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Prediction of facial attractiveness from facial proportions
Pattern Recognition 45 (2012) 2326–2334
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Pattern Recognition journal homepage: www.elsevier.com/locate/pr
Prediction of facial attractiveness from facial proportions Jintu Fan n, K.P. Chau, Xianfu Wan, Lili Zhai, Ethan Lau Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 November 2009 Received in revised form 21 September 2011 Accepted 11 November 2011 Available online 14 December 2011
This study investigates the relationship between facial attractiveness and facial proportions. Here, we generated facial images with different proportions using computer software, hence avoided the influence of hairstyle, facial expression as well as skin tone and texture on the perception of facial attractiveness. By analyzing the relationship between the facial proportions of 432 computer generated facial images and their attractiveness ratings, here we identified the optimum proportions for an attractive female face and further established a model of predicting facial attractiveness from four principle components of facial proportions with good predictability (R2 ¼ 0.64). & 2011 Elsevier Ltd. All rights reserved.
Keywords: Face recognition Facial attractiveness Facial proportion Golden ratios Neoclassical cannons
1. Introduction The question, whether there is such a thing as beauty that objectively and universally exists, has attracted debates among philosophers, psychologists and biologists for centuries [1–3]. One popular school of thought is that ‘beauty is in the eye of the beholder’, that individual attraction is a result of personal experience, cultural background and specific circumstances. Nevertheless, the assumption that beauty is just an arbitrary personal preference cannot explain the fact that even two-monthold infants prefer to gaze at faces that adults find attractive [4]. If there is objectivity in beauty assessment, what are there? Ancient Greeks believed that the world is beautiful because there is a certain measure, proportion, order and harmony between its elements [3]. For centuries, the Golden Ratio or Golden Proportion, a ratio of 1:1.618 has been considered as the perfect ratio for beauty, and widely applied for art and architectural design. With regard to human body physical attractiveness, past researches showed that the body mass index (BMI) [10,11] or more precisely the body volume divided by the square of height (VHI) [12,13] is the primary cue of body attractiveness, and there is a very good agreement on the general trend of the effect of VHI on body attractiveness among Chinese and UK viewers. There exist optimum VHI values corresponding to physical fitness and fertility, although the precise values of optimum VHI could be different for different cultures [14]. As far as facial beauty is concerned, averageness [4] and symmetry [6] were found to be the important criteria. Attractiveness increased
n
Corresponding author. Tel.: þ852 2766 6472; fax: þ852 2773 1432. E-mail address: [email protected] (J. Fan).
0031-3203/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2011.11.024
with an increasing level of averageness and symmetry. This can be understood as evolutionary pressures operate against the extremes of the population [5]. Apart from averageness and symmetry, some extreme traits, such as the peacock’s tail, can be a sign of quality and health in a mate and therefore favoured in the choice of selection. Using composites of both Caucasian and Japanese faces, Perrett et al. [7] showed that the mean shape of a set of attractive faces is preferred to the mean shape of all the faces in a sample. Attractive composite faces can be made more attractive by exaggerating the shape difference from the sample mean. Therefore, an average face shape is attractive, but not optimally attractive. Human preferences could exert a directional selection pressure on the evolution of the shape of a human face. Perrett et al. [8] further showed that more feminine female and male faces are preferred to average faces. Enhanced masculine facial characteristics increased both perceived dominance and negative attributes (for example, coldness or dishonesty) relevant to relationships and paternal investment. They believed that humans have a selection pressure that limits sexual dimorphism and encourages neoteny. It is also believed that, for attractive faces, facial measurements should follow certain defined ratios. Dated back to the renaissance period, neoclassical cannons have been proposed to be the ideal ratios of beautiful faces [16]. In popular literature, it is also claimed that beautiful faces have facial measurements close to the golden ratio [18,9,17]. Despite the popular claim that beautiful faces follow neoclassical cannons or golden ratios, it has rarely been systematically and statistically examined. The only such work is attributed to Schmid et al. [15], who recently attempted to test the above claim by relating the attractiveness of the frontal views of 420 Caucasian facial images to the ratios of facial measurements. This work was
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however not conclusive as the effects of the variations in hairstyles, skin texture, positions and expressions of the facial images on the facial attractiveness was not controlled in their experiments, and the linear regression model applied in their analysis could not model the generally nonlinear relationships and the interactions between the ratios of facial measurements. Consequently, they showed only a weak predictability (R2 ¼0.2433) of facial attractiveness even from very large number (viz. 78) of facial ratios as independent variables. The present study is aimed at investigating only the effect of facial ratios on facial attractiveness, by excluding the effects of hairstyle, skin texture and expression, so as to test whether neoclassical cannons or golden ratios are ideal ratios for beautiful faces.
2. Experimental 2.1. Preparation of facial images Poser software was used to create computer-generated facial images. In generating the facial images having varying facial dimensions and ratios, we first created a so called ‘‘original face’’ (See Fig. 1), from which some key facial dimensions are altered to generate other facial images. In order to ensure that the computer-generated facial
Fig. 1. The ‘‘Original Face’’.
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images are realistic in facial features, the ‘‘original face’’ was obtained by averaging the features of the facial images of some arbitrarily chosen famous oriental ladies (from Japanese, Korean and China) available on the internet. In order to focus our present study on the effect of facial proportions, the faces were made hairless to eliminate the effects of hairstyles. The effects of hairstyles on facial attractiveness will be an area for further investigation. In order to create sufficient, but realistic variations in the facial dimensions and ratios of the facial images for our experiments, we applied gradual alteration to the ‘‘Original Face’’ by nine different approaches, each approach created 14 different images. Approach 1: Mark 4 apex points at the left, right, top and bottom of the face as shown in Fig. 2(a). Shorten the width (i.e. the distance between the left and right apex points) by 4% of the width of the original face and the length (i.e. the distance between the top and bottom apex points) by 8% of the length of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 2: Mark 2 apex points at the left and right as shown in Fig. 2(b). Shorten the width by 4% of the width of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 3: Mark 2 apex points at the left and right as shown in Fig. 2(c). Increase the width by 9% of the width of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 4: Mark 2 apex points at the top and bottom as shown in Fig. 2(d). Shorten the length by 9% of the length of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 5: Mark 4 apex points at the left, right, top and bottom of the face as shown in Fig. 2(e). Increase the width (i.e. the distance between the left and right apex points) and the length (i.e. the distance between the top and bottom apex points) by 3% of the width and length of the original face, respectively, to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 6: Mark 2 apex points as shown in Fig. 2(f). Shorten the length of the nose by 10% of the original nose length to
Fig. 2. Alterations of ‘‘Original Face’’ to generate other Facial Images. (a) Approach 1, (b) Approach 2, (c) Approach 3, (d) Approach 4, (e) Approach 5, (f) Approach 6 (g) Approach 7, (h) Approach 8 and (i) Approach 9.
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create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other dimensions and positions were not changed. Approach 7: Mark 2 apex points as shown in Fig. 2(g). Increase the length of the nose by 14% of the original nose length to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other dimensions and positions were not changed. Approach 8: Mark 2 apex points as shown in Fig. 2(h). Move the positions of the two eyes apart by 22% of the distance of the two eyes of the original face to create a new facial image. Repeat the above procedure 13 times to create a total 14 facial images. During this alteration, all other dimensions are not altered. Approach 9: Mark 2 apex points as shown in Fig. 2(i), move the positions of the two eyes closer by 22% of the distance of the two eyes of the original face to create a new facial image. Repeat the above procedure 13 times to create a total 14 facial images. During this alteration, all other dimensions are not altered. The change in each step of the alterations in the above nine approaches were determined by, first determining the maximum possible alteration of the targeted dimension(s) without being over unrealistic, which was determined by preliminary visual assessment. The maximum possible dimensional alternation was then divided by 14 to determine the change in each step of the alterations so that we can create 14 progressive facial images in each approach. The ‘‘Original Face’’ and the 126 (9 14) images created above had wide variations in the length and width of the face, the length of the nose, and positions of the eyes, however, other facial features (viz. the size of the eyes, the width of the nose, the size and shape of the mouth) were unchanged. In order to take into account the effects of these facial features, but minimize the sample size, we applied the orthogonal experimental design to create additional 304 (16 19) facial images from the ‘‘Original Face’’ and 18 facial images selected from the above 126 images (two from each lot created by each approach with one having the maximum alteration and one having the medium alternation) by varying mouth width, eye fissure width, nose width and lip height. For each of the 19 facial images (the ‘‘Original Face’’ and the selected 18 facial images created by the above 9 approaches), the mouth width, eye fissure width, nose width and lip height were altered in the proportion as listed in Table 1, which is determined by the (4 4) orthogonal experimental design for four factors and four levels. The four levels of proportional changes of the mouth width were 10%, 5%, þ 5% and þ10%, respectively; those of
Table 1 Orthogonal design of facial alternations. No.
Mouth width
Eye fissure width
Nose width
Lip height
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
5% 5% 5% 5% 10% 10% 10% 10% þ 5% þ 5% þ 5% þ 5% þ 10% þ 10% þ 10% þ 10%
15% þ10% þ20% þ30% 15% þ10% þ20% þ30% 15% þ10% þ20% þ30% 15% þ10% þ20% þ30%
10% 20% þ10% þ20% 20% 10% þ20% þ10% þ10% þ20% 10% 20% þ20% þ10% 20% 10%
þ 5% 5% 10% 15% 10% 15% þ 5% 5% 15% 10% 5% þ 5% 5% þ 5% 15% 10%
eye fissure width were 15%, þ10%, þ20% and þ30%, respectively; those of nose width were 20%, 10%, 10% and þ20%, respectively; those of lip height were 15%, 10%, 5% and þ5%, respectively. These levels were determined so as to have wider range of variation without creating too unrealistic images. A selection of these 304 facial images is shown in Fig. 3. 2.2. Measurements of facial dimensions and calculation of facial ratios For each sample image, 29 important landmarks (See Fig. 4) were identified using Photoshop software and the vertical and horizontal distance between the landmarks were measured to derive the 21 ratios listed in Table 2. These 21 ratios were considered as important face proportions relevant to facial beauty. It was reported in the popular literature [18,17] that, attractive faces should have the first 17 ratios (r1–r17) approaching the golden ratio of 1.618, and the rest 4 ratios (representing neoclassical canons) approaching unity [16]. This myth was rarely tested experimentally except for the work by Schmid et al. (2007), who, based on 420 real facial images, showed that only some of these ratios were significant. Schmid et al.’s work was however not conclusive as the influences of the varying hairstyles, expressions, skin appearance and postures of their facial images were not excluded. Furthermore, careful examination of these ratios can reveal that r6, r7, r12, r13 and r16 are interrelated depending on the vertical position of the mouth, it is not possible to set them all at the Golden Ratio of 1.618. If r16 is set at 1.618, the closest ratios of r6, r7, r12 and r13 to the Golden Ratio are 1.280, 1.280, 1.000 and 1.036, respectively. In order to experimentally verify whether a face with almost perfect golden ratios and neoclassical canons is indeed the most attractive, we also created a face having the optimum putative ratios mostly at Golden Ratio or neoclassical canons except for r6, r7, r12, r13 (See putative optimum ratios listed in Table 2), which was later visually assessed in terms of attractiveness and compared to other facial images. 2.3. Attractiveness rating Totally 432 sample images (1 ‘‘Original Face’’, 126 faces created by the 9 approaches, 304 faces with varying mouth width, eye fissure width, nose width and lip height and 1 face with putative optimum ratios) were randomly displayed to 15 male viewers and 15 female viewers using the commercial software E-PRIME. All viewers were students of The Hong Kong Polytechnic University from Hong Kong or China. The age of the male viewers was 24.572.5 and that of female viewers was 23.071.8. The viewers were required to rate the facial images within a maximum of 4 s in terms of attractiveness in a 1–9 scale (1 being least attractive and 9 being the most attractive). The standard deviation of the average attractiveness ratings ranged from 0.12 to 0.40.
3. Results and analysis 3.1. Ranges of the facial ratios of the sample images In order to investigate the effects of facial ratios on facial attractiveness, it is important that the facial ratios of the sample images should have a wide variation. For the 432 sample images, the 21 facial ratios had 21%–123% variation over the mean and the ranges covered the golden ratios or neoclassical canons. 3.2. Differences between the attractiveness ratings of male viewers and female viewers The average attractiveness ratings (AR) by female raters were compared with those by male raters using the Wilcoxon
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Fig. 3. Examples of facial images created for the study.
signed-rank method, which is a non-parametric statistical hypothesis test for the case of two related samples. The results (z¼ 7.844 (based on positive ranks); asymp. Sig. (2-tailed)¼0.000o0.05)
indicated some statistical differences between the ARs by female and male raters. Detailed comparison of ARs by male raters (ARMs) and ARs by female raters (ARFs) revealed that ARMs are
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slightly greater (5.1% on average) than those of ARFs, However, they are strongly related (R2 ¼0.888). Since the differences between the attractiveness ratings of male viewers and female viewer are small and they are strongly correlated, only the average attractiveness ratings of all the viewers are considered in the following analysis. 3.3. Principal component analysis (PCA) The 21 ratios may be interrelated. It is therefore justifiable to use principal component analysis (PCA) to identify the unrelated principal components. The ratios were first normalized and then analyzed by PCA using SPSS. Under the minimum fraction
variance of 0.05 (or 5%), four principal components were identified with a cumulative variance of about 90%. Fig. 5 plots the cumulative variance explained by the number of principal components. Table 3 lists the transformation matrix between the four components and the 21 ratios. From the correlations between the four components and the ratios, it can be learned that the first component is largely related to the nose width or interocular distance, the second component is largely related to the ratio of nose width over the eye size, the third component is largely related to the nose length and nose to mouth distance, and the fourth component is largely related to the ratio of the length to the width of the face as well as the position of mouth in relation to the nose and chin. Fig. 6 illustrates the effect of the four principal components on the facial images. 3.4. Relationship between principal components and attractiveness ratings Each of the four principal components is plotted against the attractiveness ratings in Fig. 7(a)–(d). As can be seen, in general, there is an optimum value for each of the four components, at which the facial attractiveness is at the best. The deviation from the optimum value of each of the four components will result in a reduction in facial attractiveness, but the trend of reduction is not symmetrical at both side of the optimum value. The effect of each of the four components on the attractiveness ratings is therefore in a distorted bell shape. To capture the main trend, the following transformed Gause formula was applied to fit the data plotted in Fig. 7(a)–(d): ! xn2 a ARi ðxi Þ ¼ piffiffiffiffiffiffi exp i 2 þ ci , ði ¼ 1,2,3,4Þ ð1Þ 2si si 2p where, xni is the transformed value of each of the four components applying the Box–Cox transformation function, viz. 8 0r < ðxi Þ i 1 r a 0 i ri n xi ¼ ð2Þ : lnðx0 Þ r i ¼ 0 i
where, x0i is converted from the original value of each of the four components through Fig. 4. Feature points of a facial image.
x0i ¼
xi xm ui xm
ð3Þ
Table 2 Definition of facial ratios. Ratios
Numerator points
Denominator points
Description
Optimal putative ratios
r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21
y10–y21 y10–y21 x15–x16 x15–x16 x25–x27 y23–y29 y23–y29 x12–x13 x12–x13 x18–x20 x18–x20 x12–x11 y23–y28 y1–y29 y19–y29 x18–x20 x25–x27 y3–y19 y3–y19 x12–x13 x6–x9
x12–x13 x18–x20 x12–x13 x18–x20 x12–x13 x12–x13 x18–x20 x12–x11 y23–y28 x12–x11 y23–y28 y19–y26 y19–y26 x6–x9 y26–y29 y19–y26 x18–x20 y19–y29 y10–y21 x18–x20 (x18–x20)n4
Ear length to interocular distance Ear length to nose width Mideye distance to interocular distance Mideye distance to nose width Mouth width to interocular distance Lips–chin distance to interocular distance Lips–chin distance to nose with Interocular distance to eye fissure width Interocular distance to lip height Nose width to eye fissure width Nose width to lip height Eye fissure width to nose–mouth distance Lip height to nose–mouth distance Length of face to width of face Nose-chin distance to lips–chin distance Nose width to nose–mouth distance Mouth width to nose width Nose length to nose–chin distance Nose length to ear length Interocular distance to nose width Width of face to 4 times of nose with
1.618 1.618 1.618 1.618 1.618 1.280 1.280 1.618 1.618 1.618 1.618 1.000 1.036 1.618 1.618 1.618 1.618 1.000 1.000 1.000 1.000
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Fig. 5. Cumulative variance explained by the number of principal components.
Table 3 Transformation matrix. Ratios
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Component 1
2
3
4
0.254 0.262 0.255 0.306 0.232 0.239 0.249 0.246 0.228 0.041 0.253 0.195 0.026 0.003 0.032 0.250 0.218 0.063 0.057 0.367 0.256
0.269 0.270 0.275 0.196 0.262 0.268 0.261 0.276 0.254 0.357 0.219 0.193 0.002 0.031 0.024 0.202 0.266 0.025 0.028 0.040 0.252
0.085 0.031 0.041 0.085 0.048 0.099 0.009 0.027 0.008 0.046 0.096 0.302 0.448 0.141 0.280 0.253 0.062 0.502 0.484 0.078 0.087
0.033 0.086 0.077 0.024 0.085 0.141 0.212 0.078 0.010 0.076 0.037 0.075 0.263 0.605 0.537 0.167 0.066 0.184 0.280 0.034 0.157
Note: Component 1 relates to nose width or interocular distance. Component 2 relates to the ratio of nose width over the eye size. Component 3 relates to the nose length and nose to mouth distance. Component 4 relates to the ratio of length of face to the width of face as well as the position of mouth in relation to the nose and chin.
xm is an arbitrary value, which is chosen to ensure that the value of the independent variable of the Box–Cox function (x0 ) is positive, and ui is a value to be determined by the nonlinear regression. By the nature of the above formulae, when xi is equal to ui, the ARi ðxi Þ is at its maximum. Therefore ui is the optimum value of ith component resulting in the maximum attractiveness. By applying nonlinear regression, the values of ai, ci, ui, si and ri can be determined and they are listed in Table 4. By assuming that the overall facial attractiveness is the linear combination of the contributions of each of the four components, we have the following equation AR ¼ b0 þ b1 AR1 ðx1 Þ þb2 AR2 ðx2 Þ þb3 AR3 ðx3 Þ þ b4 AR4 ðx4 Þ
ð4Þ
The coefficients in Eq. (4), viz. b0, b1, b2, b3 and b4, can be obtained by multiple linear regression. In this study, the multiple linear regression was carried out based on the least-squares model, i.e. the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line. We obtained b0 ¼ 4.89, b1 ¼0.73, b2 ¼0.67, b3 ¼ 0.19 and b4 ¼0.69. Fig. 8 plots the mean attractiveness ratings of the 432 facial images against the predicted attractiveness rating from the four principal components (derived from the 21 facial ratios) using the regression Eqs. (1) and (4). As can be seen there is a strong one to one linear relationship between the visually assessed mean attractiveness ratings and the predicted attractiveness ratings. The percentage of fit of the model, R2, is 0.6413. The percentage of fit of our model with just four independent predictor variables is a significant improvement over that of Schmid et al.’s model, which had a percentage of fit (R2) of 0.1923 requiring 16 predictor variables. 3.5. Verification of the prediction model In order to examine whether the established model has mapped the general trends of the effects of facial proportions on the facial attractiveness, instead of memorizing the data, it is necessary to test the model against the samples not included in building model. We hence generated additional 113 facial images with randomly chosen facial dimensions using the same software and procedures as those for the previous 432 facial images. The attractiveness ratings of these 113 facial images were rated by 15 male and 15 female viewers, who were aged between 20 and 30 years. Before rating, the viewers were first trained by exposing them to a selection of facial images of varying attractiveness with known ratings rated previously. This ensures that the new group of viewers will have the equivalent scale of rating as the previous group who rated the attractiveness of the facial images used in building the prediction model. The standard deviation of the average attractiveness ratings ranged from 0.07 to 0.35. Fig. 9 plots the actual AR (attractiveness ratings) against the predicted AR using the model for the additional 113 facial images not used for building the model. It can be seen, there is a close to 1:1 relationship between the actual AR and predicted AR with a R2
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7
7
6
6
5
5 AR
AR
Fig. 6. Effect of the four principal components on the facial images.
4 3
3
2
2
1 −10
−8
−6
−4
−2 0 Component1
2
4
6
1 −10
8
7
7
6
6
5
5 AR
AR
4
4
3
2
2 −8
−6
−4
−2 0 Component3
2
4
6
−6
−4
−2 0 Component2
2
4
6
8
−2
−1
0
1 2 Component4
3
4
5
6
4
3
1
−8
1
−3
Fig. 7. Four principal components verses facial attractiveness (a: top left figure, b: top right figure, c: bottom left and d: bottom right).
J. Fan et al. / Pattern Recognition 45 (2012) 2326–2334
Table 4 Parameters obtained from nonlinear regression for each component. Component(i)
ai
ri
si
ci
ui
1 2 3 4
0.68959 0.17894 0.082028 0.76185
5.4094 22.317 25.075 2.9297
0.12881 0.030573 0.022657 0.1175
2.3933 2.7365 3.219 1.9371
1.3804 0.7103 0.2397 1.1598
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1.5151, 1.7522, 2.2138, 1.3496, 1.7127, 1.1039, 0.86206, 1.635, 1.5281, 1.4705, 1.4739, 0.77602, 0.89479, 1.3057, 1.1075). Fig. 10 shows the facial image with the optimum ratios. The face with the optimum putative ratios (mostly at the Golden Ratio except for r18–r21, i.e. the optimum ratios of which were believed to be the neoclassical canons of 1, and r6, r7, r12, r13, which cannot be set to the golden ratio when r16 is at the golden ratio) was only rated 2.6. Its image is shown in Fig. 11. Clearly, the popular claim that faces are the most beautiful when the facial ratios are at the Golden Ratio cannot be established. In Meisner’s website [18], the facial images used to illustrate the golden ratios on faces have different expressions (The male face has a neutral expression, but the female face has a big smile). As the expression of the face alter the facial ratios or proportions, it is important to define the expression when considering the effects of facial proportions on facial attractiveness. In our present investigation, the facial images were all controlled at neutral. In another Marquardt’s website [9], he claimed that beautiful faces conform to the Golden Facial Mask derived from Golden Decagon Matrices composed of complex relationship of the Golden ratio phi. Fig. 12 shows the most attractive face found in
Fig. 8. Visually assessed mean attractiveness ratings (AR) vs. the predicted attractiveness ratings (Predicted AR).
Fig. 10. Facial image with the optimum ratios (AR¼ 6.33).
Fig. 9. Actual AR (attractiveness ratings) against the predicted AR using the model for the additional 113 facial images not used for building the model.
(or percentage of fit) of 0.6463. This means that the model has effectively mapped the effects of facial proportions on the facial attractiveness for the facial images with dimensions and proportions varied from an average oriental face.
4. Discussion The prediction model enables us to find the optimum facial proportions for attractiveness. The principle component vector of the optimally attractive face is (1.3804, 0.7103, 0.2397, 1.1598) and the corresponding facial ratios for this optimum face is (r1, r2, r3,y r21) ¼(1.3631, 1.7595, 1.5565, 2.0182, 1.1425, 1.1763,
Fig. 11. Face having neoclassical canons and most possible golden ratios.
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Acknowledgment The authors would like to thank the funding support of Hong Kong Polytechnic University in the form of a Niche Area project (Project no: J-BB6T) and inter-faculty project (Project no: G-YG13). References
Fig. 12. Most attractive face found in our investigation covered with Marquardt’s Golden Facial Mask.
our investigation covered with the proposed Golden Facial Mask. It can seen that the attractive face generally conform to the Golden Facial Mask. However the claim, that beautiful faces should conform to the Golden Facial Mask, could not be statistically tested in the present investigation due to the difficulty in defining the facial landmarks to make measurements corresponding to the geometrical lines of the proposed Golden Facial Mask. In the present study, the effects of facial proportions on facial attractiveness are investigated, using computer generated facial images. Computer generated facial images have the advantage of excluding the influence of hairstyle, facial expression as well as skin tone and texture. However, the applicability of the findings and developed model to real faces should still be tested in future work, although it is foreseeable that, such work will face a big challenge in how to control and quantify the influence of facial proportion, hairstyle, facial expression, skin tone and texture.
[1] N.L. Etcoff, Beauty and the Beholder, Nature 368 (6468) (1994) 186–187. [2] N. Wolf, The Beauty Myth, Morrow, New York, 1990. [3] B. Gaut, D.M. Lopes, The Routledge Companion to Aesthetics, Chapter 20: Beauty, Routledge, London, 2001. [4] J.H. Langlois, L.A. Roggman, Attractive faces are only average, Psychological Science 1 (2) (1990) 115–121. [5] D. Symons, The Evolution of Human Sexuality, Oxford University Press, 1979. [6] G. Rhodes, F. Proffitt, J.M. Grady, A. Sumich, Facial symmetry and the perception of beauty, Psychon B Review 5 (4) (1998) 659–669. [7] D.I. Perrett, K.A. May, S. Yoshikawa, Facial shape and judgements of female attractiveness, Nature 368 (6468) (1994) 239–242. [8] D.I. Perrett, K.J. Lee, I. Penton-Voak, D. Rowland, S. Yoshikawa, D.M. Burt, S.P. Henzi, D.L. Castles, S. Akamatsu, Effects of sexual dimorphism on facial attractiveness, Nature 394 (6696) (1998) 884–887. [9] Marquardt, S., (2002) Can beauty be measured, URL: /http://www.beautya nalysis.com/index2_mba.htmS. [10] M.J. Tovee, D.S. Maisey, J.L. Emery, P.L. Cornelissen, Visual cues to female physical attractiveness, Proceedings of Royal Society of London Series B- Biological Sciences 266 (1415) (1999) 211–218. [11] M.J. Tovee, P.J.B. Hancock, S. Mahmoodi, B.R.R. Singleton, P.L. Cornelissen, Human female attractiveness: Waveform analysis of body shape, Proceedings of Royal Society of London Series B- Biological Sciences 269 (1506) (2002) 2205–2213. [12] J.T. Fan, F. Liu, J. Wu, W. Dai, Visual perception of male physical attractiveness, Proceedings of Royal Society of London Series B- Biological Sciences 271 (2004) 347–352 2004. [13] J.T. Fan, W. Dai, F. Liu, J. Wu, Visual Perception of Male Body Attractiveness, Proceedings of Royal Society of London Series B- Biological Sciences 272 (2005) 219–226 10.1098. rspb.2004.2922. [14] J.T. Fan, W. Dai, X. Qian, K.P. Chau, Q. Liu, Effects of shape parameters on the attractiveness of a female body, Perception and Motor Skills 105 (2007) 117–132. [15] K. Schmid, D. Marx, A. Samal, Computation of a face attractiveness index based on neoclassical canons, symmetry, and golden ratios, Pattern Recognition 41 (2008) 2710–2717. [16] L.G. Farkas, T.A. Hreczko, J.C. Kolar, I.R. Munro, Verical and Horizontal proportions of the face in young adult North American Caucasians: a revision of neoclassical cannons, Plastic and Reconstructive Surgery 75 (1985) 328–337. [17] Narain, D.L., (2003) The Golden Ratio, URL: /http://cuip.uchicago.edu/ dlnarain/golden/activity8.htmS. [18] Meisner, G. (2006), The Human Face, /www.goldennumber.net/face.htmS.
Jintu Fan is a professor at the Institute of Textiles and Clothing, The Hong Kong Polytechnic University. He holds a B.Sc. from Donghua University (formerly China Textile University) and Ph.D from Leeds University. His research interests include beauty psychology, thermal comfort and functional clothing.
Desmon Chau is an Instructor at the Institute of Textiles and Clothing of The Hong Kong Polytechnic University. He gained an M.Sc. in Information Technology with Internet Applications in 2003. His research interests are the Facial Beauty, CAD application, 3D Model development and Multimedia.
Xianfu Wan received his M.Sc. Degree in computer applications from Dong Hua University in 2004, and a Ph.D. Degree in Textiles and Clothing from the Hong Kong Polytechnic University in 2008. His research interests include clothing thermal comfort and visual comfort.
Lili Zhai is a Ph.D research student at the Tianjing Polytechnic University. She was a research assistant at the Institute of Textiles and Clothing, The Hong Kong Polytechnic University, when she worked on this project. Her research interests include 3D body scanning and body beauty.
Ethan Lau graduated from Institute of Textiles and Clothing of The Hong Kong Polytechnic University with a B.A. in Fashion and Textiles. His research interest is 3D modeling.
Contents lists available at SciVerse ScienceDirect
Pattern Recognition journal homepage: www.elsevier.com/locate/pr
Prediction of facial attractiveness from facial proportions Jintu Fan n, K.P. Chau, Xianfu Wan, Lili Zhai, Ethan Lau Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 November 2009 Received in revised form 21 September 2011 Accepted 11 November 2011 Available online 14 December 2011
This study investigates the relationship between facial attractiveness and facial proportions. Here, we generated facial images with different proportions using computer software, hence avoided the influence of hairstyle, facial expression as well as skin tone and texture on the perception of facial attractiveness. By analyzing the relationship between the facial proportions of 432 computer generated facial images and their attractiveness ratings, here we identified the optimum proportions for an attractive female face and further established a model of predicting facial attractiveness from four principle components of facial proportions with good predictability (R2 ¼ 0.64). & 2011 Elsevier Ltd. All rights reserved.
Keywords: Face recognition Facial attractiveness Facial proportion Golden ratios Neoclassical cannons
1. Introduction The question, whether there is such a thing as beauty that objectively and universally exists, has attracted debates among philosophers, psychologists and biologists for centuries [1–3]. One popular school of thought is that ‘beauty is in the eye of the beholder’, that individual attraction is a result of personal experience, cultural background and specific circumstances. Nevertheless, the assumption that beauty is just an arbitrary personal preference cannot explain the fact that even two-monthold infants prefer to gaze at faces that adults find attractive [4]. If there is objectivity in beauty assessment, what are there? Ancient Greeks believed that the world is beautiful because there is a certain measure, proportion, order and harmony between its elements [3]. For centuries, the Golden Ratio or Golden Proportion, a ratio of 1:1.618 has been considered as the perfect ratio for beauty, and widely applied for art and architectural design. With regard to human body physical attractiveness, past researches showed that the body mass index (BMI) [10,11] or more precisely the body volume divided by the square of height (VHI) [12,13] is the primary cue of body attractiveness, and there is a very good agreement on the general trend of the effect of VHI on body attractiveness among Chinese and UK viewers. There exist optimum VHI values corresponding to physical fitness and fertility, although the precise values of optimum VHI could be different for different cultures [14]. As far as facial beauty is concerned, averageness [4] and symmetry [6] were found to be the important criteria. Attractiveness increased
n
Corresponding author. Tel.: þ852 2766 6472; fax: þ852 2773 1432. E-mail address: [email protected] (J. Fan).
0031-3203/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2011.11.024
with an increasing level of averageness and symmetry. This can be understood as evolutionary pressures operate against the extremes of the population [5]. Apart from averageness and symmetry, some extreme traits, such as the peacock’s tail, can be a sign of quality and health in a mate and therefore favoured in the choice of selection. Using composites of both Caucasian and Japanese faces, Perrett et al. [7] showed that the mean shape of a set of attractive faces is preferred to the mean shape of all the faces in a sample. Attractive composite faces can be made more attractive by exaggerating the shape difference from the sample mean. Therefore, an average face shape is attractive, but not optimally attractive. Human preferences could exert a directional selection pressure on the evolution of the shape of a human face. Perrett et al. [8] further showed that more feminine female and male faces are preferred to average faces. Enhanced masculine facial characteristics increased both perceived dominance and negative attributes (for example, coldness or dishonesty) relevant to relationships and paternal investment. They believed that humans have a selection pressure that limits sexual dimorphism and encourages neoteny. It is also believed that, for attractive faces, facial measurements should follow certain defined ratios. Dated back to the renaissance period, neoclassical cannons have been proposed to be the ideal ratios of beautiful faces [16]. In popular literature, it is also claimed that beautiful faces have facial measurements close to the golden ratio [18,9,17]. Despite the popular claim that beautiful faces follow neoclassical cannons or golden ratios, it has rarely been systematically and statistically examined. The only such work is attributed to Schmid et al. [15], who recently attempted to test the above claim by relating the attractiveness of the frontal views of 420 Caucasian facial images to the ratios of facial measurements. This work was
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however not conclusive as the effects of the variations in hairstyles, skin texture, positions and expressions of the facial images on the facial attractiveness was not controlled in their experiments, and the linear regression model applied in their analysis could not model the generally nonlinear relationships and the interactions between the ratios of facial measurements. Consequently, they showed only a weak predictability (R2 ¼0.2433) of facial attractiveness even from very large number (viz. 78) of facial ratios as independent variables. The present study is aimed at investigating only the effect of facial ratios on facial attractiveness, by excluding the effects of hairstyle, skin texture and expression, so as to test whether neoclassical cannons or golden ratios are ideal ratios for beautiful faces.
2. Experimental 2.1. Preparation of facial images Poser software was used to create computer-generated facial images. In generating the facial images having varying facial dimensions and ratios, we first created a so called ‘‘original face’’ (See Fig. 1), from which some key facial dimensions are altered to generate other facial images. In order to ensure that the computer-generated facial
Fig. 1. The ‘‘Original Face’’.
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images are realistic in facial features, the ‘‘original face’’ was obtained by averaging the features of the facial images of some arbitrarily chosen famous oriental ladies (from Japanese, Korean and China) available on the internet. In order to focus our present study on the effect of facial proportions, the faces were made hairless to eliminate the effects of hairstyles. The effects of hairstyles on facial attractiveness will be an area for further investigation. In order to create sufficient, but realistic variations in the facial dimensions and ratios of the facial images for our experiments, we applied gradual alteration to the ‘‘Original Face’’ by nine different approaches, each approach created 14 different images. Approach 1: Mark 4 apex points at the left, right, top and bottom of the face as shown in Fig. 2(a). Shorten the width (i.e. the distance between the left and right apex points) by 4% of the width of the original face and the length (i.e. the distance between the top and bottom apex points) by 8% of the length of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 2: Mark 2 apex points at the left and right as shown in Fig. 2(b). Shorten the width by 4% of the width of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 3: Mark 2 apex points at the left and right as shown in Fig. 2(c). Increase the width by 9% of the width of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 4: Mark 2 apex points at the top and bottom as shown in Fig. 2(d). Shorten the length by 9% of the length of the original face to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 5: Mark 4 apex points at the left, right, top and bottom of the face as shown in Fig. 2(e). Increase the width (i.e. the distance between the left and right apex points) and the length (i.e. the distance between the top and bottom apex points) by 3% of the width and length of the original face, respectively, to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other facial dimensions were not changed. Approach 6: Mark 2 apex points as shown in Fig. 2(f). Shorten the length of the nose by 10% of the original nose length to
Fig. 2. Alterations of ‘‘Original Face’’ to generate other Facial Images. (a) Approach 1, (b) Approach 2, (c) Approach 3, (d) Approach 4, (e) Approach 5, (f) Approach 6 (g) Approach 7, (h) Approach 8 and (i) Approach 9.
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create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other dimensions and positions were not changed. Approach 7: Mark 2 apex points as shown in Fig. 2(g). Increase the length of the nose by 14% of the original nose length to create a new facial image. Repeat the above procedure 13 times to create a total of 14 facial images. During this alteration, all other dimensions and positions were not changed. Approach 8: Mark 2 apex points as shown in Fig. 2(h). Move the positions of the two eyes apart by 22% of the distance of the two eyes of the original face to create a new facial image. Repeat the above procedure 13 times to create a total 14 facial images. During this alteration, all other dimensions are not altered. Approach 9: Mark 2 apex points as shown in Fig. 2(i), move the positions of the two eyes closer by 22% of the distance of the two eyes of the original face to create a new facial image. Repeat the above procedure 13 times to create a total 14 facial images. During this alteration, all other dimensions are not altered. The change in each step of the alterations in the above nine approaches were determined by, first determining the maximum possible alteration of the targeted dimension(s) without being over unrealistic, which was determined by preliminary visual assessment. The maximum possible dimensional alternation was then divided by 14 to determine the change in each step of the alterations so that we can create 14 progressive facial images in each approach. The ‘‘Original Face’’ and the 126 (9 14) images created above had wide variations in the length and width of the face, the length of the nose, and positions of the eyes, however, other facial features (viz. the size of the eyes, the width of the nose, the size and shape of the mouth) were unchanged. In order to take into account the effects of these facial features, but minimize the sample size, we applied the orthogonal experimental design to create additional 304 (16 19) facial images from the ‘‘Original Face’’ and 18 facial images selected from the above 126 images (two from each lot created by each approach with one having the maximum alteration and one having the medium alternation) by varying mouth width, eye fissure width, nose width and lip height. For each of the 19 facial images (the ‘‘Original Face’’ and the selected 18 facial images created by the above 9 approaches), the mouth width, eye fissure width, nose width and lip height were altered in the proportion as listed in Table 1, which is determined by the (4 4) orthogonal experimental design for four factors and four levels. The four levels of proportional changes of the mouth width were 10%, 5%, þ 5% and þ10%, respectively; those of
Table 1 Orthogonal design of facial alternations. No.
Mouth width
Eye fissure width
Nose width
Lip height
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
5% 5% 5% 5% 10% 10% 10% 10% þ 5% þ 5% þ 5% þ 5% þ 10% þ 10% þ 10% þ 10%
15% þ10% þ20% þ30% 15% þ10% þ20% þ30% 15% þ10% þ20% þ30% 15% þ10% þ20% þ30%
10% 20% þ10% þ20% 20% 10% þ20% þ10% þ10% þ20% 10% 20% þ20% þ10% 20% 10%
þ 5% 5% 10% 15% 10% 15% þ 5% 5% 15% 10% 5% þ 5% 5% þ 5% 15% 10%
eye fissure width were 15%, þ10%, þ20% and þ30%, respectively; those of nose width were 20%, 10%, 10% and þ20%, respectively; those of lip height were 15%, 10%, 5% and þ5%, respectively. These levels were determined so as to have wider range of variation without creating too unrealistic images. A selection of these 304 facial images is shown in Fig. 3. 2.2. Measurements of facial dimensions and calculation of facial ratios For each sample image, 29 important landmarks (See Fig. 4) were identified using Photoshop software and the vertical and horizontal distance between the landmarks were measured to derive the 21 ratios listed in Table 2. These 21 ratios were considered as important face proportions relevant to facial beauty. It was reported in the popular literature [18,17] that, attractive faces should have the first 17 ratios (r1–r17) approaching the golden ratio of 1.618, and the rest 4 ratios (representing neoclassical canons) approaching unity [16]. This myth was rarely tested experimentally except for the work by Schmid et al. (2007), who, based on 420 real facial images, showed that only some of these ratios were significant. Schmid et al.’s work was however not conclusive as the influences of the varying hairstyles, expressions, skin appearance and postures of their facial images were not excluded. Furthermore, careful examination of these ratios can reveal that r6, r7, r12, r13 and r16 are interrelated depending on the vertical position of the mouth, it is not possible to set them all at the Golden Ratio of 1.618. If r16 is set at 1.618, the closest ratios of r6, r7, r12 and r13 to the Golden Ratio are 1.280, 1.280, 1.000 and 1.036, respectively. In order to experimentally verify whether a face with almost perfect golden ratios and neoclassical canons is indeed the most attractive, we also created a face having the optimum putative ratios mostly at Golden Ratio or neoclassical canons except for r6, r7, r12, r13 (See putative optimum ratios listed in Table 2), which was later visually assessed in terms of attractiveness and compared to other facial images. 2.3. Attractiveness rating Totally 432 sample images (1 ‘‘Original Face’’, 126 faces created by the 9 approaches, 304 faces with varying mouth width, eye fissure width, nose width and lip height and 1 face with putative optimum ratios) were randomly displayed to 15 male viewers and 15 female viewers using the commercial software E-PRIME. All viewers were students of The Hong Kong Polytechnic University from Hong Kong or China. The age of the male viewers was 24.572.5 and that of female viewers was 23.071.8. The viewers were required to rate the facial images within a maximum of 4 s in terms of attractiveness in a 1–9 scale (1 being least attractive and 9 being the most attractive). The standard deviation of the average attractiveness ratings ranged from 0.12 to 0.40.
3. Results and analysis 3.1. Ranges of the facial ratios of the sample images In order to investigate the effects of facial ratios on facial attractiveness, it is important that the facial ratios of the sample images should have a wide variation. For the 432 sample images, the 21 facial ratios had 21%–123% variation over the mean and the ranges covered the golden ratios or neoclassical canons. 3.2. Differences between the attractiveness ratings of male viewers and female viewers The average attractiveness ratings (AR) by female raters were compared with those by male raters using the Wilcoxon
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Fig. 3. Examples of facial images created for the study.
signed-rank method, which is a non-parametric statistical hypothesis test for the case of two related samples. The results (z¼ 7.844 (based on positive ranks); asymp. Sig. (2-tailed)¼0.000o0.05)
indicated some statistical differences between the ARs by female and male raters. Detailed comparison of ARs by male raters (ARMs) and ARs by female raters (ARFs) revealed that ARMs are
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slightly greater (5.1% on average) than those of ARFs, However, they are strongly related (R2 ¼0.888). Since the differences between the attractiveness ratings of male viewers and female viewer are small and they are strongly correlated, only the average attractiveness ratings of all the viewers are considered in the following analysis. 3.3. Principal component analysis (PCA) The 21 ratios may be interrelated. It is therefore justifiable to use principal component analysis (PCA) to identify the unrelated principal components. The ratios were first normalized and then analyzed by PCA using SPSS. Under the minimum fraction
variance of 0.05 (or 5%), four principal components were identified with a cumulative variance of about 90%. Fig. 5 plots the cumulative variance explained by the number of principal components. Table 3 lists the transformation matrix between the four components and the 21 ratios. From the correlations between the four components and the ratios, it can be learned that the first component is largely related to the nose width or interocular distance, the second component is largely related to the ratio of nose width over the eye size, the third component is largely related to the nose length and nose to mouth distance, and the fourth component is largely related to the ratio of the length to the width of the face as well as the position of mouth in relation to the nose and chin. Fig. 6 illustrates the effect of the four principal components on the facial images. 3.4. Relationship between principal components and attractiveness ratings Each of the four principal components is plotted against the attractiveness ratings in Fig. 7(a)–(d). As can be seen, in general, there is an optimum value for each of the four components, at which the facial attractiveness is at the best. The deviation from the optimum value of each of the four components will result in a reduction in facial attractiveness, but the trend of reduction is not symmetrical at both side of the optimum value. The effect of each of the four components on the attractiveness ratings is therefore in a distorted bell shape. To capture the main trend, the following transformed Gause formula was applied to fit the data plotted in Fig. 7(a)–(d): ! xn2 a ARi ðxi Þ ¼ piffiffiffiffiffiffi exp i 2 þ ci , ði ¼ 1,2,3,4Þ ð1Þ 2si si 2p where, xni is the transformed value of each of the four components applying the Box–Cox transformation function, viz. 8 0r < ðxi Þ i 1 r a 0 i ri n xi ¼ ð2Þ : lnðx0 Þ r i ¼ 0 i
where, x0i is converted from the original value of each of the four components through Fig. 4. Feature points of a facial image.
x0i ¼
xi xm ui xm
ð3Þ
Table 2 Definition of facial ratios. Ratios
Numerator points
Denominator points
Description
Optimal putative ratios
r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21
y10–y21 y10–y21 x15–x16 x15–x16 x25–x27 y23–y29 y23–y29 x12–x13 x12–x13 x18–x20 x18–x20 x12–x11 y23–y28 y1–y29 y19–y29 x18–x20 x25–x27 y3–y19 y3–y19 x12–x13 x6–x9
x12–x13 x18–x20 x12–x13 x18–x20 x12–x13 x12–x13 x18–x20 x12–x11 y23–y28 x12–x11 y23–y28 y19–y26 y19–y26 x6–x9 y26–y29 y19–y26 x18–x20 y19–y29 y10–y21 x18–x20 (x18–x20)n4
Ear length to interocular distance Ear length to nose width Mideye distance to interocular distance Mideye distance to nose width Mouth width to interocular distance Lips–chin distance to interocular distance Lips–chin distance to nose with Interocular distance to eye fissure width Interocular distance to lip height Nose width to eye fissure width Nose width to lip height Eye fissure width to nose–mouth distance Lip height to nose–mouth distance Length of face to width of face Nose-chin distance to lips–chin distance Nose width to nose–mouth distance Mouth width to nose width Nose length to nose–chin distance Nose length to ear length Interocular distance to nose width Width of face to 4 times of nose with
1.618 1.618 1.618 1.618 1.618 1.280 1.280 1.618 1.618 1.618 1.618 1.000 1.036 1.618 1.618 1.618 1.618 1.000 1.000 1.000 1.000
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Fig. 5. Cumulative variance explained by the number of principal components.
Table 3 Transformation matrix. Ratios
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Component 1
2
3
4
0.254 0.262 0.255 0.306 0.232 0.239 0.249 0.246 0.228 0.041 0.253 0.195 0.026 0.003 0.032 0.250 0.218 0.063 0.057 0.367 0.256
0.269 0.270 0.275 0.196 0.262 0.268 0.261 0.276 0.254 0.357 0.219 0.193 0.002 0.031 0.024 0.202 0.266 0.025 0.028 0.040 0.252
0.085 0.031 0.041 0.085 0.048 0.099 0.009 0.027 0.008 0.046 0.096 0.302 0.448 0.141 0.280 0.253 0.062 0.502 0.484 0.078 0.087
0.033 0.086 0.077 0.024 0.085 0.141 0.212 0.078 0.010 0.076 0.037 0.075 0.263 0.605 0.537 0.167 0.066 0.184 0.280 0.034 0.157
Note: Component 1 relates to nose width or interocular distance. Component 2 relates to the ratio of nose width over the eye size. Component 3 relates to the nose length and nose to mouth distance. Component 4 relates to the ratio of length of face to the width of face as well as the position of mouth in relation to the nose and chin.
xm is an arbitrary value, which is chosen to ensure that the value of the independent variable of the Box–Cox function (x0 ) is positive, and ui is a value to be determined by the nonlinear regression. By the nature of the above formulae, when xi is equal to ui, the ARi ðxi Þ is at its maximum. Therefore ui is the optimum value of ith component resulting in the maximum attractiveness. By applying nonlinear regression, the values of ai, ci, ui, si and ri can be determined and they are listed in Table 4. By assuming that the overall facial attractiveness is the linear combination of the contributions of each of the four components, we have the following equation AR ¼ b0 þ b1 AR1 ðx1 Þ þb2 AR2 ðx2 Þ þb3 AR3 ðx3 Þ þ b4 AR4 ðx4 Þ
ð4Þ
The coefficients in Eq. (4), viz. b0, b1, b2, b3 and b4, can be obtained by multiple linear regression. In this study, the multiple linear regression was carried out based on the least-squares model, i.e. the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line. We obtained b0 ¼ 4.89, b1 ¼0.73, b2 ¼0.67, b3 ¼ 0.19 and b4 ¼0.69. Fig. 8 plots the mean attractiveness ratings of the 432 facial images against the predicted attractiveness rating from the four principal components (derived from the 21 facial ratios) using the regression Eqs. (1) and (4). As can be seen there is a strong one to one linear relationship between the visually assessed mean attractiveness ratings and the predicted attractiveness ratings. The percentage of fit of the model, R2, is 0.6413. The percentage of fit of our model with just four independent predictor variables is a significant improvement over that of Schmid et al.’s model, which had a percentage of fit (R2) of 0.1923 requiring 16 predictor variables. 3.5. Verification of the prediction model In order to examine whether the established model has mapped the general trends of the effects of facial proportions on the facial attractiveness, instead of memorizing the data, it is necessary to test the model against the samples not included in building model. We hence generated additional 113 facial images with randomly chosen facial dimensions using the same software and procedures as those for the previous 432 facial images. The attractiveness ratings of these 113 facial images were rated by 15 male and 15 female viewers, who were aged between 20 and 30 years. Before rating, the viewers were first trained by exposing them to a selection of facial images of varying attractiveness with known ratings rated previously. This ensures that the new group of viewers will have the equivalent scale of rating as the previous group who rated the attractiveness of the facial images used in building the prediction model. The standard deviation of the average attractiveness ratings ranged from 0.07 to 0.35. Fig. 9 plots the actual AR (attractiveness ratings) against the predicted AR using the model for the additional 113 facial images not used for building the model. It can be seen, there is a close to 1:1 relationship between the actual AR and predicted AR with a R2
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7
7
6
6
5
5 AR
AR
Fig. 6. Effect of the four principal components on the facial images.
4 3
3
2
2
1 −10
−8
−6
−4
−2 0 Component1
2
4
6
1 −10
8
7
7
6
6
5
5 AR
AR
4
4
3
2
2 −8
−6
−4
−2 0 Component3
2
4
6
−6
−4
−2 0 Component2
2
4
6
8
−2
−1
0
1 2 Component4
3
4
5
6
4
3
1
−8
1
−3
Fig. 7. Four principal components verses facial attractiveness (a: top left figure, b: top right figure, c: bottom left and d: bottom right).
J. Fan et al. / Pattern Recognition 45 (2012) 2326–2334
Table 4 Parameters obtained from nonlinear regression for each component. Component(i)
ai
ri
si
ci
ui
1 2 3 4
0.68959 0.17894 0.082028 0.76185
5.4094 22.317 25.075 2.9297
0.12881 0.030573 0.022657 0.1175
2.3933 2.7365 3.219 1.9371
1.3804 0.7103 0.2397 1.1598
2333
1.5151, 1.7522, 2.2138, 1.3496, 1.7127, 1.1039, 0.86206, 1.635, 1.5281, 1.4705, 1.4739, 0.77602, 0.89479, 1.3057, 1.1075). Fig. 10 shows the facial image with the optimum ratios. The face with the optimum putative ratios (mostly at the Golden Ratio except for r18–r21, i.e. the optimum ratios of which were believed to be the neoclassical canons of 1, and r6, r7, r12, r13, which cannot be set to the golden ratio when r16 is at the golden ratio) was only rated 2.6. Its image is shown in Fig. 11. Clearly, the popular claim that faces are the most beautiful when the facial ratios are at the Golden Ratio cannot be established. In Meisner’s website [18], the facial images used to illustrate the golden ratios on faces have different expressions (The male face has a neutral expression, but the female face has a big smile). As the expression of the face alter the facial ratios or proportions, it is important to define the expression when considering the effects of facial proportions on facial attractiveness. In our present investigation, the facial images were all controlled at neutral. In another Marquardt’s website [9], he claimed that beautiful faces conform to the Golden Facial Mask derived from Golden Decagon Matrices composed of complex relationship of the Golden ratio phi. Fig. 12 shows the most attractive face found in
Fig. 8. Visually assessed mean attractiveness ratings (AR) vs. the predicted attractiveness ratings (Predicted AR).
Fig. 10. Facial image with the optimum ratios (AR¼ 6.33).
Fig. 9. Actual AR (attractiveness ratings) against the predicted AR using the model for the additional 113 facial images not used for building the model.
(or percentage of fit) of 0.6463. This means that the model has effectively mapped the effects of facial proportions on the facial attractiveness for the facial images with dimensions and proportions varied from an average oriental face.
4. Discussion The prediction model enables us to find the optimum facial proportions for attractiveness. The principle component vector of the optimally attractive face is (1.3804, 0.7103, 0.2397, 1.1598) and the corresponding facial ratios for this optimum face is (r1, r2, r3,y r21) ¼(1.3631, 1.7595, 1.5565, 2.0182, 1.1425, 1.1763,
Fig. 11. Face having neoclassical canons and most possible golden ratios.
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Acknowledgment The authors would like to thank the funding support of Hong Kong Polytechnic University in the form of a Niche Area project (Project no: J-BB6T) and inter-faculty project (Project no: G-YG13). References
Fig. 12. Most attractive face found in our investigation covered with Marquardt’s Golden Facial Mask.
our investigation covered with the proposed Golden Facial Mask. It can seen that the attractive face generally conform to the Golden Facial Mask. However the claim, that beautiful faces should conform to the Golden Facial Mask, could not be statistically tested in the present investigation due to the difficulty in defining the facial landmarks to make measurements corresponding to the geometrical lines of the proposed Golden Facial Mask. In the present study, the effects of facial proportions on facial attractiveness are investigated, using computer generated facial images. Computer generated facial images have the advantage of excluding the influence of hairstyle, facial expression as well as skin tone and texture. However, the applicability of the findings and developed model to real faces should still be tested in future work, although it is foreseeable that, such work will face a big challenge in how to control and quantify the influence of facial proportion, hairstyle, facial expression, skin tone and texture.
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Jintu Fan is a professor at the Institute of Textiles and Clothing, The Hong Kong Polytechnic University. He holds a B.Sc. from Donghua University (formerly China Textile University) and Ph.D from Leeds University. His research interests include beauty psychology, thermal comfort and functional clothing.
Desmon Chau is an Instructor at the Institute of Textiles and Clothing of The Hong Kong Polytechnic University. He gained an M.Sc. in Information Technology with Internet Applications in 2003. His research interests are the Facial Beauty, CAD application, 3D Model development and Multimedia.
Xianfu Wan received his M.Sc. Degree in computer applications from Dong Hua University in 2004, and a Ph.D. Degree in Textiles and Clothing from the Hong Kong Polytechnic University in 2008. His research interests include clothing thermal comfort and visual comfort.
Lili Zhai is a Ph.D research student at the Tianjing Polytechnic University. She was a research assistant at the Institute of Textiles and Clothing, The Hong Kong Polytechnic University, when she worked on this project. Her research interests include 3D body scanning and body beauty.
Ethan Lau graduated from Institute of Textiles and Clothing of The Hong Kong Polytechnic University with a B.A. in Fashion and Textiles. His research interest is 3D modeling.