# Symmetry Modulates the Amplitude Spectrum Slope Effect on Visual Preference

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−α}and in turn is a linear function of frequency with spectrum slope −α. In some studies, researchers discuss spectrum slope in the context of fractals, a mathematical construct that can characterize the complexity of an image. The fractal dimension (FD) of an image has a monotonic relationship with spectrum slope [5].

## 2. Materials and Methods

#### 2.1. Ethics Statement

#### 2.2. Participants

#### 2.3. Apparatus

^{2}and mean chromaticity of (0.33, 0.33) in CIE 1931-xy coordinates. The monitor’s refresh rate was 60 Hz. The viewing distance was set such that each pixel made up 1′ of visual angle. The experimental control and the stimulus generation were written in MATLAB with the Psychophysics Toolbox 3 [31].

#### 2.4. Stimuli

^{2}. We then extracted the image’s phase spectrum with a Fourier transform. We next paired this phase spectrum with a preassigned radially averaged amplitude spectrum with a predetermined slope and then applied inverse Fourier transform to convert them back to an image. The amplitude spectrum slope, −α, of the image was from −2 to 0 with steps of 0.5. This spectrum slope range was equivalent to a fractal dimension range of 1.1 to 2 as computed by applying a box-counting algorithm on our stimuli. The relationship between the spectrum slope and fractal dimension was similar to that reported by Spehar and colleagues [32,33] with the synthetic noise patterns.

#### 2.5. Procedures

## 3. Results

_{p}

^{2}= 0.35. The post hoc simple main effect analyses showed significant effects of number of axes for all of the five spectrum slope conditions (all Fs > 36.28, all ps < 0.001, and all η

_{p}

^{2}s > 0.463). On the contrary, although a spectrum slope effect existed for all conditions with symmetric images regardless of the number of axes (all Fs > 5.17, p < 0.014, η

_{p}

^{2}> 0.11), no spectrum slope effect was found in the asymmetric condition, F(1.56, 65.57) = 0.32, p = 0.67, η

_{p}

^{2}= 0.008. The main effects for number of axes, F(1.54, 64.85) = 106.42, p < 0.001, η

_{p}

^{2}= 0.72, and spectrum slope, F(1.40, 58.85) = 11.66, p < 0.001, η

_{p}

^{2}= 0.22 for slope, were also significant.

## 4. Discussion

^{2}+ c, where p is the preference rating, s is the spectrum slope, and b, a, and c are parameters that determine the shape, horizontal displacement, and vertical displacement of the function, respectively. The values of the parameters in the function are a function of the number of symmetry axes, and they can therefore help with assessing its effects. The parameter c determines the function’s vertical displacement and thus represents an additive effect. Hence, on the one hand, if the effect of the number of symmetry axes is independent from that of spectrum slope, then we should expect that only parameter c would change with the number of axes. On the other hand, if the two factors interact, then we should expect at least one of the parameters a and b to change with the number of symmetry axes because each could multiply s in the function to produce a modulation effect.

^{2}+ d × s + c, in which d is a free parameter. The linear term d × s reflects the asymmetry effect on the inverted U-shaped function. Hence, if asymmetry indeed exists in the inverted U-shaped preference function, then one should expect the parameter d to be significantly large. However, the model-fitting results showed that the parameter d for each curve was between −0.079 and 0.052, quite close to zero. Including this linear term did not significantly improve the fitting results, F(2,19) = 0.51, p = 0.81. Thus, a linear term is not necessary to account for our result. Hence, the preference function for spectrum slope is symmetric.

_{p}

^{2}= 0.325. The pairwise comparisons showed significant effects between all medium slope (−1.75 to −1.25) and shallower slope (−0.5 to 0; all ps < 0.05) pairs, except for between −1.75 and −0.5 slopes. No other pairwise comparisons were significant. This result was fitted by the quadratic function p = b(s − a)

^{2}+ c, where s is the spectrum slope, p is the preference rating, and a, b, and c are constants. We empirically found that the function p = −0.78 (s + 1.71)

^{2}+ 3.73 can fit our data well. This showed that the relationship between preference ratings and the images with different spectrum slopes was an inverted U shape that peaked at about −1.7.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Sample of the stimuli. The stimuli are the combinations of different amplitude spectrum slopes (from 0 to −2) and numbers of axes (from 0 to 16). See text for details.

**Figure 2.**Preference ratings under different combinations of spectrum slope and number of axes. The gray symbols represent the data points for the asymmetric condition, as well as red symbols for one axis, blue for two axes, green for four axes, cyan for eight axes, and pink for 16 axes. The error bars represent the standard error. The smooth curves are quadratic function fit.

**Figure 3.**Differential preference rating between the −1 and 0 spectrum slope conditions as a function of the number of axes.

**Figure 4.**Effects of number of axes on model parameters. (

**a**) The modulation depth (parameter b) decreased exponentially with the number of axes as −1.56 × exp(−1/n). (

**b**) The intercept of the quadratic function (parameter c), which represented the number of axes’ main effect, increased with the number of axes as 3.05 × exp(−1.56/n) + 1.97.

**Figure 5.**Effects of parameter b on preference ratings. When the parameter c = 0, the negative parameter b makes the quadratic functions for all symmetry axis conditions below 0.

**Figure 6.**Preference ratings under different spectrum slopes. The red symbols indicate the data points for different spectrum slopes. The error bars are the standard error. The smooth curve is the fit of the quadratic function.

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Wu, C.-C.; Chen, C.-C.
Symmetry Modulates the Amplitude Spectrum Slope Effect on Visual Preference. *Symmetry* **2020**, *12*, 1820.
https://doi.org/10.3390/sym12111820

**AMA Style**

Wu C-C, Chen C-C.
Symmetry Modulates the Amplitude Spectrum Slope Effect on Visual Preference. *Symmetry*. 2020; 12(11):1820.
https://doi.org/10.3390/sym12111820

**Chicago/Turabian Style**

Wu, Chia-Ching, and Chien-Chung Chen.
2020. "Symmetry Modulates the Amplitude Spectrum Slope Effect on Visual Preference" *Symmetry* 12, no. 11: 1820.
https://doi.org/10.3390/sym12111820