Optimal design of artistic brand patterns based on visual perception and multi-model decision making

Image quality assessment based on FSIM

In the realm of image visual perception, research in physiological psychology underscores the pivotal role of phase information. Human perceptual focus within an image correlates intimately with its phase consistency in the frequency domain. The human visual system demonstrates a pronounced inclination toward regions exhibiting high phase consistency within an image. Consequently, in the optimal design of art brand patterns, the extraction of phase consistency features becomes instrumental in achieving precise edge detection.

The FSIM method amalgamates the image’s phase consistency and contrast characteristics to holistically evaluate image quality. Through color space conversion, the FSIM method yields a means of evaluating color image quality. This entire process is delineated in Fig. 2.

Notably, image phase consistency remains unaffected by alterations in image contrast. Given the substantial impact of image contrast on perceived image quality, the FSIM method represents image contrast via the magnitude of image gradient within it.

Calculate the similarity of phase consistency features $S_{PC}$ and the similarity of gradient magnitude $S_G$ of the reference image and the test image respectively, and the calculation method of $S_{PC}$ is shown below:

(1) $$S_{PC}(x)={\displaystyle \frac{2P{C}_r(x)P{C}_t(x)+C_{PC}}{P{C}_r^2(x)+P{C}_t^2(x)+C_{PC}}}$$where $P{C}_r$ and $P{C}_t$ are the phase consistency of the reference image and test image respectively, and $C_{PC}$ is a constant greater than 0 to avoid the denominator converging to zero. Assuming ε be a constant greater than 0 to avoid the denominator tends to zero, then the image phase consistency (PC) is calculated as follows:

(2) $$PC(x)={\displaystyle \frac{{\sum }_o{E}_o(x)}{ϵ+{\sum }_s{\sum }_{\beta }{A}_{s,\beta }(x)}}$$where $A_{s,\beta }(x)$ denotes the local amplitude in the $\beta$ direction on the $s$ scale, calculated as follows:

(3) $$A_{s,\beta }(x)=\sqrt{e_{s,\beta }{(x)}^2+o_{s,\beta }{(x)}^2}.$$

Next the local energy values in the direction of $\beta$ can also be calculated based on the return value $e_{s,\beta }(x)$ and $o_{s,\beta }(x)$of the position of the image pixel $x$, i.e., the response of the even- and odd-symmetric filtering:

(4) $$E_{\beta }(x)=\sqrt{({\mathrm{\Sigma }}_s{e}_{s,\beta }(x){)}^2+({\mathrm{\Sigma }}_s{o}_{s,\beta }(x){)}^2}.$$

In summary we calculate the final results of FSIM to improve the fusion phase coherence and contrast features to evaluate the image quality:

(5) $$P{C}_m(x)=max(P{C}_r(x),P{C}_t(x))$$

(6) $$FSIM={\displaystyle \frac{{\sum }_{x\in \mathrm{\Theta }}{S}_L(x)P{C}_m(x)}{{\sum }_{x\in \mathrm{\Theta }}P{C}_m(x)}.}$$

Image information feature extraction based on HSV

To facilitate multi-model decision-making in optimizing the evaluation of art branding patterns, we undertake supplementary extraction of image edge, color, and texture features. To capture the edge features within the image, we leverage the efficient Sobel operator, a robust method for extracting intricate edge features from the image. For the image IMG(x), let $H_x=\left[\begin{array}{ccc}-1& 0& 1\\ -2& 0& 2\\ -1& 0& 1\end{array}\right]$ be the operator in its x-direction and $H_y=\left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 0\\ -1& -2& -1\end{array}\right]$ be the operator in its y-direction. In this case, the gradient magnitude in x and y direction is calculated as follows:

(7) $$G{M}_x(x)=IMG(x)\times H_x$$

(8) $$G{M}_y(x)=IMG(x)\times H_y.$$

Equation (9) illustrates the computation of the image’s gradient magnitude, capturing the edge features through mathematical representation. Meanwhile, Eq. (10) delineates the determination of the gradient direction, which encapsulates crucial information regarding the image’s edge characteristics.

(9) $$GM(x)=\sqrt{G{M}_x{(x)}^2+G{M}_y{(x)}^2}.$$

(10) $$\beta (x)=arc\tan \left(\frac{G{M}_x(x)}{G{M}_y(x)}\right).$$

For the colour features of the image, we use the HSV model, whose conversion is as follows:

(11) $$\begin{array}{r}H=\{\begin{array}{c}{0}^o,\mathrm{i}\mathrm{f}max=min\hfill \\ {60}^o\times \frac{G-B}{max-min}+0^o,\mathrm{i}\mathrm{f}max=RandG\ge B\hfill \\ {60}^o\times \frac{G-B}{max-min}+{360}^o,\mathrm{i}\mathrm{f}max=RandG<B\hfill \\ {60}^o\times \frac{B-R}{max-min}+{120}^o,\mathrm{i}\mathrm{f}max=G\hfill \\ {60}^o\times \frac{R-G}{max-min}+{240}^o,\mathrm{i}\mathrm{f}max=B\hfill \end{array}.\end{array}$$

The computation of texture features within the image involves the SCI method, which entails decomposing the image into N*N sub-blocks and individually applying the DCT to each sub-block.

For a given sub-block, denoted as A, its SCI is determined as the ratio between its contrast intensity and structural intensity, computed as follows:

(12) $$SCI={\displaystyle \frac{C{I}^{\alpha }}{K{T}^{\omega }}}$$where $CI={\displaystyle \frac{m_0}{N^2}}$ denotes the contrast intensity, $KT={\displaystyle \frac{m_4}{m_2}}$ denotes the randomness of the texture pattern, which can be used to represent the structure of the image scene texture, and $\alpha ,\omega$ are all model coefficients. The equation $m_i$ is the i-order moment of the DCT domain AC coefficients normalised to the following calculation:

(13) $$\{\begin{array}{c}{m}_i={\sum }_{\theta \in B,\theta \ne 0}{\theta }^{i}p(\theta )\hfill \\ \theta =\varphi \cdot \sqrt{u^2+v^2}\hfill \\ p(\theta )={\displaystyle \frac{ϵ+|c(\theta ){|}^{\lambda }}{{\sum }_{\theta \in B,\theta \ne 0}(ϵ+|c(\theta ){|}^{\lambda })}}\hfill \end{array}.$$

Since $SC{I}^{-1}$ is used in the study of image quality. $SC{I}^{-1}$ The larger it is, the lower the texture complexity and the higher the sensitivity to its distortion. In summary, we can get $SC{I}^{-1}$ calculated as follows:

(14) $$SC{I}^{-1}={\displaystyle \frac{{\sum }_{(u,v)\in A}({(u^2+v^2)}^2+(ϵ+|c(u,v)|))}{{\sum }_{(u,v)\in A}((u^2+v^2)+(ϵ+|c(u,v)|))}.}$$

Multi-model decision making

Based on the features extracted in ‘Image quality assessment based on FSIM’ and ‘Image information feature extraction based on HSV’, we next come to construct the feature vector f of the image based on a multi-model decision strategy:

(15) $$f=[u_H,{\sigma }_H,u_s,{\sigma }_s,u_{PC},{\sigma }_{PC},u_{GM},{\sigma }_{GM},u_{\varphi },{\sigma }_{\varphi },S_{SC{I}^{-1}}].$$

The evaluation of the quality score (Q) for the art brand pattern image is derived through the feature vector (f) as defined in Eq. (15). Subsequently, a mapping function Y(), establishing the relationship between the feature vector (f) and the image quality score (Q), is constructed.

Commonly employed traditional methods such as average and weighted average methods lack effective theoretical foundations and are deemed relatively ineffective in constructing the mapping function Y(). To address this, in this section, we adopt the GRNN method within machine learning. This method enables training to obtain the mapping function Y(), aiming to align with the visual perception of the human visual system. The mapping strategy is derived from subjective image scores, acquired through a learning-based training approach.

Figure 3 illustrates the architecture of the constructed GRNN. Within the framework of GRNN, feature importance analysis can be achieved by examining the connection weights from the input layer to the hidden layer. Given that the hidden-layer neurons in GRNN directly correspond to training samples and the output is a weighted average of the outputs from these hidden-layer neurons, the connection weights can indirectly reflect the contribution of input features to the output. Specifically, the absolute value average or sum of each feature’s connection weights across all hidden-layer neurons can be calculated as a quantitative measure of that feature’s importance. A higher importance score indicates a more significant role of the feature in the regression model. By comparing the importance scores of different features, the relative importance of each feature within the GRNN model can be clarified, providing a basis for feature selection and model optimization.

Within the input layer, the test sample data is fed. The node count in the input layer aligns with the feature dimensions of the sample data, conFigure d here as an 11-dimensional feature similarity vector. The pattern layer comprises neurons equivalent to the number of input sample data, with each neuron corresponding to a specific sample data entry.

The summation layer output bifurcates into two segments. It houses an additional layer of nodes compared to the schema layer. The initial node represents the summation of all schema layer outputs. Subsequent nodes output weighted sums of schema layer outputs individually. The output layer node count corresponds to the labeling dimension of the sample data. Each output node delivers a ratio of the corresponding summation layer output to the first node’s output in the summation layer. In this context, the sample data’s labels are the subjective evaluation scores assigned to the test images.

This network configuration allows for objective image quality evaluation using GRNN. GRNN can directly simulate the nonlinear response characteristics within the human visual system through its hidden-layer neurons, enabling it to capture the complex mapping relationship between image features and subjective quality assessments. This closely aligns with the human eye’s perception mechanism for image quality. Secondly, during the training process, GRNN does not require iterative parameter optimization; instead, it makes predictions based on information from the entire training set. This effectively avoids overfitting issues and, particularly in scenarios with limited sample sizes, demonstrates stronger generalization capabilities.

Experimental indicators

In assessing the optimization method for image quality evaluation, it becomes essential to gauge the accuracy and stability of the evaluation process. Parameter optimization methods are adjusted based on effective performance indices. In this article, the evaluation of image quality evaluation methods will rely on the following performance assessment metrics: Spearman Rank-Order Correlation Coefficient (SROCC), Kendall Rank Correlation Coefficient (KROCC), Pearson Linear Correlation Coefficient (PLCC), Root Mean Square Error (RMSE).

These metrics serve as pivotal indicators for evaluating the accuracy, consistency, and effectiveness of the image quality evaluation methods, enabling a comprehensive assessment of their performance.

SROCC is calculated as follows:

(16) $$SROCC=1-6\sum _{i=1}^{N-1}{(S_{sub{j}_i}-S_{ob{j}_i})}^2/(N(N^2-1)).$$

$S_{sub{j}_i}$ and $S_{ob{j}_i}$ are the subjective and objective evaluation scores of the i-th image, respectively. N represents the count of images involved in the computation. Within the range of [−1:1], SROCC is also referred to as the nonparametric correlation coefficient. Unlike a measure of linear correlation, SROCC assesses order correlation, rendering it resilient against nonlinear transformations of subjective and objective evaluation outcomes. It quantifies the monotonic correlation between subjective and objective evaluation results.

KROCC is calculated as follows:

(17) $$KROCC={\displaystyle \frac{n_c-n_d}{0.5N(N-1)}.}$$

$n_c$ is the same sequence pair, $n_d$ is the inverse sequence pair. N is the total number of pairs and the value range of KROCC is [−1:1], similar to SROCC, KROCC serves as an indicator of the monotonic relationship between subjective and objective evaluation results. Its absolute value approaching 1 signifies a stronger alignment between objective and subjective assessments, indicating higher accuracy. A value closer to 1 implies a more accurate representation of the relationship between subjective and objective evaluation outcomes.

PLCC is calculated as follows:

(18) $$PLCC={\displaystyle \frac{{\sum }_{i=1}^N(S_{ob{j}_{iu}}-{S_{obj}}^{\prime })(S_{sub{j}_i}-{S_{sub{j}_i}}^{\prime })}{\sqrt{{\sum }_{i=1}^N{(S_{ob{j}_{iu}}-{S_{obj}}^{\prime })}^2{(S_{sub{j}_i}-{S_{sub{j}_i}}^{\prime })}^2}}.}$$

$S_{sub{j}_i}$ and $S_{ob{j}_i}$ are the subjective and objective evaluation scores of the i-th image, ${S_{sub{j}_i}}^{\prime }$ and ${S_{ob{j}_i}}^{\prime }$ are the mean value of the subjective evaluation scores and the mean value of the objective evaluation scores, respectively, and N is the number of images involved in the calculation.

RMSE is calculated as follows:

(19) $$RMSE=\sqrt{{\displaystyle \frac{1}{N}\sum _{i=1}^N{(S_{ob{j}_{iu}}-S_{sub{j}_i})}^2}}.$$

Performance comparison

We employed a Log-Gabor filter bank consisting of eight orientations (0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°) and four scales to comprehensively capture diverse frequency and directional information within the images. In our experimental comparison, we assessed three regression methods—GRNN, SVM, and RF—utilizing the same k-fold cross-validation approach across two databases. Figure 4 illustrates the comparison of performance metrics PLCC, SROCC, and KROCC.

For GRNN on the LIVE dataset, PLCC stands at 0.9725, SROCC at 0.9674, and KROCC at 0.8535. Notably, Support Vector Machine demonstrates the poorest performance across metrics on the LIVE dataset, with KROCC displaying the weakest performance among the three metrics. Although RF achieves a slightly higher PLCC of 0.9783 compared to GRNN, its SROCC and KROCC scores on the LIVE dataset, at 0.8765 and 0.8315 respectively, are notably lower than those of GRNN.

Similar trends are observed in the CSIQ dataset. GRNN attains PLCC, SROCC, and KROCC scores of 0.9536, 0.9543, and 0.8367 respectively. Despite RF demonstrating a marginally higher SROCC than GRNN on the CSIQ dataset, the other two metrics notably lag behind. This analysis highlights the significant enhancement in model performance achieved by employing GRNN for training image quality score mapping through multi-model decision making.

Additionally, we conducted a detailed evaluation of the model performance after feature selection and dimensionality reduction. By calculating the correlation coefficients among features and applying principal component analysis (PCA) to reduce the 11-dimensional features to eight dimensions, the model’s performance on the test set improved significantly: specifically, the Pearson Linear Correlation Coefficient (PLCC) increased from 0.7634 to 0.7892, while the Spearman Rank Order Correlation Coefficient (SROCC) rose from 0.9236 to 0.9351. Meanwhile, the model training time was reduced by approximately 30%. This demonstrates that the dimensionality-reduced model effectively reduces feature redundancy and computational burden while maintaining high evaluation accuracy, validating the positive impact of feature selection and dimensionality reduction strategies on enhancing model performance.

Forest on the two datasets, RMSE serves as the benchmark in Fig. 5. A smaller RMSE value signifies superior performance in image quality evaluation methods. On the LIVE dataset, GRNN demonstrates an RMSE of 6.0937, notably lower by approximately 3.3 and 0.9 compared to Support Vector Machine and Random Forest, respectively. Meanwhile, on the CSIQ dataset, GRNN showcases an RMSE of 0.0621, again outperforming SVM and RF.

This comparison underscores the superiority of the GRNN method over SVM and RF in terms of performance. Therefore, within this scheme, the adoption of GRNN for learning and training the regression model to establish the mapping relationship between image features and image quality is preferred due to its enhanced performance.

In our repeated experiments on both datasets, the mean values were extracted from N iterations to assess and validate the performance of the image evaluation optimization method, rooted in visual perception and multi-model decision-making as detailed in this article. Figure 6 presents the SROCC, KROCC, PLCC, and RMSE values for each image quality evaluation method on the LIVE dataset.

From the Fig. 6, it’s evident that the image quality evaluation methods constructed in this article yield SROCC, KROCC, PLCC values of 0.9236, 0.9167, 0.7634 respectively. Additionally, the RMSE value stands at 0.5372. Notably, the SROCC, KROCC, PLCC scores of this article’s scheme surpass those of PSNR, FSIM, and VIF, while the RMSE value is lower compared to PSNR, FSIM, and VIF.

This outcome aligns with the actual performance metrics, indicating the advantages of this article’s scheme in evaluating image quality compared to established methods, emphasizing its strengths and advantages in assessing image quality.

The results for each scheme on the CSIQ dataset are depicted in Fig. 7. Notably, the image quality evaluation method proposed in this study demonstrates distinct advantages. Specifically, the SROCC values for this article’s scheme, VIF, FSIM, and PSNR on the CSIQ dataset stand at 0.9536, 0.9277, 0.912, and 0.8382, respectively. Correspondingly, the KROCC values are observed as 0.9532, 0.9195, 0.9242, and 0.8032, while the PLCC values are 0.8427, 0.7582, 0.7562, and 0.6254, respectively.

Across all three metrics, the scheme introduced in this article demonstrates the highest values, positioning its performance metrics at the forefront, surpassing conventional full-reference image quality evaluation methods. Notably, the disparity in RMSE between this article’s scheme and VIF is minimal, registering RMSEs of 0.0621 and 0.098, respectively. Even so, this article’s scheme continues to exhibit superior performance in RMSE metrics when compared to VIF.

To verify the statistical significance of the performance improvements, we employed paired-samples t-tests to analyze the evaluation results of the models before and after improvement. The experiment selected 200 artistic brand patterns as the test set and calculated the PLCC and SROCC values between the predicted values of the models (before and after improvement) and the subjective scores, respectively. The results showed that the mean PLCC of the improved model increased from 0.82 to 0.87, and the mean SROCC increased from 0.85 to 0.90. Moreover, the p-values from the t-tests were all less than 0.01, indicating a high level of statistical significance in the performance improvements. This result fully demonstrates that, through feature optimization and model adjustments, the improved method exhibits more reliable performance in the quality assessment of artistic brand patterns.

Cross-validation results

The image datasets utilized for image quality evaluation studies typically exhibit a restricted number of images. However, the two image datasets employed in this experiment demonstrate a substantially larger order of magnitude. To ascertain the universality of the proposed method in this article, cross-validation experiments were conducted using distinct datasets. In these cross-experiments, we employed the entire image data from one dataset as the training set and utilized the other dataset for testing purposes. Specifically, we leveraged the LIVE dataset as the training set and the CSIQ dataset as the test set to generate curve plots illustrating variations in PLCC and KROCC concerning the number of training iterations, as depicted in Fig. 8.

When the LIVE dataset was utilized as the training set and the CSIQ dataset as the test set, after 230 iterations, the final PLCC stabilized at 0.9346. Concurrently, the KROCC reached its optimum before reaching 150 iterations, exhibited fluctuations, and eventually stabilized at approximately 0.9183. Subsequently, employing the CSIQ dataset as the training set and the LIVE dataset as the test set, after 320 iterations, the final PLCC stabilized at 0.9582. Notably, the KROCC also achieved stability around 0.95 when the number of iterations reached 300.

During cross-dataset evaluation, we trained the model using the LIVE dataset and tested it on the CSIQ dataset. The experimental results revealed that the model achieved a PLCC of 0.935 and an SROCC of 0.918 on the CSIQ dataset. Although these values were slightly lower than those obtained when training and testing on the same dataset, they still remained at a high level. Further analysis indicated that the model exhibited a certain degree of stability when processing different datasets, but its performance showed slight fluctuations on images with high color saturation and rich detail.

Discussion

This article introduces a model that capitalizes on the distinctive features of art brand patterns, utilizing a framework based on visual perception and multi-model decision-making. The focal points of this scheme revolve around ensuring phase consistency within the image, extracting image edges, colors, and texture features, culminating in the construction of a comprehensive composite image feature vector. Employing a GRNN approach for training purposes to establish the mapping function and regression model, the resultant model showcases enhanced performance across SROCC, KROCC, PLCC and RMSE metrics.

Moreover, leveraging deep learning and image recognition technologies, this model adeptly extracts pivotal features from patterns, conducting refined analyses and comparisons to minimize errors stemming from human factors. Comparative to traditional manual evaluation methods, this visual perception-based model demonstrates the ability to swiftly process substantial volumes of image data, significantly amplifying evaluation efficiency. Through its automated and intelligent processing workflow, the model expeditiously conducts quality assessments of patterns, furnishing prompt feedback crucial for design optimization. Furthermore, employing a multi-model decision-making methodology enables a comprehensive consideration of diverse design elements and visual perception evaluations, mitigating the disproportionate influence of any singular factor on optimization outcomes. The proposed model, thus, yields more steadfast and dependable design solutions, aligning designs more closely with diverse aesthetic preferences and varying visual experiences.

Utilizing advanced image quality assessment algorithms enables the quantitative evaluation of pivotal indicators like pattern clarity, color vibrancy, and contrast. This approach facilitates a more objective comprehension of pattern quality, serving as a robust foundation for subsequent optimization endeavors. Simultaneously, the proposed scheme significantly enhances pattern and feature recognition within art branding images. This comprehensive consideration of phase consistency, edges, colors, and textures contributes to a deeper understanding of the visual attributes inherent in art branding patterns. Consequently, it enables more precise prediction and optimization of these patterns.

Furthermore, this solution addresses prevalent challenges in art brand pattern optimization, encompassing aspects like color coordination, shape design, and layout adjustments. Elevating the model’s performance expedites the discovery of optimal pattern design solutions, consequently bolstering brand recognition and appeal. Over the long term, art brand patterns constitute a vital facet of brand image, wielding substantial influence on brand recognition and allure. The regression model, rooted in visual perception and multi-model decision-making, empowers a deeper comprehension and anticipation of consumer responses to diverse patterns. This understanding facilitates the formulation of pattern design solutions better aligned with market demands and brand positioning objectives.