Pencil Tracing Mirage: Principle and its Evaluation
Katsuhisa Kanazawa†‡†Tokyo Healthcare UniversitySetagaya 3 11 3 Setagaya-ku Tokyok-kanazawa@thcu.ac.jp

Yuma Sakato‡Tokiichiro Takahashi‡*‡Tokyo Denki University/*UEI Research
Senju Asahi-cho 5 Adachi-ku Tokyo{kanazawa, sakato, toki}@vcl.im.dendai.ac.jp

Figure 1. Images of square sun mirage generated by our method.
matrix to 5x5. Extended system matrix and one for theperturbation
is as follows:To generate realistic representation of the nature scene is one of
the most challenging areas in the computer graphics community.
Ray tracing[1] is the most well-known technique to synthesize arealistic image. Since ray tracing is the most suitable method for
simulating reflection and refraction of the light, it has been used
for simulating atmospheric optical phenomena due to reflection
and refraction of the light. The mirage is a kind of atmospheric
optical phenomenon. Therefore, it is possible to synthesize
mirages in 3DCG by simulating or modeling condition of the air.
We focus on pencil tracing technique[2] that is an extention of
conventional ray tracing technique based on the paraxial
approximation theory. Our simple method based on pencil tracing
can efficiently generate an appearance of mirage without any
complex thermodynamic simulation.

1. Introduction

2. Proposed Method

We extend the pencil tracing method to synthesize mirages. Ouridea is to introduce a special kind of object in scene, and integratea perturbation factor into image generation process of the pencil
tracing. Our method is useful because it can treat perturbation ofthe air as simply matrix product.

2.1 Pencil Tracing

The pencil tracing is a method for accelerating image generationof the ray tracing method by tracing a pencil (or bundle) of rays
instead of an individual ray. Thus, a ray is not traced per a pixel ora sub-pixel, it is traced per domain of pixels. An axial ray (centerray of domain) is traced in the same manner as ray tracing, but
paraxial rays (surrounding rays of the axial ray) are omitted to be
traced by using result of tracing their axial ray. Therefore, pencil
tracing is faster than the conventional ray tracing. In this method,an optical system affecting a path of the axial ray is represented as
4x4 matrix called system matrix, and a ray is represented as 4D
vector called ray-vector. Paraxial rays changed by the opticalsystem are approximated by the transformation of their ray-vector
representation with the system matrix. This is formulated asfollows:=Here, , ′ are ray-vectors of an original ray and of a transformed
ray, respectively. is a system matrix.

2.2 Perturbation Object and Matrix

Most of mirages have fluctuated appearance due to perturbationof the air. It is one of the most important factors of the mirageappearance. To synthesize this, we introduce a perturbation
object. This is a special kind of scene object that adds positionaland directional noise to a ray. To integrate this operation of
addition into the form of matrix product, we extend a system

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SIGGRAPH 2013, July 21 – 25, 2013, Anaheim, California.
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ACM 978-1-4503-2261-4/13/07

Figure 2.

Image Generation Time.

′=,
=1
1
Here is a 4x4 system matrix and ′is its equivalent extended
matrix, is a 4x4 identity matrix,
is noise ray-vector, and is
4D column zero vector.

3. Results

As shown in Fig.1, images of square sun mirage are generated byour method. These images are a part of the short video (450frames). Compared to the actual video (see the supplementalvideo), realistic images are successfully generated.
All the images of Fig.1 are generated by the following computer
environments: CPU - Intel Core i7-920 (2.66GHz), systemmemory - 16GB. We measured the image generation time of avariety of sizes of images and sizes of domains. Figure 2 shows
the generation time of 450 images of square sun mirage.
Measurement results verified that our method can generate images
about 1.5-5 times as fast as the conventional method.

4. Conclusions

We proposed a pencil tracing technique for synthesizing mirages.
Our method generated images of square sun mirage fast andefficiently.

References[1] WHITTED, T, 1980. An Improved Illumination Model for
Shaded Display. Comm. ACM. 23, 6, June 1980, 343-349.
[2] SHINYA, M., TAKAHASHI, T., NAITO, S, 1987. Principles andApplications of Pencil Tracing. SIGGRAPH Comput. Graph.
21, 4 (August 1987), 45-54. DOI=10.1145/37402.37408