CURVATURE-APPROXIMATED ESTIMATION OF REAL-TIME

AMBIENT OCCLUSION

Tomohito Hattori, Hiroyuki Kubo and Shigeo Morishima

Waseda University, Tokyo, Japan

Keywords:

Real-time, Ambient Occlusion, For Game, Curvature.

Abstract:

We present a novel technique for computing ambient occlusion (Akenine-M¨oller et al., 2008) on real-time

graphics hardware. Our method approximates the occlusion for a local illumination model by introducing

curvature-dependent function. Using our method, we are able to acquire occlusion at lower computational cost

than conventional methods such as SSAO (Bavoil et al., 2008). Our method requires a multi-pass algorithm

with the graphics processing unit (GPU). In the ﬁrst pass curvature is acquired, and in the second pass the

occlusion is computed from the curvature. In the calculating occlusion from the curvature, we approximate

the geometric shape by a quadric surface function, and then obtain a curvature dependent function which is an

approximation of geometric surface. This function depends only on local variables and we are able to calculate

the ambient occlusion for the local illumination model. According to our results, both the computational

and storage costs are sufﬁciently low for the technique to be applied to current graphics hardware supported

computer games.

1 INTRODUCTION

Currently, a large number of methods such as soft

shadows, subsurface scattering, high-dynamic range

effects, are being applied in real-time graphics appli-

cations.

Ambient occlusion (Akenine-M¨oller et al., 2008)

is one such method which is as a special case of the

obscurances technique. This method was ﬁrst pre-

sented by Zhukov et al. and popularized by Landis

and Christensen in the movie industry. Ambient oc-

clusion provides a distinctive edge or soft shadow, by

taking account of the attenuation of ambient light due

to the presence of nearby geometrical features. The

method has been used in ﬁlms as an alternative to

large computational global illumination methods, be-

cause, it can imitate indirect lighting effects such as

global illumination at minimum computational cost.

The ambient occlusion A at a point

P with surface

normal~n is given by

P,~n) =

Ω

(V(

ω,

P)⌊

ω·~n⌋)dω (1)

Where

ω represents random directions from

P on the

surface of the unit hemisphere Ω

. V(

ω,

P) is the vis-

ibility function which has the value 0 when no geo-

metrical feature is visible in the direction

ω from

and the value 1 otherwise.

In the past the ambient light was given by a con-

stant value in many cases. Multiplying the general

ambient term (Phong, 1975) with (1-A(

P,~n)) gener-

ates a much better looking result than the constant am-

bient term only. The ambient occlusion solution bears

comparison with the full global illumination solution,

because ambient occlusion is able to approximate the

ambient term attained by analyzing the global behav-

ior of light.

For this reason, ambient occlusion is also now

gaining interest in the game industry. Several ap-

proaches have been applied in 3D games, baking am-

bient occlusion to create texture, ambient occlusion

ﬁelds (Kontkanen and Laine, 2005), screen space am-

bient occlusion (Bavoil et al., 2008) and the like.

However, those approaches are based on various

direction sampling and the computational cost to im-

plement them on computer video games is high. We

are able to reduce the computational cost more than

with current ambient occlusion techniques, by intro-

ducing an algorithm which computes the occlusion

from a curvature dependent function.

The basic idea is as follows. We approximate a

geometrical shape at a point

P by a curvature depen-

dent function f(

P) and we employ the maximum and

minimum principal curvatures as this function’s vari-

268

Hattori T., Kubo H. and Morishima S..

CURVATURE-APPROXIMATED ESTIMATION OF REAL-TIME AMBIENT OCCLUSION.

DOI: 10.5220/0003848402680273

In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2012), pages 268-273

ISBN: 978-989-8565-02-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Our ambient occlusion sample applied to the Stan-

ford dragon model.

ables. Then comparing f(

P) with Ω

, we detect an

occluded direction from f(

P) at

P. In the next sec-

tion we discuss the previous work. The concept of the

curvature dependent ambient occlusion is explained

in Section 3, the curvature estimation in Section 4 and

the implementation in Section 5.

2 PREVIOUS WORK

Ambient occlusion was originally regarded as a sim-

pliﬁed case of the obscurances illumination model

(Zhukov et al., 1998), in which the visibility function

contains distance information of the occlusive. In this

method, the obscurance W is given as follows.

P,~n) :=

Ω

ρ(d(

ω,

P)⌊

ω·~n⌋)dω (2)

where d(

ω,

P) refers to the distance of the ﬁrst inter-

ception along a ray direction

ω from

P. The function

ρ ?maps the appropriate distance. Especially in the

game industry, there is a great demand for a real-time

ambient occlusion computing technique, and there are

various attractive methods.

Mendez et al. (M´endez et al., 2003) introduced a

real-time computingambient occlusion method which

dynamically updates the obscurance information in

the presence of deforming objects. In a moving scene,

obscurances are re-sampled only in a selected region

of the scene by utilizing temporal coherence. How-

ever, the method requires a huge number of patches

to give high quality contact shadows.

Ambient occlusion has been popularized in pro-

duction rendering (Landis, 2002) and has been

adopted in the movie, television and game industries.

When ambient occlusion is implemented to create

content, it is typically computed on the surface of

each object in a scene and baked on a texture map to

reduce the computational cost in the run-time. How-

ever, a huge number of occlusion maps have to be

prepared for a movie creation, because one occlusion

map of all the objects in each frame is needed. If the

computing time per map is long, huge timing com-

plexity arises. Therefore many derivative methods

of ambient occlusion have been suggested and devel-

oped. In this case, the goal is to evaluate equation(1)

on the surface of an object and to implement the am-

bient occlusion speedily.

Bavoil et al. (Bavoil et al., 2008) introduced

screen space ambient occlusion (SSAO) which is a

multi-pass rendering technique using a z-buffer. A z-

buffer is one of the back-buffers that contains normal

vectors and the depth of pixels in the screen space.

In SSAO, the occlusion at a point is computed by the

GPU from this z-buffer. The basic idea is as follows.

The simpliﬁed shape of the nearby geometry is de-

tected by the depth and normal vector direction of

one (or two) pixels neighboring the depth buffer. The

Monte-Carlo collision detection of the ambient occlu-

sion is calculated from this shape information. Thus,

Bavoil et al. (Bavoil and Sainz, 2009) introduced a

developed version of SSAO for improving the quality

of the ambient occlusion effect.

SSAO is employed in the computer game industry

more than any other ambient occlusion technique, be-

cause it is easy to implement on graphics hardware,

and requires no precomputing for rendering. De-

spite the merits and superiority of SSAO, it has not

been adopted for real computer games, because the

ray sampling method inevitable with SSAO requires

a large computational cost for current graphics pro-

cessors. Therefore, in the next section, we introduce

a novel method for computing high quality ambient

occlusion at a lower cost than SSAO.

3 CURVATURE DEPENDED

AMBIENT OCCLUSION

Previous real-time ambient occlusion techniques need

a great number of functions due to the Monte-Carlo

collision detection algorithms. Our goal is to com-

pute ambient occlusion by the GPU in real-time and

implement our method at a lower computational cost

than previous works.

For this purpose, we approximate the occlusion at

a point

P into a function model which depends on

variables given by local features. A technique fea-

turing local variables only is usually called a local

illumination model as opposed to a global illumina-

tion model. The advantages of the local illumination

CURVATURE-APPROXIMATED ESTIMATION OF REAL-TIME AMBIENT OCCLUSION

269

Table 1: List of used symbols.

Variable meaning

3D position

direction vector

~n normal vector on surface

ω) visibility function

P,~n) ambient occlusion

A(κ

max

,κ

min

) our ambient occlusion

Ω

hemisphere area

r radius of hemisphere

,κ

curvatures

max

,κ

min

maximum, minimum principal

curvatures

model are lower computational cost and the ease with

which it can be implemented on the GPU. In this sec-

tion, we introduce our curvature dependent function

which computes the occlusion at

The main features described in this section are as

follows.

• Approximating the shape of a geometrical surface

at point

P by a function f(

P).

• Obtaining the occlusion at point

P by comparing

P) with the unit hemisphere Ω

3.1 Approximation of Geometry

Current ambient occlusion techniques employ ray

tracing algorithms to compute the occlusion at a point

P, because the geometric shape is undetectable by

variables at this point. We may regard the geometric

surface at

P as a function f(

P) and we detect the oc-

clusion by analyzing this function over the unit hemi-

sphere Ω

. However it is difﬁcult to identify f(

P) by

the variables at

P, because the geometric shape can, in

many cases, be very complex and unpredictable. To

analyze this geometrical function easily, we converted

the function f(

P) into a Taylor series and considered

only the lower terms in the series to simplify it. Fur-

thermore, we put f(

P) in the tangential vector space,

by a coordinate transformation. Then the Taylor ex-

pansion becomes a Maclaurin series.In the tangential

vector space the z axis of the Cartesian coordinates is

normal to the surface.

f(x, y) = f(0) + f

(0)x+ f

(0)y

(0)x

+ 2

(0)xy

(0)y

...(3)

(x,y) is the ﬁrst order differentiation of f(x, y) with

respect to x. The x and y axes are in the tangential and

bi-normal vector directions, respectively. The f(0)

is equal to 0 because it is the origin of the coordi-

nates. Both f

(0) and f

(0) are also equal to 0, be-

cause f(x, y) is tangential plane at (x,y) = (0, 0).

Furthermore f

(0) and f

(0) are the curvatures of

the lines of intersection between the x −z(or y −z)

plane and f (x,y) at (x,y) = (0,0). We deﬁne the di-

rection of each curvature as κ

and κ

. The κ

and κ

intersect each other. We can transform equation(3) as

follows.

f(x, y) =

+ 2

(0)xy

... (4)

Then f

(0) is equal to 0, provided that x and y belong

to the axes in which κ

and κ

are the maximum and

minimum principal curvatures(κ

max

, κ

min

). Addition-

ally, by removing the higher than second order terms,

we obtain the simpliﬁed f(x, y) as follows.

f(x, y) =

(κ

max

+ κ

min

) (5)

This function is a quadric surface function and it de-

pends only on the points x and y and the maximum

and minimum principal curvatures. Therefore, it is

very easy to analyze this function. We regard this

simpliﬁed f(x,y) as the geometric shape at point

and we analyze this function to obtain the occlusion

at this point. In the next subsection, we analyze this

function on the surface of the unit hemisphere Ω

and

acquire a curvature dependent occlusion estimation.

3.2 Approximation of Occlusion

In the previous subsection, we introduced a curvature

dependent function which estimates the geometrical

shape at the point

P. We analyze this function along

the surface of the unit hemisphere Ω

to estimate the

occlusion at

P. Before the estimation, we have to

transform f(x, y) into polar coordinates, as follows.

f(r, θ,φ) :rcosθ =

sin

θ(κ

max

cos

φ+κ

min

cos

φ)

(6)

For the unit hemisphere the radius r is constant, and

the range of θ and φ are (−

≤ θ ≤

) and (0 ≤ φ ≤

2π) respectively.

Previous ambient occlusion techniques require

great computational cost because they adopt ray trac-

ing algorithms for collision detection. The Monte-

Carlo ray sampling algorithm needs to sample more

than 16 rays per pixel, to acquire a practical quality.

The reason why ray tracing algorithms are employed

is that it is difﬁcult to detect occlusions from the ge-

ometry shape surrounding

P. However, now we have

the shape of the surrounding geometry and the sur-

face of the hemisphere as functions, we are able to use

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

270

Hemisphere area ¼

 

yxz NN 

Surface

0),( PV

1),( PV

Figure 2: The surface of the hemisphere Ω

(blue line) and the quadric surface function f(r, θ,φ) (red line). In the occluded

area, the visibility function V(

ω,

P) is equal to 0, and is 1 otherwise. We will obtain the occluded area by analyzing this ﬁgure

estimation.

these analytically to acquire the occlusion. We deﬁne

the hemisphere Ω

to be centered at the origin of the

tangential space, and the function f(x,y) to be an oc-

clusive geometry. We estimate the visible parts on Ω

from origin as in ﬁg(). Then, we obtain the curvature

dependent visibility function V

′

(

ω,

P) as follows.

′

(

ω,

P) =

(

1 (

(κ

max

′2

+ κ

min

′2

) > z

′

)

(κ

max

′2

+ κ

min

′2

) ≤ z

′

)

′

andz

′

refer to the points of intersection between

Ω

and f(x, y) =

(κ

max

+ κ

min

). As a result, we

obtain the following occlusion estimation function.

A(κ

max

,κ

min

) =

2π

φsinθ

′

dθ

′

dφ (7)

this θ = arccos(

−1±

√

1+ A

)

(A = R(κ

max

cos

φ+ κ

min

sin

φ)

This function, depending on curvature alone, is a lo-

cal illumination approximation of occlusion. Adopt-

ing this approximation, we can obtain an ambient oc-

clusion effect at low cost, thus the implementation is

very easy. When we obtain the occlusion from a static

object, we do not need any other process. In this pa-

per, we more simplify equation(8) as follows by using

a symmetry between x and y, because it is difﬁcult to

implement equation(8) for current computer.

A(κ

max

,κ

min

) =

2arccos(

−1±

√

1+(κ

max

+κ

min

)

(κ

max

+κ

min

)

(8)

This is an important update from ”Curvature de-

pended local illumination approximation of ambi-

ent occlusion” (Hattori et al., 2010). In the paper,

we have to adopt some look-up table(LUT) or quite

rough approximated function which ignore physical

property to implement equation(8). Now, using this

equation(8), we are able to implement our ambient oc-

clusion correctly and in real-time. However, our goal

is to compute the ambient occlusion for a deformable

object, so we introduce a real-time curvature comput-

ing technique in the next section.

4 IMPLEMENTATION

In the previous section, we obtained the curvature de-

pendent function for estimating an occlusion at

P. By

converting this function using the current rendering

process, we can acquire an ambient occlusion effect

as a local illumination model. However, it is rare for

the curvature to be given by current polygon model

formats, so we have to calculate the curvature in real-

time. In previous works, many real-time estimations

of curvature have been suggested and the methods can

be simply divided into two categories as follows.

• Estimations in the screen space with no pre-

computing.

• Estimations in world space with some pre-

computing.

In our method, we employ the screen space algorithm,

because the purpose of our work is to obtain an am-

bient occlusion effect, at lower cost than in previous

works and by a method that is easy to implement.

It has already been shown that screen space estima-

tion is easy to implement in computer games on cur-

rent graphics hardware. Wesley et al. (Grifﬁn et al.,

2011) introduced real time curvature estimation in the

world space. They employed a programmable geom-

etry shader to detect the normal curvature by analyz-

ing the normal and position vectors of neighboring

polygons. Yu et al. (Yu et al., 2007) applied their

screen space caustics estimation to compute curvature

in real time. They used pre-computed shape infor-

CURVATURE-APPROXIMATED ESTIMATION OF REAL-TIME AMBIENT OCCLUSION

271

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Figure 3: The ﬂow of our implementation.

(a) Lambert. (b) Our method. (c) Our method + Lambert. (d) Lambert + SSAO.

Figure 4: Curvature dependent ambient occlusion images(b)(c) and SSAO(d).

mation to compute curvature with low computational

running cost. Employing these methods we can ac-

quire a greater variety of ambient occlusion effects.

When we adopt Wesleyfs technique, we can obtain

the characteristics of a three-dimensional geometric

shape. In some cases, this is a great advantage, so we

shall be implementing this method in our future work.

5 RESULT AND DISCUSSION

We show a result obtained by employing our method

in Figure(4). (a) shows a Lambert-rendered image,

(b) shows our method, and (c) shows an image com-

posed of (a) and (b). (d) shows the image ren-

dered by SSAO. The speed of rendering a 1280×960

pixel image by applying our method is 60fps and

applying SSAO(16 rays sampling with no blur) is

26fps using an Intel 1.6GHz dual core cpu and a

nVIDIA GeForce 320M machine. The result shows

our method runs 130 percent faster than SSAO. Fur-

thermore, Our method is a local illumination method

to realize a fast run time and easy implementation. It

is extremely easy to implement, and, in addition, by

adopting the curvature dependent method, we can ob-

tain a more detailed occlusion estimate than with the

conventional method using collision detection. The

following ﬁgure, Figure (5) shows close up images, in

which we can see the effect of having more detailed

occlusion information.

We can see that even for a fairly small shape

changes in geometry are detected. From these results,

we ﬁnd that our method is effective for determining

small changes in geometry.

Thus, by adjusting the scale of the curvature, cre-

ators are able to change the strength of the ambient

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

272

(a) Our method. (b) SSAO.

Figure 5: More closed consideration between our method

and SSAO.

occlusion. The results demonstrate that our method

is easy to implement and runs fast on current graph-

ics hard ware, thus making it effective for rendering

small changes in geometry and adjusting the ambient

occlusion intensity.

6 CONCLUSIONS

In this paper, we introduced a curvature dependent

ambient occlusion estimation algorithm. Our method

approximates the ambient occlusion in a local illumi-

nation model by introducing a curvature-dependent

occlusion estimation function. Since the function is

a local illumination model, we are able to render am-

bient occlusion in real-time, and the proposed method

can be easily implemented. Furthermore, the render-

ing speed is twice as fast as SSAO. Using this method,

smoother, more detailed and equal or better quality

occlusion can be achieved than by using the conven-

tional method (Figure 5), because the curvature is

continuous on the surface. These results demonstrate

that our method is effective for implementing in com-

puter games. Adopting our method, we can reduce

the personnel and computational cost, which would

be beneﬁcial for the game industry.

REFERENCES

Akenine-M¨oller, T., Haines, E., and Hoffman, N. (2008).

Real-Time Rendering 3rd Edition. A. K. Peters, Ltd.,

Natick, MA, USA.

Bavoil, L. and Sainz, M. (2009). Multi-layer dual-

resolution screen-space ambient occlusion. In SIG-

GRAPH 2009: Talks, SIGGRAPH ’09, pages 45:1–

45:1, New York, NY, USA. ACM.

Bavoil, L., Sainz, M., and Dimitrov, R. (2008). Image-

space horizon-based ambient occlusion. In ACM SIG-

GRAPH 2008 talks, SIGGRAPH ’08, pages 22:1–

22:1, New York, NY, USA. ACM.

Grifﬁn, W., Wang, Y., Berrios, D., and Olano, M. (2011).

Gpu curvature estimation on deformable meshes. In

Symposium on Interactive 3D Graphics and Games,

I3D ’11, pages 159–166, New York, NY, USA. ACM.

Hattori, T., Kubo, H., and Morishima, S. (2010). Cur-

vature depended local illumination approximation of

ambient occlusion. In ACM SIGGRAPH 2010 Posters,

SIGGRAPH ’10, pages 122:1–122:1, New York, NY,

USA. ACM.

Kontkanen, J. and Laine, S. (2005). Ambient occlusion

ﬁelds. In Proceedings of ACM SIGGRAPH 2005 Sym-

posium on Interactive 3D Graphics and Games, pages

41–48. ACM Press.

Landis, H. (2002). Production-Ready Global Illumination.

In Siggraph Course Notes, volume 16.

M´endez, A., Sbert, M., and Cat´a, J. (2003). Real-time ob-

scurances with color bleeding. In Proceedings of the

19th spring conference on Computer graphics, SCCG

’03, pages 171–176, New York, NY, USA. ACM.

Phong, B.-T. (1975). Illumination for Computer Generated

Pictures. 18(6):311–317.

Yu, X., Li, F., and Yu, J. (2007). Image-space caustics and

curvatures. Computer Graphics and Applications, Pa-

ciﬁc Conference on, 0:181–188.

Zhukov, S., Iones, A., and Kronin, G. (1998). An ambient

light illumination model. In Rendering Techniques,

pages 45–56.

CURVATURE-APPROXIMATED ESTIMATION OF REAL-TIME AMBIENT OCCLUSION

273