Mathematical Optimization in Computer Graphics and Vision

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Mathematical optimization is used in nearly all computer graphics applications, from computer vision to animation. This book teaches readers the core set of techniques that every computer graphics professional should understand in order to envision and expand the boundaries of what is possible in their work.

Study of this authoritative reference will help readers develop a very powerful tool- the ability to create and decipher mathematical models that can better realize solutions to even the toughest problems confronting computer graphics community today.

• Distills down a vast and complex world of information on optimization into one short, self-contained volume especially for computer graphics
• Helps CG professionals identify the best technique for solving particular problems quickly, by categorizing the most effective algorithms by application
• Keeps readers current by supplementing the focus on key, classic methods with special end-of-chapter sections on cutting-edge developments

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Product Details

Meet the Author

Luiz Velho works with Jonas Gomes at IMPA, also as a computer graphics researcher.

Jonas Gomes is a computer graphics researcher at the Institute of Pure and Applied Mathematics (IMPA), Brazil.

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Read an Excerpt


By Clyde H. Moore


Copyright © 2001 Elsevier B.V.
All right reserved.

ISBN: 978-0-08-087858-4

Chapter One



While this book is concerned primarily with porosity evolution and diagenesis in carbonate reservoirs, the reader and the author must ultimately share a common understanding of the fundamental characteristics of the carbonate realm. Therefore, these first introductory chapters are designed to highlight general concepts unique to the carbonate regimen, such as: 1) the biological origin enjoyed by most carbonate sediments; 2) the complexity of carbonate rock classification; 3) the bathymetric framework of carbonate depositional facies and environments; 4) the response of carbonate depositional systems to changes in relative sea level, i.e. concepts of sequence stratigraphy as applied to carbonates; 5) the diagenetic consequences of the high chemical reactivity of carbonates.

Readers desiring an in-depth review of carbonate depositional environments should ' read the review of carbonate facies by James, (1979) the extensive compilation edited by Scholle et al. (1983) and the most recent review presented in Carbonate Sedimentology by Tucker and Wright (1990). The recent book entitled Sequence Stratigraphy, edited by Emery and Myers (1996), will give the reader a reasonable overview of the basic concepts of sequence stratigraphy as well as its application to carbonate rock sequences.



The basic characteristics of carbonate sediments can generally be traced to the overwhelming biological origin of carbonate sediments and the influences that this origin exerts on sediment textures, fabrics, and depositional processes such as the ability of certain organisms to build a rigid carbonate framework. The following section outlines these broad biological influences on carbonate sediments and sedimentation.

Origin of carbonate sediments

Well over 90% of the carbonate sediments found in modem environments are thought to be biological in origin and form under marine conditions (Milliman, 1974; Wilson, 1975; Sellwood, 1978; Tucker and Wright, 1990). Distribution of most carbonate sediments is directly controlled by environmental parameters favorable for the growth of the calcium carbonate secreting organisms. These parameters include temperature, salinity, substrate, and the presence/absence of siliciclastics (Lees, 1975). Schlanger (1981) beautifully illustrated the latitudinal control over the distribution of carbonate secreting organisms and its impact on total carbonate production. Carbonate production in tropical waters is much faster than in temperate waters because high carbonate producers such as hermatypic corals thrive in the warm, clear waters of the tropics (Fig. 1.1).

Since many hermatypic corals and other reef organisms are depth and light sensitive, maximum carbonate production is generally attained in the first 10m water depth in the marine environment (Fig. 1.2) (Schlager, 1992). Most carbonate sediments, therefore, were initially deposited or formed in shallow marine waters. The origin of larger carbonate grains is relatively easy to determine. The grain ultra-structure and overall shape allows the worker to easily determine the grain origin such as a fragment of an oyster, a whole foraminifer, or, perhaps, a chemically precipitated ooid. The difficulty arises when one considers the origin of the carbonate mud fraction. Does carbonate mud consist of fine particles derived by erosion of larger bioclastic material or is it a chemical precipitate (such as the whitings common to tropical carbonate environments) (Shinn et al., 1989; Macintyre and Reid, 1992)? Another possibility is that of aragonite needles released from green algae on death (Perkins et al., 1971; Neumann and Land, 1975). Finally, do bacteria play a role in the formation of carbonate muds (Drew, 1914; Berner, 1971; Chafetz, 1986; Folk, 1993)? Today most carbonate workers would concede that carbonate muds probably originate from all of the above mechanisms but are uncertain about the relative importance of each. We will re-visit these questions later when we consider mineralogical composition of carbonate sediments and in particular the role bacteria may play in marine diagenesis. Finally, it should be noted that most carbonate sediments are generally deposited near the site of their origin. This is in sharp contrast to siliciclastics, which are generally formed outside the basin of deposition, and are transported to the basin, where physical processes control their distribution. For siliciclastics, climate is no constraint, for they are found worldwide and are abundant at all depths, in fresh water as well as marine environments.

The reef: a unique depositional environment

The ability of certain carbonate-secreting organisms to dramatically modify their environment by encrusting, framebuilding and binding leads to the depositional environment unique to the carbonate realm, the reef (Fig. 1.3). In this discussion, the term reef will be used in its genetic sense, i.e. a solid organic framework that resists waves (James, 1983). In a modem reef, there is an organism-sediment mosaic that sets the pattern for organic framework reef sequences. There are four elements: 1) the framework organisms, including encrusting, attached, massive, and branching metazoa; 2) internal sediment, filling primary growth as well as bioeroded cavities; 3) the bioeroders, which break down reef elements by boring, rasping, or grazing, thereby contributing sediment to peri-reef as well as internal reef deposits; and 4) cement, which actively lithities and may even contribute to internal sediment (Fig. 1.3).While the reef rock scenario is complex, it is consistent. Chapter 5 presents a comprehensive treatment of marine cementation associated with reef depositional environments.

Today, corals and red algae construct the reef frame. Ancillary organisms such as green algae contribute sediment to the reef system. Reef organisms have undergone a progressive evolution through geologic time, so that the reef-formers of the Lower and Middle Paleozoic (i.e. stromatoporoids) are certainly different from those of the Mesozoic (rudistids and corals) and from those that are observed today (James, 1983). Indeed, there were periods, such as the Upper Cambrian, Mississippian, and Pennsylvanian, when the reef-forming organisms were diminished, or not present, and major reef development did not occur (James, 1983).

Framework reefs, then, while certainly influenced by physical processes are dominated by a variety of complex, unique, biological and diagenetic processes that have no siliciclastic counterpart.

Unique biological control over the texture And fabric of carbonate sediments

The biological origin of most carbonate sedimentary particles places severe constraints on the utility of textural and fabric analysis of carbonate sediments and rocks. Size and sorting in siliciclastics are generally indicators of the amount and type of physical energy (such as wind, waves, directed currents and their intensity) influencing sediment texture at the site of deposition (Folk, 1968). Size and sorting in carbonate sediments, however, may be more influenced by the population dynamics of the organism from which the particles were derived, as well as the peculiarities of the organism's ultrastructure. Folk and Robles (1964) documented the influence of the ultrastructure of coral and Halimeda on the grainsize distribution of beach sands derived from these organisms at Alacran Reef in Mexico (Fig. 1.4).

In certain restricted environments, such as on a tidal fiat, it is not uncommon to find carbonate grains composed entirely of a single species of gastropod (Shinn et al., 1969; Shinn, 1983). The mean size and sorting of the resulting sediment is controlled by natural size distribution of the gastropod population and tells us little about the physical conditions at the site of deposition. For example, one commonly encounters large conchs living in and adjacent to mud-dominated carbonate lagoons in the tropics. Upon death, these conchs become incorporated as large clasts in a muddy sediment. This striking textural inversion does not necessitate, as it might in siliciclastics, some unusual transport mechanism such as ice rafting, but simply means that the conch lived and died in a depositional environment dominated by carbonate mud.

Other textural and fabric parameters such as roundness also reflect biological control. Roundness in siliciclastic grains is generally thought to indicate distance of transport, and/ or the intensity of physical processes at the site of deposition (Blatt et al., 1980; Blatt, 1982). Roundness in carbonate grains, however, may well be controlled by the initial shape of the organism from which the grain is derived (for example, most foraminifera are round). In addition, an organism's ultrastructure, such as the spherical fiber fascioles characteristic of the coelenterates, may also control the shape of the grains derived from the coral colony (Fig. 1.5).

Finally, some grains such as oncoids, rhodoliths and ooids are round because they originate in an agitated environment where sequential layers are acquired during the grain's travels over the bottom, with the final product assuming a distinctly rounded shape (Bathurst, 1975; Tucker and Wright, 1990; Sumner and Grotzinger, 1993). Great care must be used when interpreting the textures and fabrics of carbonate sediments and rocks as a function of physical conditions at the site of deposition.

Carbonate grain composition

Skeletal remains of organisms furnish most of the coarse-grained sediments deposited in carbonate environments. It follows that the grain composition of carbonate sediments and rocks often directly reflects their environment of deposition because of the general lack of transport in carbonate regimens and the direct tie to the biological components of the environment. A number of workers have documented the close correlation between biological communities, depositional environment and subsequent grain composition in modem carbonate depositional systems (Ginsburg, 1956; Swinchatt, 1965; Thibodaux, 1972) (Fig. 1.6). Our ability to determine the identity of the organism from which a grain originates by its distinctive and unique ultrastructure is the key element to the usefulness of grain composition for environmental reconstruction in ancient carbonate rock sequences (Bathurst, 1975). Carozzi (1967) and Wilson (1975) based their detailed microfacies studies on the thin section identification of grain composition, including detailed identification of the biological affinities of bioclasts.

In contrast, grain composition in siliciclastics is related to ultimate provenance of the sediment, climate, and stage of tectonic development of the source, rather than to conditions at the site of deposition (Krynine, 1941; Folk, 1954; Pettijohn, 1957; Blatt, 1982).

Carbonate rock classification

Carbonate rock classification parallels, in some respects, the classifications commonly used to characterize siliciclastics (Folk, 1968). Siliciclastics are normally classified on the basis of composition or texture, or both. Compositional classifications of sandstones generally are based on three end-members, Quartz+Chert, Feldspars and Unstable Rock Fragments (Blatt, 1982). As noted above, sandstone compositional classes generally reflect the tectonic setting, provenance, and climate of the extrabasinal source of the sandstones. In his textural classification, Folk (1968) uses three end-members, sand, mud, or gravel, or sand, silt, and clay, in the absence of gravel. The sediment and rock types recognized by such a texturally based classification are thought to reflect the general level of energy present at the site of deposition (Folk, 1968). This concept of interdependency of sediment texture and energy at the site of deposition has been incorporated into the two most widely accepted carbonate classifications. The first was published by Folk (1959) and the second by Dunham (1962).


Excerpted from CARBONATE RESERVOIRS by Clyde H. Moore Copyright © 2001 by Elsevier B.V.. Excerpted by permission of ELSEVIER. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Computer Graphics
Optimization: an overview
Optimization and computer graphics
Variational optimization
Continuous optimization
Combinatorial optimization
Global Optimization
Probability and Optimization

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