[d_1,d_2,\dots ,d_n]$ is a vector of input data (data in spatial domain) and $C=[c_1,c_2,\dots ,c_n]$ is a vector of coefficient (data in innovator domain). $\Psi$ is a set of orthogonal function and ${\Psi }^{-1}$ its inverse (also a set of orthogonal function) and ${⊛}_{i=1}^n$ is the convolution process.

The orthogonality of any transform, constraints it to use a unified type of function and therefore a limited number of useful patterns in 2D images will be resulted. Chimera Transform overcame this challenge by proposing a hybrid set of patterns in form of small image blocks called code-book to increase the information in Chimera domain [12].

2.1. The Concept of Multi-Chimera Transform (MCT)

Chimera Transform (CT) generated only one coefficient for each block. This could be considered as a weakness issue of Chimera Transform as it used a small block size of $4\times 4$. This small block size is beneficial for image compression; however, it generates a large amount of redundant information for feature detection. Therefore, a large block size of $16\times 16$ could reduce the number of features; however, the reconstructed image would be degraded. To enhance the reconstructed image, the Chimera Transform can be applied many times to generate what is called a Multi-Chimera Transform (MCT).

Equations (3) and (4) are the forward process, while, Equation (5) represents the inverse process.

$$\begin{array}{c}{a}_1=min\left\{D\right\}\\ b_i=max\{D-R_i\},i=1,2,...,n\\ c_i=argmax\left\{{\Psi }_i⊛(D-R_i)\right\},i=1,2,...,n\end{array}$$

(3)

$$R_i=\left\{\begin{array}{cc}0,\hfill & i=1.\hfill \\ a_1+\sum _{j=1}^{i-1}\left[b_j\times f\left(c_j\right)\right],\hfill & i=2,...,n.\hfill \end{array}\right.$$

(4)

$$D_r=a_1+\sum _{i=1}^n\left[b_i\times f\left(c_i\right)\right]$$

(5)

where $a_1$, $b_i$, and $c_i$ are the MCT parameters, D is input sub-image (block), $f\left(c_j\right)$ is the intended function of the codebook library for the index $c_j$, and $D_r$ is the reconstructed block, ${\Psi }_i$ is the codebook library, and n is the number of iterations in which two or three iterations were used. Figure 1 shows the whole process of the proposed scheme of a Multi-Chimera Transform (MCT).

2.2. Codebook Generation of MCT

The codebook library (${\Psi }_i$) is a set of $2D$ functions $f\left(c_j\right)$, which were stated in Equations (3)–(5). These equations represent the core of MCT. Consequently, this library determines the tendency of the MCT field. The essential concept of generating these function sets are the Vector Coding (VC) technique. VC selects some samples of blocks from a set of images to generate the codebook. This approach selects the most repeated patterns of standard images. Thereafter, some blocks will be reconstructed without error while the majority of the blocks will be approximated and replaced with the most similar available pattern in the codebook. In general, VC is considered a simple approach but with high error in the reconstructed image; thus, it needs a large codebook library to reduce the aforementioned error.

MCT introduces a novel technique to generate new patterns for optimizing the output error in the reconstructed image. The codebook was generated from 16 selected images of the same field. A total of 16 facial images were collected from different ethnicity and light conditions to construct the codebook which was used for facial biometric modality. A large set of data was composed from these selected images to obtain a small subsets of data. These subsets contain the most similar patterns, with one pattern kept from each subset. As a result, a diverse pattern is maintained in the codebook. This process will be discussed in the following section.

3. The Proposed Approach

The overall steps within the proposed work are shown in Figure 2. These consist of mask bank generation, parameter generation, and image restoration, which will be explained in the next sections.

3.1. Mask Bank Generation

The first stage in the proposed approach is mask bank creation. This stage involves progressive steps to create a mask bank, which is used to generate the required parameters for the proposed work in the next stage. The steps are block generation, similarity score creation, and mask bank implementation. Figure 3 shows the steps of the mask bank generation stage, where R1 × C1 is the image dimension, and Ch1 is the gray value of colored image.

Initially, sixteen images of human faces of different ethnicity and varying light conditions were read (as shown in Figure 4A) and resized to $256\times 256$ to generate three masks bank of size $256\times 16\times 16$ by end of this stage. The entire human face image was divided into 256 blocks of size $16\times 16$, as shown in Figure 4B. This procedure was repeated on the sixteen facial images to obtain a generated block of size $4096\times 16\times 16$. The 4096 generated blocks were collected from these 16 facial images as a result. Consequently, a 256 blocks were constructed from each image as shown in Figure 4. Then, a Gaussian low pass filter was applied on each created block to smooth each individual entity. Then, the filtered blocks were normalized using min-max normalization according to the Equation (6).

$$BLKnor{m}_i=\frac{Bl{k}_i-min\left(Blk\right)}{max\left(Blk\right)-min\left(Blk\right)}$$

(6)

where $Blk$ is the input block, $BLKnor{m}_i$ is the output normalized block and $Bl{k}_i$ is the $16\times 16$ block.

Subsequently, a Mean Absolute Error (MAE) was calculated for these 4096 normalized generated blocks of size $16\times 16$, in order to build the MAE matrix of size ($4096\times 4096$), as outlined in the Equation (7). The MAE matrix was implemented to allocate the similarity between the generated blocks which were acquired from these sixteen facial images.

$$MA{E}_{i,j}=\frac{{\displaystyle \sum _{i=1,j=1}^{Bn}}abs(B_i-B_j)}{\left|\right|B_j\left|\right|}$$

(7)

where $MA{E}_{i,j}$ is the generated mean absolute error, $B_i$ and $B_j$ are block vector of size ($16\times 16$), and $Bn$ is the number of blocks.

A total of three masks of size ($256\times 16\times 16$) were selected for each row in MAE matrix. The 16 most similar values out of 4096 were selected according to the following strategies:

• The most similar (minimum MAE) value was kept in the first mask bank while the other 15 values were excluded.
• The most similar (minimum MAE) value was subtracted from the most dissimilar (maximum MAE) value; thereafter, the normalized subtracted value was kept in the second mask bank while the other 14 values were excluded.
• The seventh similar value was subtracted from eighth similar value then the subtracted value was kept in the third mask bank and the other 14 values were excluded.

Figure 5 shows these three masks of size $256\times 16\times 16$, which were concatenated into a $4D$ matrix of size $256\times 16\times 16\times 3$, to form the resultant mask bank which was utilized by the next two successive stages.

3.2. Parameter Generation

The second stage of MCT is the parameter generation, which is shown in Figure 6 (the second stage of multi-chimera scheme). This stage consists of four steps which are; block generation, block normalization, similarity allocation, and statistical calculation. To this end, three parameters were generated from each acquired block of the input test image. These parameters are minimum value (P1), maximum value (P2), and the index (P3) of most similar block of the mask bank, where P1 is the matrix of $a_1$, P2 is the matrix of $b_i$, and P3 is the matrix of $c_i$.

In this stage, the test facial images were obtained from the AR database [22] with two different image size sets, which were the warp faces (2599 facial images) and full faces (3313 facial images). Firstly, the test image of size $R2\times C2$ was converted into $Bn=R2\times C2\times 256$ blocks of size $16\times 16$, then the generated blocks were normalized. Then, the similarity was allocated between the mask bank and the recently generated normalized block. Finally, the column index of most similar blocks which had the lowest value of MAE among other values, was saved into the P3 parameter.

On the other side, a statistical calculation was applied on the generated blocks. Two statistical calculations were used in this stage which firstly included the minimum value of each block of size $16\times 16$ (to obtain P1 parameter). A second statistical calculation considered the maximum of each block which was subtracted from the minimum applied on the same block to get the P2 parameter.

Typically, the procedure of this stage could be repeated N times. However, the procedure was repeated three times for this proposed scheme, and for each iteration a certain mask bank of size $256\times 16\times 16$ was used. For instance, for the first iteration, the first mask (out of three masks that were stored in the bank) was used with the test image to allocate the similarity between the input image and the saved mask bank. In the same manner, for the second iteration, the second mask bank was used with the new subtracted image instead of the test image to allocate the similarity between the new subtracted image and the stored mask bank. This subtracted image was obtained from the next stage (explained in the next section). Figure 7 shows the processes of parameter generation with the first iteration.

3.3. Image Restoration

The final stage of the proposed method was the image restoration, as shown in Figure 8. This stage involved a number of subsequent steps to achieve two goals. The first goal was to restore the test image, and the second one was to evaluate the proposed method.

The first step of image restoration was the construction of the selected mask. To this end, the block index parameter P3 allocated to the most similar block in the mask bank of size $256\times 16\times 16$. Then, this picked block of size $16\times 16$, was repeatedly accumulated to obtain the selected blocks of size $Bn\times 16\times 16$, where, $Bn$ represents the number of collected blocks and the calculation of $Bn$ was outlined in Equation (8).

$$B_n=\lceil \frac{R2\ast C2}{256}\rceil$$

(8)

where R2 and C2 are the number of rows and columns of the input image, respectively.

Thereafter, the resultant blocks $Bl{k}_{Rest}$ of size $Bn\times 16\times 16$ were restored using the selected mask. $Mas{k}_{Selected}$, P1, and P2 are as outlined in Equation (9). Consequently, the restored blocks of size $Bn\times 16\times 16$ were arranged into the original size of the input image (R1 × C1).

$$Bl{k}_{Rest}=Mas{k}_{Selected}\times P2+P1$$

(9)

The restored image was evaluated visually using two metrics; namely, Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM). The evaluation process will be explained in detail in the next section.

As outlined in the previous stage (parameter generation), the image restoration stage was repeated N times. However, this stage was designed to be repeated three times. For each iteration, the recent restored image was subtracted from original image; then, the resultant image was normalized and sent to previous stage to generate new parameters ($P_1$, $P_2$, and $P_3$). Figure 9 shows a visual sample for this process. In addition, the coded image was constructed using Equation (9).

4. Experiments

This section describes the AR database [22] that was used for evaluation. A comparative evaluation was applied between the proposed approach and the standard transforms such as Karhunen–Loève Transform (KLT) [23] and Wavelet Transform (WT) [24]. Further, two matrices were used to measure the performance of our proposed method and to compare the proposed MCT with the aforementioned transforms. All the experiments have been implemented by dividing each image of size $R2\times C2$ into $Bn\times 16\times 16$ input blocks. Only three parameters of size $Bn\times 3$ were used to obtain the reconstructed blocks.

4.1. Databases

The AR database [22] contains over 4000 colored facial images. These images include different facial expressions which were collected from 126 subjects (70 men and 56 women) during different illumination conditions and occlusions. A total of three occlusions were involved in this database; namely, prescription eyeglasses, sunglasses, and scarf. These occlusions represent the main challenge for facial biometric modality since they represent important parts of the human face. In addition, four facial expressions were captured including neutral expression, smile, anger, and scream. Additionally, three light conditions were applied to the database; namely, left light, right light, and all side light. This database was collected in two sessions separated by 14 days. Two subset were included in this database, firstly, 2600 warp facial images of size $120\times 165\times 3$ as (bmp) image format. The second subset was 3315 facial images of size $768\times 576\times 3$ as (raw) image format. Moreover, 16 images of different sizes and image formats were collected randomly, to expand the visual evaluation.

4.2. Visual Comparative Evaluation

A visual comparative evaluation was applied on the warp subset of AR database as shown in Figure 10. This evaluation involved different subjects (people), facial expressions, occlusions, sessions and light conditions. The proposed scheme (MCT) demonstrates powerful visual reconstruction versus the KLT and WT transforms.

Figure 10A shows an evaluation of the scheme, which was applied to neutral facial expression without any occlusion. Similarly, Figure 10B involves a prescription eyeglass occlusion while Figure 10C,D show other types of occlusion including sunglasses and scarf. Figure 10E–G show evaluations which were applied on three facial expressions—screaming, smiling, and anger, respectively. Figure 10H was applied on two types of occlusion; namely, prescription eyeglasses and scarf.

The suggested scheme shows a reasonable visual performance on the solid biometric modalities such as human face and ocular region. In addition, the suggested scheme maintains the visual features of the soft biometric modalities such as eyebrows, prescription eyeglasses, sunglasses, and scarf. As result, the suggested scheme could be applied as an information extractor method for solid and soft biometric identification.

In the same manner, Figure 11 shows a visual comparative evaluation applied on the full image of the AR database. This evaluation included the similar diversity shown in Figure 10; that is, different people, facial expressions, occlusions, sessions, and light conditions. It is felt that the proposed scheme (MCT) shows a visual robustness in comparison with KLT and WT transforms.

Figure 11A shows a reconstruction for a neutral face expression, while Figure 11B–D show a reconstruction for an occluded face with thin prescription eyeglasses, sunglasses and scarf, respectively. Figure 11E–G show a reconstruction for three facial expressions, namely, screaming, smiling, and anger, respectively. Finally, Figure 11H shows a reconstruction for double occlusions, prescription eyeglasses and scarf.

In the a similar manner, Figure 12 shows a visual comparative evaluation was applied to the random full image selected from websites. This evaluation included four facial expressions that are mentioned in previous figures (Figure 10 and Figure 11). In addition, two occlusions were included in this evaluation that were prescription eyeglasses and hat.

Figure 12A shows a reconstruction for neutral face expression, while Figure 12B–D shows a reconstruction for three facial expressions; namely, screaming, smiling, and anger face. Figure 12E,F shows a construction for two occluded faces by thin prescription eyeglasses and head hat, respectively.

The suggested scheme (MCT) indicates the superiority of reconstruction for solid biometric modalities (face and ocular region) and for soft biometric modalities (eyebrow, prescription eyeglasses, sunglasses, scarf, and head hat) over the KLT and WT. Meanwhile, the aforementioned evaluations were applied on 5931 facial images which included 2600 warp faces of AR database, 3315 full images of AR database, and 16 random full images from different websites.

4.3. Metric Comparative Evaluation

Figure 13 shows an evaluation applied on 2600 warp facial images inside the AR database. In this evaluation, two metrics were used to measure the transforms performance which were Peak Signal to Noise Ratio (PSNR) and Structure Similar Index Measure (SSIM). In Figure 13A, the PSNR for MCT of the 2600 warp facial images surpassed the existed methods of KLT and WT for all images. Also, the average PSNR for MCT was $27.2268$ while the average PSNR for KLT and WT was $24.6990$ and $21.7893$, respectively.

On the other hand, Figure 13B shows a powerful structure similarity for MCT between the 2600 reconstructed warp facial images and the original images with an average SSIM of $0.80936$. In comparison, the average SSIM for the existing KLT and WT methods was $0.73963$ and $0.40922$, respectively. Further, the reconstructed MCT kept about $81\%$ of warp facial information, while the KLT and WT kept about $74\%$ and $41\%$ of facial information, respectively.

Figure 14 shows a random view from the previous evaluation shown in Figure 13. Eight random warp facial images were selected out of 2600 facial images of the AR database to show a representative view of the difference between the proposed MCT and the existing KLT and WT methods in respect to PSNR and SSIM. Additionally, sensitivity of PSNR and SSIM for different facial expression and facial occlusion are shown. Images 1, 2, 5, and 7 show the metric sensitivity for screaming, neutral, anger, and smile face. Meanwhile, images 3 and 8, show the metric sensitivity for persons wearing a prescription eyeglasses as a software biometric.

Figure 15 shows an evaluation applied on the 3315 full facial images inside the AR database. Two metrics were used; namely, PSNR and SSIM. PSNR evaluation is shown in Figure 15A in which the KLT outperforms MCT and WT with a PSNR average of $22.3969$ while the other two methods had values of $21.7271$ and $20.0283$, respectively. However, the structure similarity of the proposed MCT still had the highest SSIM over the other two existing methods for the entire 3315 full facial images. In Figure 15B, MCT had an average SSIM of $0.86693$ while KLT and WT had a SSIM of $0.81153$ and $0.67574$, respectively. Therefore, the reconstructed full facial images kept about $87\%$ of the original facial information under the proposed method, while the KLT and WT kept about $81\%$ and $68\%$, respectively.

Figure 16 shows an evaluation applied on 16 random full facial images selected from different websites. A similar manner of evaluation which was applied as per Figure 14; this demonstrates a close sensitivity of PSNR and SSIM for different facial expressions and facial occlusions in the full facial images. For instance, images 2, 5, 7, and 15 show sensitivity for PSNR and SSIM for different facial expressions; namely, neutral, anger, smile, and screaming face. Meanwhile, the images 4 and 9 show sensitivity to PSNR and SSIM for two facial occlusion; namely, prescription eyeglasses and head hat. In most cases shown in Figure 16, the proposed MCT shows superior outstanding performance over KLT and WT.

Figure 17 shows a metric comparative evaluation for datasets with respect to PSNR and SSIM. Figure 17A shows a PSNR comparison between three datasets in which the proposed MCT of warp facial images got a highest value of $27.2268$. The MCT, which was applied on full images of the AR database and websites, obtained values of $22.3996$ and $26.0928$, respectively. The MCT application to warp facial images overcame the KLT and WT of warp facial images within the same dataset. However, the MCT of the full images of AR database was dropped by $0.6725$ in comparison with KLT of the full images of AR database. This was affected by the image background, which could be similar in some cases to human skin color and to fabrics of some clothes (these two parameters do not exist in the warp or the AR database for the same subjects). Additionally, the MCT was superior in comparison to both KLT and WT for full images of websites.

In addition, Figure 17B shows the superiority of the proposed MCT over KLT and WT in respect of SSIM. Therefore, the structural similarity between reconstructed images of MCT and the original images in the entire used datasets was the highest for all methods. It was noted, for the warp face of AR, the SSIM for MCT was $81\%$, while KLT and WT were $74\%$ and $49\%$, respectively. In respect of the full images of AR, SSIM for MCT was $87\%$ while it was $81\%$ and $68\%$ for KLT and WT. Finally, the MCT of full images from websites had $77\%$, while KLT and WT had $74\%$ and $51\%$, respectively.

4.4. Complexity and Time Comparative Evaluation

In general, the calculation of the transform complexity is manifold and ambiguous; so, in this work, the complexity metric will be summarized as a number of the mathematical operations. This factor is ambiguous since it comprises mathematical operations such as addition and multiplication, and the input of these operations could be an integer or floating numbers. As a result, the number of mathematical operations is not an even-handed evaluation metric however, it can be considered as an acceptable metric for transform complexity [25].

At this point, the complexity could be estimated for each block of size $N\times N$ as follows, for $2D$ four levels, Haar WT the complexity (C(WT)) has $O\left(8N^2\right)$ and the KLT complexity (C(KLT)) has $O\left(N^3\right)$ for coding and decoding sides. The coding complexity of MCT (C(MCTc)) has $O\left(2N^3\right)$ and the decoding complexity of MCT (C(MCTd)) has $O\left(4N\right)$. To recap, the complexity of coding operation is sorted as C(MCTd) < C(WT) < C(KLT) < C(MCTc).

However, because modern computers use parallel processing to execute the $N\times N$ block operation simultaneously, the transform, which is dependent on block operation (which both MCT and KLT and are dependent on), could be time superior.

On the other side, some transforms that depend on point operation (such as WT) rather than block operation could be inferior. Therefore, the time execution of MCT and KLT would be faster than WT.

Table 1. Execution time for the proposed MCT and the existed methods KLT and WT.

Dataset	Coding Time			Decoding Time
	MCT	KLT	WT	MCT	KLT	WT
Warp Face AR	0.574	0.041	2.284	0.008	0.013	0.395
Full Image AR	2.801	0.040	9.959	0.048	0.027	1.728
Full Image Web	0.918	0.023	3.591	0.015	0.013	0.633
Average	1.431	0.035	5.278	0.024	0.018	0.919

shows the coding, decoding, and the total time for the proposed MCT, KLT, and WT transforms on three datasets. It was calculated using an Intel(R) Core(TM) $i7-4712HQ$ CPU $2.30$ Ghz, and 16 GB of RAM.

Figure 18A shows the execution time of coding processes with an average of execution time for WT which was higher than the MCT by 4 times. The average execution time of MCT was higher than the KLT by 40 times. Figure 18B shows the execution time of the decoding processes for which the average of execution time for WT was higher than the MCT by 38 times, while the average execution time of MCT was higher than the KLT by $1.3$ times. Moreover, the decoding time is considered more valuable than the coding time, since the decoding operation could be used more frequently whilst the coding operation could be used just once.

5. Conclusions and Future Work

In this paper, a new transform was proposed for data reduction and was applied on large evaluation of the AR database (warp face and full image face) along with selected facial images from websites. The reconstructed images from the proposed transform have a higher SSIM and PSNR values over the KLT and WT methods.

A large set of data was represented by a small number of parameters with valuable information preserved. The proposed MCT can be used as an initial stage evaluation for different applications such as biometric verification system, object recognition, semantic segmentation, and image retrieval.

MCT is considered superior to other ranking transforms such as KLT and WT within appointed pattern (warp faces), since KLT and WT are general transforms, whilst MCT is a transform that depends on a particular codebook that is related to a particular field. Specifically, the proposed scheme could be improved through the constructed codebook unique to operation, in contrast with KLT and WT which have a stationary process.