History
Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Appro-ximation using superposition of functions has existed since the early 1800’s, when Joseph Fourier discovered that he could superpose sines and cosines to represent other functions. However, in wavelet analysis, the scale that we use to look at data plays a special role. Wavelet algorithms process data at different scales or resolutions. If we look at a signal with a large “window”, we would notice gross features. Similarly, if we look at a signal with a small
“window”, we would notice small features. The result in wavelet analysis is to see both small and large “window”.
The Wavelets were proposed in the 1980’s as an alternative to the Fourier transforms for signal analysis, and were initially used for representing seismic signal. In fact, the term wavelet was coined by Ricker in 1940 ([28]) to de-scribe the disturbance that proceeds outward from a sharp seismic impulse or explosive charge. However, only in the middle of 1980’s their mathematic theory was formalized in a rigorous manner.
The Wavelets are now used as a mathematical tool for hierarchically decom-posing functions. They allow any function to be decomposed in terms of a coarse overall shape, plus details that range from broad to narrow.
Mathematical definition
Definition 1.2.1. A wavelet is a function ψ(t) ∈ L2(R) such that the family of functions
ψj,k(t) =: 2j/2ψ(2jt− k)
where j and k are arbitrary integers, is an orthonormal basis in the Hilbert space L2(R).
An orthonormal, compactly supported wavelet basis of L2(R) is formed by the dilation and translation of a single function ψ that satisfies the following
“two-scale” difference equation
ψ(t) =X
i
piψ(2t − i) (1.2.1)
where pi is a set of coefficients.
The wavelet function has a companion, the scaling function, which also forms a set of orthonormal bases of L2(R)
φj,k(t) =: 2j/2φ(2jt− k)
The scaling function φ also satisfies the “two-scale” difference equation ψ(t) =X
i
qiψ(2t − i) (1.2.2)
where qi is a set of coefficients.
Good wavelet are usually constructed starting from a multiresolution analysis.
Let us give a definition of this concept.
Definition 1.2.2. A multiresolution analysis is a sequence {Vj}j∈Z of sub-spaces of L2(R) such that
i) . . . ⊂ V−1⊂ V0 ⊂ V1 ⊂ . . . ii) spanS
j∈ZVj = L2(R)
iii) f (x) ∈ Vj if and only if f (2−jx) ∈ V0,i.e. the spaces Vj are scaled versions of the central space V0
iv) f ∈ V0 if and only if f (x − m) ∈ V0 for all m ∈ Z,that is, V0 (and hence all the Vj) is invariant under translation.
v) there exists a function φ ∈ V0, called scaling function, such that the system {φ(t − m)}m∈Zis an orthonormal basis in V0
Wavelet for Computer Graphics
Several researches in CG have been centered around the use of wavelets. The ideas of multiple levels of resolution, or so called “level of detail” modeling, have been largely used in CG as in many other disciplines. Thus, when wavelets moved from a mathematical analysis tool to a practical computa-tional tool, they were quickly put to use in CG.
The Wavelets can be applied to a wide variety of objects used in graphics, as we point out in the following list:
• Image Processing: some of most powerful compression techniques are based on wavelet transforms;
• Global Illumination: radiosity and radiance algorithms, that are based on wavelets are asymptotically faster than other finite element methods;
• Hierarchical Modeling: the use of multiresolution representations for curves and surfaces accelerates and simplifies many applications of edit-ing;
• Animation: the large constrained optimization tasks which arise in physically based modeling and animation subject to goal constraints can be resolved faster and more robustly with wavelets;
• Volume Rendering and Processing: the wavelets can simplify the treatment of very large data sets because they can be used as well as feature detection and enhancement;
• Multiresolution Painting: the use of multiresolution analysis one can build efficient “infinite” resolution paint systems;
• Image Query: the use of a small number of the largest wavelet coeffi-cients of an image, gives the opportunity of accelerating searching and retrieval systems.
There are several reasons for the enthusiastic welcome wavelet based algo-rithms have received in CG. Foremost stand the huge computational demands of CG applications. Geometric models, for example, are often built from a large number of primitive elements in order to achieve some desired level of fi-delity with respect to a real world object. At the same time, users would desire that all the manipulations and the computations which involve these objects occur interactively, that is, with the screen updates occurring multiple times per second. Similarly, the shading and the motion of these objects should appear realistic. Depending on the number and the complexity of objects appearing in a scene, this can lead to very expensive algorithms. Some exam-ples include the modeling of constrained dynamical systems, the deformation of objects under forces, and the indirect illumination effects. Performing the necessary computations under the constraints of interactivity demands the as efficient algorithms as possible, and the application of some shrewd appro-ximations which permit to preserve a perceived realism. Due to the ability of wavelets to represent compactly functions and data sets within user spe-cified error bounds they are a natural tool to be considered. Their zooming
in ability allows the use of “just enough” precision in a given region of inte-rest, allowing at the same time coarse representations in regions outside the immediate area of interest. Even more importantly, they facilitate a smooth tradeoff between computation time and resulting simulation quality. Often the algorithms which manipulate these hierarchical objects become asympto-tically faster as well, allowing the use of much larger and more complicated scenes with respect to what is possible to treat without this tool.
Wavelet as new language
The Wavelet take some advantages on the traditional methods like the Fourier analysis especially when the real signal presents some discontinuities. There has been a large interest in studying those functions which approximate dis-continuous signals more appropriate with respect to the sin and the cosine functions adopted in the Fourier transform.
In fact the deficiencies of Fourier techniques have led researchers in a variety of disciplines to develop various hierarchical representations of functions. This enormous interest, on wavelet theory, in many and different research fields has lead researches to comunicate necessarily between themselves.
The mathematician Ingrid Daubechies describes in clear manner what the Wavelets are and how this term became a new communication language among all scientists ([37]).
. . . Mathematicians are like the French, the German poet Goethe once remarked. “They take whatever you tell them and translate it into their own language - and from then on it is something entirely different”.
Goethe’s observation is as true now as ever. But times may be changing. In the last ten years, mathematicians and researchers in diverse areas of science, engineering and even art have discovered and begun to develop a theoretical language they can all understand.
This new common language is sparking new collaborations. Many mathematicians are now crossing over into such applied areas as signal processing, medical imaging, and speech synthesis. At the same time, much deep but abstract - sounding mathematics is be-coming accessible to researchers in fields from geophysics to elec-trical engineering.
The new language is wavelet theory. Those who speak it it describe
wavelets as powerful new tools for analyzing data. Wavelet theory serves as a kind of numerical zoom lens, able to focus tightly on interesting patches of data - but without losing sight of mathema-tical forest while attending to the trees, twigs, buds, and grains of pollen....
Mathematically, wavelets are an offshoot of the theory of Fourier analysis. Introduced by the French mathematician Joseph Fourier in his essay “Theorie analytique de la chaleur” (analytic theory of heat), published in 1822. Fourier analysis seeks, with great suc-cess, to understand complicated phenomena by breaking them into mathematically simple components. The fundamental idea is to take a function and express it as a sum of trigonometric sine and cosine waves of various frequencies and amplitudes. The familiar and well- understood trigonometric functions are easy to analyze.
By combining information about a function’s sine and cosine com-ponents, properties of the function itself are easily deduced, at least in principle. Fourier analysis is among mathematics’ most widely used theories. It is especially suited to analyzing periodic pheno-mena, periodicity being the most prominent property of sines and cosines. But even so, the theory has its limitations and its pit-falls. The main problem is that finding detailed information about a function requires looking at a huge number of its infinitely many Fourier components. For example, a transient “blip” obvious in a graph, is impossible to recognize from its effect on a single com-ponent. The reason, in essence, is that each sine and cosine wave undulates infinitely in both directions: thus a single wave can’t help locate anything. Indeed the sharper the blip, the more Fourier com-ponents are needed to describe it.
Wavelet theory takes a different approach. Instead of working with the infinitely undulating sine and cosine waves, wavelet analysis relies on translations and dilations of a suitably chosen “mother wavelet” that is concentrated in a finite interval. Almost any func-tion can serve as the mother wavelet: this makes wavelet the-ory more flexible than traditional Fourier analysis. “Daughter”
wavelets are formed by translating, or shifting, the mother wavelet by unit steps and by contracting or expanding it by powers of two (see Figure 1.3).
One then expresses other functions as combinations of wavelets,
Figure 1.3: A “mother” wavelet and two “daughters”
just as Fourier analysis represents functions by combining sines and cosines.
The fact that the mother wavelet is concentrated in a finite inter-val gives wavelet theory its zoom-in capability:an interesting blip in a function can be analyzed by looking at increasingly contracted copies of the mother wavelet in the vicinity of the blip.
Many of the ideas underlying wavelet theory have been around for decades, but the subject itself got off the ground only recently. The story starts in the early 1980s in France, when wavelets were in-troduced by geophysicist Jean Morlet and mathematical physicist Alexander Grossmann. In 1985, mathematician Yves Meyer con-structed a family of wavelets with two highly desirable mathema-tical properties called smoothness and orthogonality.(Interestingly, J. O. Stromberg at the University of Tromso in Norway had con-structed such a family several years earlier, but the connection with the nascent theory of wavelets was not realized until after Meyer’s work.)
The following year, Meyer and Stephane Mallat gave the subject a solid foundation with a theory of “multiresolution analysis”. Then in 1987, Daubechies constructed a family of wavelet that, in addi-tion to being smooth and orthogonal, were identically zero outside a finite interval. Daubechies ’s construction opened up the field.
“Compactly supported” wavelets are now easy to come by, and are
among the most commonly used in applications. And applications are abundant. Wavelets are being tested for use in everything from digital image enhancement- making blurry pictures sharp- to new methods in numerical analysis (itself widely used in scientific com-puting). “They’re a very versatile tool” says Daubechies. Not all the applications will pan out, but many will, and some already have.
“There are some very nice success stories” Daubechies adds.
One such story may have far- reaching effects, especially for the next generation of criminals. The Federal Bureau of Investigation has adopted a wavelet-based standard for computerizing its finger-print files. The FBI has around 200 million fingerfinger-print cards on file, according to Peter Higgins, is to digitize the files, store them electronically, and “put [them] in something that would fit in a 20 × 20-foot room”.
It sounds easy: after all, entire encyclopedias now fit on a compact disk with room to spare. But that’s words. Images are something else. At a resolution of 500 pixels per inch. a standard finger-print card contains nearly 10 megabytes of data. Transmitting that much information over a modem-something the police would like to be able to do takes hours at today’s transmission rates. . . What’s needed is some way to compress the data on a fingerprint card without distorting the picture. That’s where wavelets come in. By treating the fingerprint image as a two- dimensional function, it’s possible to represent it with a combination of wavelets. With a suit-ably chosen family of wavelets, only a relative handful are needed to represent a fingerprint, and the contribution of each wavelet can be rounded off, or “quantized”, which reduces the amount of data that needs to be stored or transmitted. . .