# Influence of 3D Centro-Symmetry on a 2D Retinal Image

## Abstract

**:**

## 1. Introduction

## 2. Theorems and Proofs

#### 2.1. A 2D Orthographic Projection of 3D Centro-Symmetry

_{i}and Q

_{i}is at the symmetry point ${O}_{3D}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{t}$.

_{i}and Q

_{i}can be written as:

_{i}is the projection of P

_{i}and q

_{i}is the projection of Q

_{i}. The relation between p

_{i}and q

_{i}can be represented as ${p}_{i}=-{q}_{i}$. It is equivalent with a 2D rotation for 180° around ${O}_{2D}={\left[\begin{array}{cc}0& 0\end{array}\right]}^{t}$ to which the symmetry point O

_{3D}is projected. This implies that a 2D orthographic image of a 3D centro-symmetrical object is always 2D 2-fold rotation-symmetrical around a projection of the symmetry center of the object. This means that the image is invariant when it rotates 180° around the projection of the symmetry center.

#### 2.2. Recovering a 3D Centro-Symmetrical Shape from Its 2D Orthographic Image

_{i}and d

_{i}can be constructed as:

_{i}is the 3D interpretation of b

_{i}, D

_{i}is the 3D interpretation of d

_{i}, and Z

_{i}is an arbitrary real number. The midpoint between B

_{i}and D

_{i}is at ${O}_{3D}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{t}$. This construction process can be applied to all 2D rotation-symmetrical pairs of points in the image. For each pair of points, the Z coordinates of their constructed 3D points can be set independently from the other pairs of points. When this is done, the set of pairs of constructed 3D points is 3D centro-symmetrical about O

_{3D}. Each pair of the constructed points can be connected by a symmetry line segment that is bisected by O

_{3D}.

#### 2.3. A 2D Perspective Projection of 3D Centro-Symmetry

_{i}, q

_{i}). A line segment connecting p

_{i}and q

_{i}passes c

_{s}. This pair of points can be written as:

_{i}is the orientation of the line segment connecting p

_{i}and q

_{i}, and $\left|{l}_{pi}\right|$ and $\left|{l}_{qi}\right|$ are the distance of p

_{i}and q

_{i}from c

_{s}. Note that the line segment connecting p

_{i}and q

_{i}is a projection of a symmetry line segment, and c

_{s}is a projection of the symmetry point. The symmetry line segment is parallel to a vector S

_{i}connecting the center of projection [0 0 0]

^{t}and the vanishing point v

_{i}of the symmetry line segment. The vanishing point v

_{i}of the symmetry line segment should be collinear with p

_{i}, q

_{i}, and c

_{s}, and it can be written as:

_{i}from c

_{s}. The endpoints of the symmetry line segment and the symmetry point bisecting the symmetry line segment are projected to p

_{i}, q

_{i}, and c

_{s}. Then, the relation between p

_{i}, q

_{i}, c

_{s}, and v

_{i}can be represented by the following equation (see [25,35]):

_{i}connecting the center of projection [0 0 0]

^{t}and its vanishing point v

_{i}in the 2D image:

_{i}and q

_{i}be P

_{i}and Q

_{i}and the symmetry point be C

_{s}:

_{s}is a free parameter representing the distance of C

_{s}from the center of projection. Note that P

_{i}and Q

_{i}in the 3D scene are connected by the symmetry line segment that is bisected by C

_{s}:

_{i}| is the length of the symmetry line segment. From Equations (3) and (7)–(9), we have:

_{i}can be computed as:

_{s}is the distance of the symmetry point C

_{s}from the center of projection. It determines the size of the constructed 3D centro-symmetrical interpretation. This construction process can be applied to all of the corresponding pairs of points in the 2D image. Δ

_{s}should be the same among all of the pairs of points. When this has been done, the set of pairs of constructed 3D points is 3D centro-symmetrical about C

_{s}. Each pair of constructed points can be connected by a symmetry line segment that is bisected by C

_{s}.

#### 2.4. Model-Based Invariant of 3D Centro-Symmetry with Planar Contours

#### 2.4.1. Model-Based Invariant of 3D Centro-Symmetry with Planar Contours under a 2D Orthographic Projection

_{i}is the 3D interpretation of p

_{i}and Q

_{i}is the 3D interpretation of q

_{i}. The parameter Z

_{i}is computed as:

_{X}, N

_{Y}, N

_{Z}, and k

_{Φ}are free parameters. This construction process can be applied to all corresponding pairs of points on ϕ and ψ. N

_{X}, N

_{Y}, N

_{Z}, and k

_{Φ}should be the same among all of the pairs of points. When this is done, a 3D centro-symmetrical pair of planar curves Φ and Ψ can be constructed from ϕ and ψ. Equation (12) represents a plane of Φ. The plane of Φ is parallel to a plane of Ψ.

#### 2.4.2. Model-Based Invariant of 3D Centro-Symmetry with Planar Contours under a 2D Perspective Projection

_{Φ}of Φ and plane Π

_{Ψ}of Ψ are parallel to one another. Set the 2D Cartesian coordinate system of a 2D image and the 3D Cartesian coordinate system of a 3D scene in which the 3D interpretation will be constructed as follows: (i) the origin of the 3D coordinate system is at the center of projection, (ii) the Z-axis of the 3D coordinate system coincides with the principal axis and it is perpendicular to the image plane Z = f, where f is the focal distance, (iii) the Z-axis passes the origin of the 2D coordinate system, and (iv) the X- and Y-axes of the 3D coordinate system are parallel to the x- and y-axes of the 2D coordinate system.

_{Φ}and Π

_{Ψ}are frontoparallel:

_{Φ}, k

_{Ψ}are constants. Let the symmetry center of Φ and Ψ be ${C}_{s}={\left[\begin{array}{ccc}{X}_{C}& {Y}_{C}& {Z}_{C}\end{array}\right]}^{t}$. From Equation (13), $\left({k}_{\Phi}+{k}_{\Psi}\right)/2={Z}_{C}$. Now, consider a corresponding pair of points P

_{i}on Φ and Q

_{i}on Ψ. P

_{i}and Q

_{i}can be written as: ${P}_{i}={\left[\begin{array}{cc}{X}_{C}+{X}_{i}& {Y}_{C}+{Y}_{i}\hspace{1em}{k}_{\Phi}\end{array}\right]}^{t}$ and ${Q}_{i}={\left[\begin{array}{ccc}{X}_{C}-{X}_{i}& {Y}_{C}-{Y}_{i}& 2{Z}_{C}-{k}_{\Phi}\end{array}\right]}^{t}$. Their perspective projections can be written as:

_{i}and q

_{i}is:

_{S}. This shows that the relation between φ and ψ can be represented by a sub-group of the 2D affine transformation, namely scaling by a factor of −k

_{Ψ}/k

_{Φ}around c

_{s}.

_{1}, b

_{2}, … b

_{n}) on β and (d

_{1}, d

_{2}, … d

_{n}) on δ where n ≥ 3. They satisfy Equation (15): (b

_{i}– c

_{βδ}) = −k

_{βδ}(d

_{i}– c

_{βδ}), where i is an integer (1 ≤ i ≤ n), k

_{βδ}is a constant, and c

_{βδ}is an intersection of line segments connecting the corresponding pairs of points between β and δ. From Equation (3), b

_{i}and d

_{i}can be written as b

_{i}= c

_{βδ}+ l

_{bi}[cosθ

_{i}sinθ

_{i}]

^{t}and d

_{i}= c

_{βδ}+ l

_{di}[cosθ

_{i}sinθ

_{i}]

^{t}, where θ

_{i}is the orientation of the line segment connecting b

_{i}and d

_{i}, and $\left|{l}_{bi}\right|$ and $\left|{l}_{di}\right|$ are the distance of b

_{i}and d

_{i}from c

_{βδ}. Note that l

_{bi}/l

_{di}is k

_{βδ}for all of the corresponding pairs of points, because β and δ satisfy Equation (15). Let B

_{i}and D

_{i}be the 3D centro-symmetrical interpretations of b

_{i}and d

_{i}and Z

_{Bi}and D

_{Bi}be their Z-coordinates. From Equations (7)–(11), the Z-coordinates of B

_{i}and D

_{i}are:

_{s}is a free parameter representing the distance of the symmetry point from the center of projection. It should be the same among all of the pairs of points. Note that $\left({l}_{bi}+{l}_{di}\right)/\left({l}_{di}-{l}_{bi}\right)$ is also the same among all of the pairs of points, because l

_{bi}/l

_{di}is constant (k

_{βδ}). Equation (16) shows that the 3D centro-symmetrical interpretations of β and δ are a pair of contours that are individually frontoparallel and planar.

_{Φ}and Π

_{Ψ}are not frontoparallel:

_{Φ}, k

_{Ψ}, N

_{X}, N

_{Y}, and N

_{Z}are constants, and N

_{X}

^{2}+ N

_{Y}

^{2}= 1. Note that [N

_{X}N

_{Y}N

_{Z}]

^{t}is normal to Π

_{Φ}and Π

_{Ψ}. Let C

_{S}= [X

_{C}, Y

_{C}, Z

_{C}]

^{t}be their symmetry center. Note that a line connecting two of the points P

_{i}and P

_{j}of Φ, and a line connecting their corresponding points Q

_{i}and Q

_{j}of Ψ, are parallel to one another. Let φ, ψ, and c

_{s}be 2D perspective projections of Φ, Ψ, and C

_{S}. This general case can be transformed to a simple case in which Π

_{Φ}and Π

_{Ψ}are frontoparallel by rotating the principal axis and the image plane around the center of projection [0 0 0]

^{t}. The perspective image after this rotation can be computed by using Kanatani’s transformation [37].

_{1}, p

_{2}, … p

_{n}) on φ and (q

_{1}, q

_{2}, … q

_{n}) on ψ where n ≥ 3 (Figure 4). They are projections of points (P

_{1}, P

_{2}, … P

_{n}) on Φ and (Q

_{1}, Q

_{2}, … Q

_{n}) on Ψ. Draw a line connecting two of the points p

_{i}and p

_{j}on φ and a line connecting their corresponding points q

_{i}and q

_{j}on ψ (the dashed gray lines in Figure 4). The intersection h

_{ij}of these two lines is a vanishing point of two parallel lines connecting P

_{i}and P

_{j}and connecting Q

_{i}and Q

_{j}. These lines are on the planes Π

_{Φ}and Π

_{Ψ}individually, and they are parallel to one another. It follows that h

_{ij}is on a line H

_{s}(the solid red line in Figure 4) that is a common horizon of Π

_{Φ}and Π

_{Ψ}. Another vanishing point can be found by drawing a line connecting different points on φ and by drawing a line connecting their corresponding points on ψ. An additional corresponding pair of points on φ and ψ can be determined by drawing a line (the dotted green line in Figure 4) passing c

_{s}and finding its intersection with φ and ψ if necessary.

_{s}. H

_{s}can be determined by finding a line that connects these vanishing points.

_{s}of Π

_{Φ}and Π

_{Ψ}, a vanishing point of a line normal to Π

_{Φ}and Π

_{Ψ}can be found. Now, draw a line that passes the principal point [0 0]

^{t}in the image and that is perpendicular to H

_{s}. Let h

_{v}be an intersection of this line with H

_{s}. Next, find a point v

_{Π}on this line such that the visual angle between h

_{v}and v

_{Π}from the center of projection is perpendicular. The line connecting v

_{Π}and the center of projection is normal to Π

_{Φ}and Π

_{Ψ}and it is parallel to [N

_{X}N

_{Y}N

_{Z}]

^{t}(Equation (17)).

_{Φ}and Π

_{Ψ}and Π

_{Φ}and Π

_{Ψ}become frontoparallel. This rotation is around a vector [N

_{Y}− N

_{X}0]

^{t}and the degree of the rotation is $\theta ={\mathrm{cos}}^{-1}\left({N}_{Z}/\sqrt{{N}_{Z}{}^{2}+1}\right)$.

_{Π}representing this rotation can be formulated by using Rodrigues’ rotation formula:

_{θ}= cosθ, s

_{θ}= sinθ, and r

_{θ}= 1−cosθ. After the rotation R

_{Π}, a point [x y]

^{t}in the original image is transformed to [x′ y′]

^{t}[37]:

_{Φ}and Π

_{Ψ}become frontoparallel after R

_{Π}. With this done, the transformed image satisfies Equation (15). From Equations (15) and (19), the relation between φ and ψ is:

_{i}′, q

_{i}′, and c

_{s}′ are the transformations of p

_{i}, q

_{i}, and c

_{s}by using Equation (19). This shows that the relation between φ and ψ can be represented by the transformation of Equation (19) and by a sub-group of the 2D affine transformation that is scaling.

## 3. General Discussion

## Funding

## Conflicts of Interest

## References

- Biederman, I. Recognition-by-components: A theory of human image understanding. Psychol. Rev.
**1987**, 94, 115–147. [Google Scholar] [CrossRef][Green Version] - Feldman, J. Formation of visual “objects” in the early computation of spatial relations. Percept. Psychophys.
**2007**, 69, 816–827. [Google Scholar] [CrossRef][Green Version] - Koffka, K. Principles of Gestalt Psychology; Harcourt, Brace, & World: New York, NY, USA, 1935. [Google Scholar]
- Kubilius, J.; Sleurs, C.; Wagemans, J. Sensitivity to nonaccidental configurations of two-line stimuli. i-Perception.
**2017**, 8, 1–12. [Google Scholar] [CrossRef] - Leeuwenberg, E.; van der Helm, P.A. Structural Information Theory: The Simplicity of Visual Form; Cambridge University Press: New York, NY, USA, 2013. [Google Scholar]
- Metzger, W. Laws of Seeing Translated by L Spillman & S Lehar; MIT Press: Cambridge, MA, USA, 2009. [Google Scholar]
- Wagemans, J. Perceptual use of nonaccidental properties. Can. J. Psychol.
**1992**, 46, 236–279. [Google Scholar] [CrossRef] [PubMed] - Witkin, A.P.; Tenenbaum, J.M. On the Role of Structure in Vision. In Human and Machine Vision; Beck, J., Hope, B., Rosenfeld, A., Eds.; Academic Press: New York, NY, USA, 1983; pp. 481–543. [Google Scholar]
- van der Helm, P.A.; Leeuwenberg, E.L.J. Goodness of visual regularities: A nontransformational approach. Psychol. Rev.
**1996**, 103, 429–456. [Google Scholar] [CrossRef] [PubMed] - Swaddle, J. Visual signaling by asymmetry: A review of perceptual processes. Philos. Trans. Biol. Sci.
**1999**, 354, 1383–1393. [Google Scholar] [CrossRef] [PubMed] - Treder, M.S. Behind the looking-glass: A review on human symmetry perception. Symmetry
**2010**, 2, 1510–1543. [Google Scholar] [CrossRef] - Wagemans, J. Detection of visual symmetries. Spat. Vis.
**1995**, 9, 9–32. [Google Scholar] [CrossRef] - Liu, Y.; Hel-Or, H.; Kaplan, C.S. Computational symmetry in computer vision and computer graphics. Found. Trends Comput. Graph. Vis.
**2010**, 5, 1–195. [Google Scholar] [CrossRef] - Stewart, I.; Golubitsky, M. Fearful Symmetry: Is God a Geometer? Penguin Books: London, UK, 1993. [Google Scholar]
- Weyl, H. Symmetry; Princeton University Press: Princeton, NJ, USA, 1952. [Google Scholar]
- Kahn, J.I.; Foster, D.H. Horizontal-vertical structure in the visual comparison of rigidly transformed patterns. J. Exp. Psychol. Hum. Percept. Perform.
**1986**, 12, 422–433. [Google Scholar] [CrossRef] - Mach, E. The Analysis of Sensations and the Relation of the Physical to the Psychical; Dover: New York, NY, USA, 1959. [Google Scholar]
- Makin, A.D.J.; Rampone, G.; Pecchinenda, A.; Bertamini, M. Eletrophysiological responses to visuospatial regularity. Psychophysiology
**2013**, 50, 1045–1055. [Google Scholar] [PubMed] - Makin, A.D.J.; Wilton, M.M.; Pecchinenda, A.; Bertamini, M. Symmetry perception and affective responses: A combined EEG/EMG study. Neuropsychologia
**2012**, 50, 3250–3261. [Google Scholar] [CrossRef] [PubMed] - Wagemans, J.; Van Gool, L.; Swinnen, V.; Van Horebeek, J. Higher-order structure in regularity detection. Vis. Res.
**1993**, 33, 1067–1088. [Google Scholar] [CrossRef] - Funk, C.; Liu, Y. Beyond planar symmetry: Modeling human perception of reflection and rotation symmetries in the wild. In Proceedings of the IEEE International Conference on Computer Vision 2017, Venice, Italy, 22–29 October 2017; pp. 793–803. [Google Scholar]
- Rothwell, C.A. Object Recognition through Invariant Indexing; Oxford University Press: Oxford, UK, 1995. [Google Scholar]
- Sawada, T.; Li, Y.; Pizlo, Z. Shape Perception. In Oxford Handbook of Computational and Mathematical Psychology; Busemeyer, J., Townsend, J., Wang, Z.J., Eidels, A., Eds.; Oxford University Press: New York, NY, USA, 2015; pp. 255–276. [Google Scholar]
- Horaud, R.; Brady, M. On the geometric interpretation of image contours. Artif. Intell.
**1988**, 37, 333–353. [Google Scholar] [CrossRef] - Sawada, T.; Zaidi, Q. Rotational symmetry in a 3D scene and its 2D image. J. Math. Psychol.
**2018**, 87, 108–125. [Google Scholar] [CrossRef] - Wong, K.K.; Mendoça, P.R.S.; Cipolla, R. Reconstruction of surfaces of revolution from single uncalibrated views. Image. Vis. Comput.
**2004**, 22, 829–836. [Google Scholar] [CrossRef][Green Version] - Hong, W.; Yang, A.Y.; Huang, K.; Ma, Y. On symmetry and multiple-view geometry: Structure, pose, and calibration from a single image. International J. Comput. Vis.
**2004**, 60, 241–265. [Google Scholar] [CrossRef][Green Version] - Vetter, T.; Poggio, T. Symmetric 3D objects are an easy case for 2D object recognition. Spat. Vis.
**1994**, 8, 443–453. [Google Scholar] - Yang, A.Y.; Huang, K.; Rao, S.; Hong, W.; Ma, Y. Symmetry-based 3-D reconstruction from perspective images. Comput. Vis. Image Underst.
**2005**, 99, 210–240. [Google Scholar] [CrossRef][Green Version] - Pizlo, Z.; Li, Y.; Sawada, T.; Steinman, R. Making a Machine that Sees Like Us; Oxford University Press: New York, NY, USA, 2014. [Google Scholar]
- Sawada, T. Visual detection of symmetry of 3D shapes. J. Vis.
**2010**, 10, 4. [Google Scholar] [CrossRef] - Hong, W.; Ma, Y.; Yu, Y. Reconstruction of 3-D deformed symmetric curves from perspective images without discrete features. In Lecture Notes in Computer Science: Computer Vision-ECCV; Pajdla, T., Matas, J., Eds.; Springer-Verlag: Berlin, Germany, 2004; Volume 3023, pp. 533–545. [Google Scholar]
- Sawada, T.; Li, Y.; Pizlo, Z. Any pair of 2D curves is consistent with a 3D symmetric interpretation. Symmetry
**2011**, 3, 365–388. [Google Scholar] [CrossRef] - Sugihara, K. Anomalous mirror symmetry generated by optical illusion. Symmetry
**2016**, 8, 21. [Google Scholar] [CrossRef][Green Version] - Erkelens, C.J. Equidistant intervals in perspective photographs and paintings. i-Perception
**2016**, 7, 4. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sawada, T.; Li, Y.; Pizlo, Z. Detecting 3-D mirror symmetry in a 2-D camera image for 3-D shape recovery. Proc. IEEE
**2014**, 102, 1588–1606. [Google Scholar] [CrossRef] - Kanatani, K. Constraints on length and angle. Comput. Vis. Graph. Image Process
**1988**, 41, 28–42. [Google Scholar] [CrossRef] - Chen, C.C.; Sio, L.T. 3D surface configuration modulates 2D symmetry detection. Vis. Res.
**2015**, 107, 86–93. [Google Scholar] [CrossRef] - Farell, B. The perception of symmetry in depth: Effect of symmetry plane orientation. Symmetry
**2015**, 7, 336–353. [Google Scholar] [CrossRef][Green Version] - Locher, P.; Smets, G. The influence of stimulus dimensionality and viewing orientation on detection of symmetry in dot patterns. Bull. Psychon. Soc.
**1992**, 30, 43–46. [Google Scholar] [CrossRef][Green Version] - Treder, M.S.; van der Helm, P.A. Symmetry versus repetition in cyclopean vision: A microgenetic analysis. Vis. Res.
**2007**, 47, 2956–2967. [Google Scholar] [CrossRef]

**Figure 1.**Objects with 3 types of 3D symmetry: (

**a**) a 3D mirror-symmetrical object, (

**b**) a 3D rotation-symmetrical object, and (

**c**) a 3D centro-symmetrical object.

**Figure 2.**An orthographic image of a 3D centro-symmetrical object (Figure 1c) with its surface transparent. The 2D image of the 3D centro-symmetrical shape itself is 2D rotational symmetrical with 2 folds, once we ignore the occlusion by the opaque surface of the shape.

**Figure 3.**(

**a**) The 3D mirror- and (

**b**) the 3D rotation-symmetrical objects in Figure 1 viewed from different directions. Thick black lines represent the viewing directions for images in Figure 1, and the dotted lines represent the viewing directions of their virtual images that were generated from the images in Figure 1.

**Figure 4.**A pair of 2D curves ϕ and ψ satisfying the following two conditions: (i) line segments connecting the corresponding pairs of points on ϕ and ψ (the dotted gray line) intersect with one another at a point c

_{s}(the red diamond), (ii) a line connecting two of the points on φ and a line connecting their corresponding points on ψ always intersect on H

_{s}.

**Figure 5.**Stereoscopic images of random dot patterns with 2D rotational symmetry (for both crossed and uncrossed fusion). The depth distributions of the patterns were (

**a**) 3D centro-symmetrical and (

**b**) 3D rotation-symmetrical.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sawada, T.
Influence of 3D Centro-Symmetry on a 2D Retinal Image. *Symmetry* **2020**, *12*, 1863.
https://doi.org/10.3390/sym12111863

**AMA Style**

Sawada T.
Influence of 3D Centro-Symmetry on a 2D Retinal Image. *Symmetry*. 2020; 12(11):1863.
https://doi.org/10.3390/sym12111863

**Chicago/Turabian Style**

Sawada, Tadamasa.
2020. "Influence of 3D Centro-Symmetry on a 2D Retinal Image" *Symmetry* 12, no. 11: 1863.
https://doi.org/10.3390/sym12111863