Complex Plane Transformations for

Manipulation and Visualization of Panoramas

Leonardo Sacht and Luiz Velho

IMPA - Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brazil

Keywords:

Panoramic Images, Conformal Projections.

Abstract:

We present a method for manipulation and visualization of wide-angle images using transformations deﬁned

on the complex plane C. We map the unit sphere S

to C using the stereographic projection, multiply the

complex plane by a given complex number, and map the result back to the sphere using the inverse of the

stereographic projection. Since all these transformations preserve angle, we obtain a result containing only

distortions due to the latitude/longitude representation of the sphere, which were already present in the input

image. We then explore the possibility given by our technique of mapping wide ﬁelds of view to narrower ones.

This makes possible to apply perspective projection to wider ﬁelds of view, leading to a natural generalization

of the perspective projection in the context of panoramic images. Our results are generated in real-time and

compare competitively with state-of-the-art methods used to project the viewing sphere to the image plane.

1 INTRODUCTION

Wide-angle images have become increasingly popular

during the last years due to the development of com-

putational photography techniques and equipment to

capture this kind of images. Although they represent

much better the information of a scene, they usually

present artifacts such as bending of straight lines.

A format commonly used to represent images

of the full sphere around a viewpoint is the equi-

rectangular format (Figure 1 (a), bottom). Each point

on the the viewing sphere is represented by its longi-

tude/latitude coordinates on the equi-rectangular im-

age. Thousands of images in this format are available

on the Internet, for example in (Flickr, 2012).

Our method allows the possibility of applying

conformal transformations to these images, i.e., trans-

formations that preserve angles on the sphere. Due to

this angle-preserving property, the resulting modiﬁed

equi-rectangular images (Figure 1 (d)) could be pro-

jected on a spherical surface such as a dome and the

perceived result would not present unpleasant distor-

tions. We see this possibility as a promising future

application of our technique.

Other possibility than projecting the modiﬁed

spherical image on a dome, is to apply some planar

projection and map it to a new image. It is well known

that the only projection that preserves all possible

straight lines in the scene is the perspective projection,

but it distorts objects for ﬁelds of view wider than 90

degrees. Our technique allows mapping a wide ﬁeld

of view on the sphere to a narrower one, and then ap-

plying the perspective projection to this narrow ﬁeld

of view (Figure 1 (e)). This process leads to high-

quality results, naturally extends the perspective pro-

jection and is very simple to implement in real-time,

making it possible to be incorporated to panorama

viewers such as Google Street View (Google Street

View, 2012) and ﬁeldOfView (ﬁeldOfView, 2012).

Incorporating our technique to these viewers would

improve the navigation offered by them, since their

visualization is limited to a narrow ﬁled of view to

avoid the distortions of the standard perspective pro-

jection.

Figure 1 describes the pipeline of our method:

it receives as input an equi-rectangular image repre-

senting the full viewing sphere around a given view-

point (Figure 1 (a)). We map this sphere to the com-

plex plane using the stereographic projection (Figure

1 (b)), which is known to be conformal and surjective.

We then transform the complex plane by multiplying

each point by some given real number (Figure 1 (c)).

The transformation that we explore most is uniform

scaling of the complex plane, but our technique can

handle full spatially-varying Mobius transformations

as well. This transformed complex plane is mapped

back to the unit sphere using the inverse stereographic

projection, leading to a transformed equi-rectangular

179

Sacht L. and Velho L..

Complex Plane Transformations for Manipulation and Visualization of Panoramas.

DOI: 10.5220/0004197701790184

In Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information

Visualization Theory and Applications (GRAPP-2013), pages 179-184

ISBN: 978-989-8565-46-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

(a) (b) (c) (d) (e)

Figure 1: The input image of our method is a viewing spehre at a viewpoint, which is represented by an equi-rectangular

image (a). We map the sphere to the complex plane using the stereographic projection (ﬁgure (b) shows only a limited part

of the complex plane), apply a transformation to the complex plane (c) and map the result back to the sphere (d). Finally,

perspective projection is applied to the modiﬁed viewing sphere, allowing for visualization of wide ﬁelds of view with a good

balance between straight line and conformality (ﬁgure (e) shows a 160 degree ﬁeld of view).

image (Figure 1 (d)). We then apply a perspective

projection to the resulting sphere (Figure 1 (e)).

All these transformations are independent for each

sphere vertex, allowing us to implement them in par-

allel and obtain real-time interaction. As a limitation,

our results may present straight line bending for some

scenes and especially for ﬁelds of view that are too

wide (wider than 180 degrees).

To summarize, the main contributions of our work

are the following:

• Use of complex transformations to map wide

ﬁelds of view on the sphere to narrower ones in

a conformal way;

• Generalization of the perspective projection to vi-

sualize wide-angle images real-time.

The rest of the paper is organized as follows:

in section 2 we review the panoramic image liter-

ature and relate previous works to ours; In section

3 we describe in detail our technique for manipula-

tion of equi-rectangular images and perspective re-

projection; in section 4 we present implementation

details; in section 5 we describe our results and com-

pare with previous works; Finally, in section 6 we de-

scribe limitations and possibilities for future work.

2 RELATED WORK

Due to their increasing popularity and interest,

panoramic images have become a theme of intense

discussion in the Computer Graphics and Image Pro-

cessing communities in the last twenty years.

The impossibility of obtaining a global projection

from the sphere to the plane that preserves all possible

straight lines and object shapes was shown in the sem-

inal work by Zorin and Barr (Zorin and Barr, 1995).

The alternative to this problem proposed in

(Zelnik-Manor et al., 2005) was to use different per-

spective projections in the same scene. In that work

the user speciﬁed different projection planes and view

directions to deﬁne the different projections. The dis-

continuities caused by using different projections for

different regions of the panorama were hidden (if pos-

sible) by choosing the projection planes in a way that

ﬁt well orientation discontinuities that were already

present in the scene.

The work in (German et al., 2007) explored con-

formal mappings to preserve the shape of the objects

in a panoramic image. They investigated the stereo-

graphic projection and scaling of the complex plane

but only for artistic and exploratory purposes. Since

their focus was on shape preservation, their results

present bent lines. In our work we go beyond and im-

prove these ideas to map wide ﬁelds of view to nar-

rower ones and add a perspective re-projection step,

which allows for better quality results because the

straight lines are less bent.

Other methods relied on both user interaction and

energy-minimization formulations. (Carroll et al.,

2009) used the important lines in the scene provided

by the user and detected faces to control straight line

preservation and conformality in these regions. (Kopf

et al., 2009) used regions speciﬁed by the user where

the projection should be nearly planar to formulate

their optimization framework. In these methods, user

interaction was usually laborious and the optimiza-

tion formulations made them impossible to be imple-

mented in real-time.

Another important technique was proposed in

(Kopf et al., 2007). Their viewer changes the pro-

jection depending on the ﬁled of view, what can be

achieved by our viewer by applying different scales to

the complex plane as the ﬁeld of view of the perspec-

tive projection changes. But in our work, we let the

user specify both parameters, what gives her or him

more control. Also, our visualization simulates better

camera movements since it is a natural generalization

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

180

Figure 2: Left: perspective projection applied to a 160 degree ﬁeld of view. Center: stereographic projection. Right: Our

result using the value s = 0.8 to scale the complex plane.

of the perspective projection.

Another advantage of our method compared to

previous ones is that we do not rely on heavy user

interaction. The user is only asked to vary the ﬁeld

of view of the perspective projection and the scale to

be applied to the complex plane in our viewer, what

makes our panorama viewer a pleasant experience,

instead of laborious. Also, our formulation is sim-

ple and does not rely on heavy optimizations, which

makes our method possible to be implemented in real-

time. We elaborate more on comparisons with previ-

ous works in Section 5.

A work that is not related to panoramic images

but has some connection with ours is (Crane et al.,

2011). In this work, the authors look for conformal

transformations of surfaces by associating R

with

the imaginary part of the Quaternions Im(H). We

work with the complex plane C instead of Im(H),

since our surface of interest is the unit sphere S

and

\{(0, 0, −1)} is confomally equivalent to the com-

plex plane. Applying their work to our context would

produce conformal results, but straight lines in the

scene would appear bent.

3 METHOD

In this section we explain details of our method to ma-

nipulate equi-rectangular images and visualize them

with less distortions. We ﬁrst give an overview and

motivation of our technique and then we present the

Mathematics involved in each transformation of our

pipeline.

3.1 Overview

The importance of conformal mappings is well known

in geometry applications. Intuitively, a mapping from

a surface to another is conformal if it is locally a com-

position of a scale and a rotation. This property pre-

vents the map to shear the surface, an important prop-

erty for mesh quality and texture mapping of the ﬁnal

surface (for more details, see the work (Crane et al.,

2011)). Regarding panoramic images, conformality is

also essential (Carroll et al., 2009).

In this work, we apply a sequence of conformal

transformations to a given viewing sphere. The only

transformation that is not conformal is the ﬁnal per-

spective projection. Although this seems to be a prob-

lem, we apply this ﬁnal mapping to a narrow ﬁeld

of view of the (conformally) transformed sphere, and

perspective projection does not deviate too much from

conformality for narrow ﬁelds of view. The sequence

of transformations we use is described in Figure 1 and

detailed in the next sub-section.

The choices we have made in our method are justi-

ﬁed in Figure 2. We ﬁrst show the results of applying

the perspective (left) and stereographic (center) pro-

jections to a 160 degree ﬁeld of view. The ﬁrst result

preserves straight lines but presents excessive shear-

ing especially in the periphery of the image. On the

other hand, the stereographic projection is conformal,

but lines are clearly bent. We propose to combine the

good properties of both methods. We ﬁrst observe in

the detail of Figure 2 (left) that the perspective projec-

tion is almost conformal for narrow ﬁelds of view. By

applying the sequence of conformal transformations

described in Figure 1 using the value s = 0.8 to scale

the complex plane, we are able to map 160 degree

FOV to a narrower FOV and then apply perspective

transformation to this narrow ﬁeld of view. Figure 2

(right) shows the ﬁnal result of our method.

We leave to the user the speciﬁcation of the ﬁeld

of view of the original sphere to visualized and the

scale of the complex plane to be applied. Of course,

we recommend that these parameters are such that the

ﬁnal perspective projection is applied to a narrow ﬁeld

of view.

3.2 Transformations

We ﬁrst map the unit sphere to the complex plane us-

ing the stereographic projection, which is illustrated

in Figure 1 (b). This projection consists of map-

ComplexPlaneTransformationsforManipulationandVisualizationofPanoramas

181

ping each point on S

\{(0, 0, −1)} to the z = 1 plane

through lines emanating from the pole opposite to the

point of tangency (0, 0, −1), for which the projection

is not well-deﬁned. The expression for this projection

is given by:

S : S

\{(0, 0, −1)} → C

(x, y, z) 7→ (u, v) =



z+1



(1)

It is well known that the stereographic projection is

a conformal mapping from the sphere to the complex

plane (Snyder, 1987).

The next step of our method is to multiply the

complex plane by some given complex number. We

ﬁrst write the points in their polar form

u + iv = (u, v) 7→ (

√

+ v

, arctan2(u, v)) =

= (r, θ) = re

iθ

(2)

In this form, multiplication by a given complex num-

ber se

iα

is given by the simple expression

(r, θ) 7→ (˜r,

θ) = (rs, θ + α) (3)

From complex analysis (Conway, 1978), it is known

that multiplication by a complex number is a confor-

mal mapping from the complex plane onto itself. We

show in Fig. 1 (c) the effect of multiplying the plane

by 0.6, i.e., a uniform scale of 0.6 and no rotation.

Finally, we rewrite the complex plane in Cartesian

coordinates

(˜r,

θ) 7→ ( ˜u, ˜v) = (˜r cos(

θ), ˜r sin(

θ)) (4)

and map the transformed points back to the unit

sphere using the inverse stereographic projection,

which is also conformal:

−1

: C → S

\{(0, 0, −1)}

( ˜u, ˜v) 7→



4 ˜u

˜u

+ ˜v

4 ˜v

˜u

+ ˜v

˜u

+ ˜v

−4

˜u

+ ˜v



(5)

The result of this transformation is presented in Fig-

ure 1 (d).

At this point we have a modiﬁed viewing sphere

containing the information of a wide FOV represented

in a narrower one. Since all the manipulations per-

formed until now were conformal, one could project

this modiﬁed sphere in a spherical surface such as a

dome and see it without angle distortions, allowing

for acceptable visualizations of wide ﬁelds of view on

a dome.

In this work we apply a perspective transforma-

tion to the modiﬁed sphere, in order to see it in a ﬂat

surface such as a computer screen:

P : S

\{z < 0} → C

(x, y, z) 7→





. (6)

The result of this projection is shown in Figure 1 (e).

4 IMPLEMENTATION

To implement all the transformations we have just

described, we represent the unit sphere as a triangle

mesh. Since all operations from (1) to (6) can be per-

formed independently for each vertex on the mesh, we

implemented them as a GLSL vertex shader.

The shader is loaded by a Qt Viewer Application

that implements our technique. This application con-

sists of an interface where the user speciﬁes the view

direction, the ﬁeld of view of the input viewing sphere

to be visualized and the scale to be applied to the com-

plex plane. The interface also has an option to cal-

culate a good scale value depending on the speciﬁed

ﬁeld of view, so the user speciﬁes only one of the two

parameters.

To illustrate the real-time performance of our

viewer, we show some timing numbers in Table 1.

All these numbers were generated with a screen res-

olution of 1024 ×768 pixels in a PC with an Intel

Xeon Quad Core 2.13GHz and 12 GB of RAM and

a GeForce GTX 470 GPU. We emphasize that mesh

and image resolutions shown in Table 1 generate vi-

sualizations without discretization artifacts.

Table 1: Frame rates generated by our technique while the

user interacts, for different mesh resolutions (vaying on the

rows) and different equi-rectangular panorama image reso-

lutions (varying on the columns).

Vertices \ Pixels 4000 ×2000 8000 ×4000

200 ×200 93 fps 89 fps

400 ×400 85 fps 84 fps

800 ×800 35 fps 33 fps

5 RESULTS AND DISCUSSION

In the accompanying video of this paper we show

three results of our technique. The ﬁrst one shows

the effect of applying different scales to the view-

ing sphere, without applying a perspective transfor-

mation. The proportions of the objects in the equi-

rectangular image change due to the scaling param-

eter, but the angles are well preserved. We also ob-

serve that the transformations are bijective, i.e., the

whole content of the sphere appear in all transformed

results.

The next two results in the accompanying video

show the result of applying a scale to the sphere, fol-

lowed by a perspective transformation. Our interface

allows the user to look around the sphere and control

the ﬁeld of view and scaling factor. In both results, we

zoom out until the perspective projection (our result

with scale s = 1) becomes too stretched. To correct

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

182

(a) Equi-rectangular (b) Perspective (c) Mercator (d) Ours, s = 0.75

Figure 3: Comparison with standard projections for a 160 degree ﬁeld of view. Our result is the only one with a good balance

between straight line and object shape preservation.

this problem, we decrease the scale s which makes

possible to see a much wider ﬁeld of view with less

distortions. We observe that the FOV shown in the

video is the one used to deﬁne the perspective pro-

jection on the modiﬁed viewing sphere, which is not

the same as the original viewing sphere. For instance,

when we have a FOV of 130 degrees and s = 0.5 we

are seeing a FOV wider than 180 degrees in the origi-

nal sphere.

We also compare our method with standard pro-

jections to map the sphere to the image plane in Fig-

ure 3. We observe that for the ﬁeld of view of 160

degrees the perspective projection (b) distorts objects

too much but preserves all straight lines. On the other

hand, Mercator (c) projection preserves object shapes

because it is conformal, but bends lines. The result

obtained with our method (d) using a scale parameter

s = 0.75 is the only one that presents a good balance

between these two properties.

Finally, we compare our method with recent

works on the same topic in Figure 4. The result ob-

tained by the technique proposed in (Zelnik-Manor

et al., 2005) (a) shows discontinuities on the ﬂoor pro-

duced by using different projections for different ar-

eas of the image. This strategy works successfully for

the building in the image, since these discontinuities

are hidden by natural discontinuities in the scene, but

it fails to ﬁt the geometry of the ﬂoor. In Fig. 4 (b) we

show a result produced by our implementation of the

technique in (Carroll et al., 2009). All straight lines

speciﬁed by the user (please see their work for more

details) are well-preserved, but the lines on the ﬂoor

appear bent, since they are too many to be marked by

the user. Although their energy minimization formu-

lation guarantees conformality and smoothness of the

ﬁnal result, it has the problem of taking some seconds

to be performed. Our result ((c), for which we apply

a scale s = 0.8 to the complex plane) does not rely on

heavy user interaction nor on any optimization and is

not restricted to scenes with any particular geometry.

6 LIMITATIONS AND FUTURE

WORK

A limitation of our method appears when one uses our

interface to visualize too wide ﬁelds of view (Figure

5). Applying standard perspective projection (s = 1)

is prohibitive in this case, and applying a small scale

to the complex plane makes the ﬁnal result of our

method deviate too much from the standard perspec-

tive projection and present bent lines. One possibility

to overcome this limitation would be to apply content-

dependent scale to the complex plane, i.e., regions

with straight lines would be forced to have a scaling

parameter close to 1. However, the speciﬁcation of

important lines in the scene could be laborious, as al-

ready happened in previous works.

We also intend to use our technique to map wide

ﬁelds of view to narrower ones and project the re-

sulting viewing sphere on a spherical dome, instead

of applying a planar perspective projection as we de-

scribed in Section 3. This visualization would not

have the distortions caused by the planar projections,

and would beneﬁt from our conformal pipeline.

Another interesting direction for future work is to

extend our technique to panoramic videos, i.e., tem-

porally varying viewing spheres. Distortions as the

ones observed in (Sacht et al., 2011) would have to be

considered and a time-varying warping could have to

be formulated.

Since the technique we have presented in this pa-

per is simple and can be implemented in real-time,

we think it can be easily incorporated to current

panorama viewers such as Google Street View.

ACKNOWLEDGEMENTS

The ﬁrst author thanks CNPq for the ﬁnancial sup-

port. The authors thank the following Flickr users

ComplexPlaneTransformationsforManipulationandVisualizationofPanoramas

183

(a) (Zelnik-Manor et al., 2005) (b) (Carroll et al., 2009) (c) Ours, s = 0.8

Figure 4: Comparison with other methods to project the viewing sphere to the image plane for a ﬁeld of view of 150 degrees.

In the result obtained by the method in (Zelnik-Manor et al., 2005) (a), the different perspective projections used for different

areas of the image appear clear and unpleasant on the ﬂoor of the scene. The method by (Carroll et al., 2009) (b) preserves

all straight lines marked by the user, but fails to preserve the ones on the ﬂoor (which are too many to be marked by the user).

Our result (c) has all straight lines in the scene with very little bending.

Figure 5: Top-left: input equi-rectangular image. Top-

right: result of applying a scale s = 0.05 on the viewing

sphere. Bottom: result of perspective projection applied to

the transformed viewing sphere (showing only the area in-

side the red rectangle of the top-right ﬁgure). Although al-

most the entire viewing sphere is being shown, unpleasant

straight line distortions appear.

for making available their images under the Creative

Commons license: Janne., gadl and HamburgerJung.

REFERENCES

Carroll, R., Agrawal, M., and Agarwala, A. (2009). Opti-

mizing content-preserving projections for wide-angle

images. In ACM SIGGRAPH 2009 papers, SIG-

GRAPH ’09, pages 43:1–43:9, New York, NY, USA.

ACM.

Conway, J. B. (1978). Functions of One Complex Variable.

Springer-Verlag, New York.

Crane, K., Pinkall, U., and Schr

oder, P. (2011). Spin trans-

formations of discrete surfaces. ACM Trans. Graph.,

40.

ﬁeldOfView (2012). Interactive panoramas - ﬁeldofview.

http:// ﬁeldofview.com/panoramas.

Flickr (2012). Flickr: Equirectangular. http://

www.ﬂickr.com/groups/equirectangular/.

German, D. M., Burchill, L., Duret-Lutz, A., Prez-Duarte,

S., Prez-Duarte, E., and Sommers, J. (2007). Flat-

tening the viewable sphere. In Cunningham, D. W.,

Meyer, G. W., Neumann, L., Dunning, A., and Pari-

cio, R., editors, Computational Aesthetics, pages 23–

28. Eurographics Association.

Google Street View (2012). Street view - google maps.

http:// maps.google.com/streetview.

Kopf, J., Lischinski, D., Deussen, O., Cohen-Or, D., and

Cohen, M. F. (2009). Locally adapted projections to

reduce panorama distortions. Comput. Graph. Forum,

28(4):1083–1089.

Kopf, J., Uyttendaele, M., Deussen, O., and Cohen, M. F.

(2007). Capturing and viewing gigapixel images. In

ACM SIGGRAPH 2007 papers, SIGGRAPH ’07, New

York, NY, USA. ACM.

Sacht, L., Velho, L., Nehab, D., and Cicconet, M. (2011).

Scalable motion-aware panoramic videos. In SIG-

GRAPH Asia 2011 Sketches, SA ’11, pages 37:1–

37:2, New York, NY, USA. ACM.

Snyder, J. P. (1987). Map projections – a working manual.

Technical Report 1395, U. S. Geological Survey.

Zelnik-Manor, L., Peters, G., and Perona, P. (2005). Squar-

ing the circles in panoramas. In Proceedings of

the Tenth IEEE International Conference on Com-

puter Vision - Volume 2, ICCV ’05, pages 1292–1299,

Washington, DC, USA. IEEE Computer Society.

Zorin, D. and Barr, A. H. (1995). Correction of geomet-

ric perceptual distortions in pictures. In Proceedings

of the 22nd annual conference on Computer graph-

ics and interactive techniques, SIGGRAPH ’95, pages

257–264, New York, NY, USA. ACM.

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

184