Inequalities for convex sequences and their applications Using the theory of majorization, new inequalities for convex sequences are proved. A necessary and sufficient condition for a convex sequence is established. This reveals an important relationship between the convex sequence and the majorized inequality. Finally, some applications of our new inequalities and some improvements of certain known results are presented. Considerable attention has been given to the study of convex sequences and their properties, and the corresponding inequalities with applications. In general, convex sequences as discrete versions of convex functions play an important role in mathematical analysis and in the theory of inequalities. Inequalities for convex sequences provided considerable interest in proving a large number of elegant results with applications (see Wu and Shi [1] and Mercer [2] ). In addition, several authors including Mitrinović and Vasić [3] , Roberts and Varberg [4] , and Mitrinović et al. [5] presented a large number of major results for convex sequences and related inequalities.. The major objective of this paper is to prove some new inequalities for convex sequences with applications. Using analytic techniques and the theory of majorization (see Marshall and Olkin [6] and Wu [7, 8] ), a necessary and sufficient condition for a convex sequence is proved. An important relationship between convex sequences and majorized inequalities is obtained. This result can be effectively used to discover and prove new inequalities for convex sequences. Moreover, many useful inequalities related to convex functions as well as convex sequences are established. Some of these inequalities provide a refined estimation for the sum of certain finite sequences. In addition, examples of applications of the obtained results are presented in the last section of this paper.. We denote by R and N the set of real numbers and the set of positive integers respectively. I denotes interval. Let x = (x 1 , x 2 , . . . , x n ) denote n-tuple (n-dimensional real vector). We denote the vector space R n by. Definition 1 (Mitrinović and Vasić [3] , p. 11). A real sequence {a i } n i=1 is said to be a convex sequence if 2a i a i−1 + a i+1 for all i = 2, 3, . . . , n − 1.. (1). A real sequence {a i } n i=1 is said to be a concave sequence if the reverse inequality in (1) holds.. Definition 2 (Marshall and Olkin [6] , p. 7). For any x = (x 1 , x 2 , . . . , x n ) ∈ R n , and y = (y 1 , y 2 , . . . , y n ) ∈ R n , let x [1] x [2] · · · x [n] and y [1] y [2] · · · y [n] denote the components of x and y in decreasing order, respectively. The n-tuple y is said to majorize x (or x is to be majorized by y) in symbols. Definition 3 (Marshall and Olkin [6] , p. 78). Let x ∈ R n , we define the kth symmetric function as follows

O O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A framework for evaluating the data-hiding capacity of image sources An information-theoretic model for image watermarking and data hiding is presented in this paper. Recent theoretical results are used to characterize the fundamental capacity limits of image watermarking and data-hiding systems. Capacity is determined by the statistical model used for the host image, by the distortion constraints on the data hider and the attacker, and by the information available to the data hider, to the attacker, and to the decoder. We consider autoregressive, block-DCT and wavelet statistical models for images and compute datahiding capacity for compressed and uncompressed host-image sources. Closed-form expressions are obtained under sparse-model approximations. Models for geometric attacks and distortion measures that are invariant to such attacks are considered. Data hiding refers to the nearly invisible embedding of information within a host data set such as text, audio, image, or video [1, 2, 3, 4, 5, 6, 7] . There is a broad variety of data one may want to hide in such data sets. In watermarking applications, the hidden data represent authorship information, a time stamp, or copyright information. In steganographic applications, the hidden data are a secret message whose mere presence within the host data set should be undetectable; a classical example is that of a prisoner communicating with the outside world under the supervision of a prison warden. In image database applications, the hidden data could represent a textual description of image features, or some complementary information (words, sound, etc.) about the original scene. In this context, data hiding represents a useful alternative to the construction of a hypermedia document, which may be less convenient to manipulate. A related application is in-band captioning of data such as foreign-language movie subtitles, stock-market data, etc.. In some of these applications, a malicious opponent may apply various data processing operations to interfere with a decoder's ability to recover the hidden data. In other applications, there is no malicious opponent, but various manipulations of the modified data set may result in similar unfortunate effects. Hence there is a need not only to develop ways to embed information in a nearly invisible fashion, but also to do so in a way that is robust against a broad variety of data processing operations, whether they are intentional or not. In this paper, we assume the presence of an attacker that attempts to disrupt the communication of hidden data, subject to certain constraints. We tend to use the terminology data hiding over watermarking, as the latter often suggests that a very limited amount of data is to be hidden within the data set; whereas our focus is on communicating significant amounts of information to a decoder.. Most research papers in the watermarking and data-hiding literature have focused on novel ways to hide data and to remove hidden data [1, 8, 2, 3, 4, 5, 6, 7] . However, the information-theoretic analysis that describes the fundamental limits of any watermarking or data-hiding system and can ultimately guide the design of efficient watermarking algorithms has been emerging only recently [9, 10, 11, 12] . Information theory yields results that are substantially different from those obtained so far in the data-hiding literature. A typical approach in this literature is to assume that the dataembedding and -decoding functions take a particular form, and that a particular type of attack is used [13, 14, 15] . Restrictions on the embedding and decoding functions can have a severe impact on performance, as evidenced by the large gap to capacity for commonly-used systems [11] . Dually, a system analysis that assumes a suboptimal attack yields overly optimistic results. To derive the fundamental limits of watermarking and data-hiding systems, one should avoid making a priori assumptions about the embedding and decoding functions, analogously to Shannon's analysis of the fundamental limits of communication systems [16] ; and one may want to assume an intelligent opponent, as is done in some game-theoretic analyses of jamming systems [17, 18] . We have characterized data-hiding capacity for realistic image sources, by application of recent theoretical results on parallel Gaussian channels. Several statistical image models have been considered, ranging from conventional ones such as autoregressive processes and block-DCT models, to more advanced wavelet models that capture the spatial clustering of wavelet coefficients. The results obtained are guaranteed to be upper bounds on actual capacity if the true model deviates from the assumed parallel Gaussian model. The analysis demonstrates the essential role of image sparsity in determining capacity. The EQ wavelet model we considered captures sparsity much better than does the simpler block-DCT and autoregressive models, and yields better capacity estimates.. Our basic capacity results have been derived under MSE constraints on the data hider and on the attacker. We have also considered two types of perceptual distortion metrics: a weighted MSE, and a modified MSE metric that is invariant to geometric attacks. It appears that the dependence of capacity on the actual distortion metric used is quite weak. Moreover, we have shown that the loss in capacity due to geometric attacks may be quite benign.. Research is already underway on data-hiding codes that approach capacity for Gaussian channels. The results presented in this paper should likewise help design data-hiding codes that approach the capacity limits for images. The performance of several watermarking codes has been studied in several recent papers [30, 43, 44, 45] .. Closely related to these efforts is the development of optimal decoders. This requires moving beyond the conventional correlation rules and normalized correlation rules that have often been used in the image watermarking literature, due to their simplicity. An optimal decoder needs to be designed jointly with the encoder, and may need to be provided with the means to reliably estimate various unknown source and channel parameters.

O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


State Agnostic Planning Graphs: Deterministic, Non-Deterministic, and Probabilistic Planning Planning graphs have been shown to be a rich source of heuristic information for many kinds of planners. In many cases, planners must compute a planning graph for each element of a set of states, and the naive technique enumerates the graphs individually. This is equivalent to solving a multiple-source shortest path problem by iterating a single-source algorithm over each source.We introduce a data-structure, the state agnostic planning graph, that directly solves the multiple-source problem for the relaxation introduced by planning graphs. The technique can also be characterized as exploiting the overlap present in sets of planning graphs. For the purpose of exposition, we first present the technique in deterministic planning to capture a set of planning graphs used in forward chaining search. A more prominent application of this technique is in belief state space planning, where each search node utilizes a set of planning graphs; an optimization to exploit state overlap between belief states collapses the set of sets of planning graphs to a single set. We describe another extension in probabilistic planning that reuses planning graph samples of probabilistic action outcomes across search nodes to otherwise curb the inherent prediction cost associated with handling probabilistic actions. Our experimental evaluation (using many existing International Planning Competition problems) quantifies each of these performance boosts, and demonstrates that heuristic belief state space progression planning using our technique is competitive with the state of the art. Heuristics derived from planning graphs [4] are widespread in planning [19, 22, 43, 33, 8] . A planning graph represents a relaxed look-ahead of the state space that identifies propositions reachable at different depths. Planning graphs are typically layered graphs of vertices (P 0 , A 0 , P 1 , A 1 , ..., A k−1 , P k ), where each level t contains a proposition layer P t and an action layer A t . Edges between the layers denote the propositions in action preconditions (from P t to A t ) and effects (from A t−1 to P t ).. In many cases, heuristics are derived from a set of planning graphs. In deterministic (classical) planning, progression planners typically compute a planning graph for every search state in order to derive a heuristic cost to reach a goal state. (The same situation arises in planning under uncertainty when calculating the heuristic for a belief state.) A set of planning graphs for related states can be highly redundant. That is, any two planning graphs often overlap significantly. As an extreme example, the planning graph for a child state is a sub-graph of the planning graph of the parent, left-shifted by one step [44] . Computing a set of planning graphs by enumerating its members is, therefore, inherently redundant.. Consider progression planning in a classical planning formulation (P, A, I, G) of a Rovers domain, described in Figure 1 . The formulation (discussed in more detail in the next section) defines sets P of propositions, A of actions, I of initial state propositions, G of goal propositions. In the problem, there are two locations, L1 and L2, and an image I1 can be taken at L2. The goal is to achieve comm(I1), having communicated the image back to a lander. There are four actions drive(L1, L2), drive(L2, L1), sample(I1, L2), and commun(I1). The rover can use the plan: (drive(L1, L2), sample(I1, L2), commun(I1)) to achieve the goal. The state sequence corresponding to this plan is: of the same actions appear in the first action layer of the planning graph for each state. Figure 2 (described in detail below) depicts the search tree (top) and planning graphs for several states (bottom). A common task in many planners is to compute a set of planning graphs. The naive approach fails to take account of the redundant sub-structure of planning graphs. We developed the SAG as an extension of prior work on the LUG. The SAG employs a labeling technique which exploits the redundant sub-structure, if any, of arbitrary sets of planning graphs.. itive results in applying SAG, we see promise in applying SAG to other planning formalisms. The state agnostic planning graph is similar to many previous works that use common structure of planning problems within planning algorithms, use precomputed heuristics during search, or speed up the construction or use of planning graphs.

O O B O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O B O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O B O O B O O O B O O O O O O O O O O B O


Bag-of-colors

for

improved

image

search

This

paper

investigates

the

use

of

color

information

when

used

within

a

state-of-the-art

large

scale

image

search

system.

We

introduce

a

simple

yet

effective

and

efficient

color

signature

generation

procedure.

It

is

used

either

to

produce

global

or

local

descriptors.

As

a

global

descriptor,

it

outperforms

several

state-of-the-art

color

description

methods,

in

particular

the

bag-of-words

method

based

on

color

SIFT.

As

a

local

descriptor,

our

signature

is

used

jointly

with

SIFT

descriptors

(no

color)

to

provide

complementary

information.

This

significantly

improves

the

recognition

rate,

outperforming

the

state

of

the

art

on

two

image

search

benchmarks.

We

provide

an

open

source

package

of

our

signature

We

consider

the

problem

of

content-based

image

retrieval,

for

applications

such

as

object

recognition

or

similar

image

retrieval.

This

problem

requires

to

produce

a

description

of

the

image,

typically

a

fixed-size

vector.

This

description

must

be

discriminative,

but

sufficiently

invariant

to

handle

the

transformations

the

image

may

have

undergone

(cropping,

different

viewpoints,

rotation,

etc)..

Most

of

the

recent

state-of-the-art

large

scale

image

search

systems

rely

on

local

descriptors,

in

particular

the

SIFT

descriptors

and

its

variants.

They

are

extracted

from

regions

of

interest

using

a

scale-invariant

detector

These

descriptors

are

often

used

jointly

with

the

bag-of-words

(BOW)

framework

which

produces

a

single

vector

from

a

set

of

local

descriptors..

Surprisingly

and

despite

its

assessed

relevance

for

image

classification

most

of

the

recent

papers

on

large

scale

image

search

ignore

color

information.

The

best

results

reported

for

this

task

on

Area

chair:

Bernard

Merialdo

Permission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page.

To

copy

otherwise,

to

republish,

to

post

on

servers

or

to

redistribute

to

lists,

requires

prior

specific

permission

and/or

a

fee.

MM

November

Scottsdale,

Arizona,

USA.

Copyright

ACM

popular

benchmarks,

such

as

the

UKB

or

Holidays

datasets

which

both

contain

colorful

images,

are

reported

for

techniques

based

on

grey-level

SIFT

descriptors

only.

We

have

proposed

a

color

signature

inspired

by

the

bag-of-words

framework,

and

used

it

either

as

a

global

or

a

local

descriptor.

Our

experiments

show

that

color

information

significantly

improves

the

retrieval

quality

in

large

scale

image

search,

although

it

is

often

ignored

in

this

context.

Our

global

descriptor

exhibits

very

competitive

performance

compared

to

more

sophisticated

methods.

Used

as

a

local

descriptor,

the

results

are

better

than

those

obtained

by

a

state-of-the-art

system

based

on

SIFT

or

color

SIFT

descriptors.

As

in

Section

our

evaluation

is

performed

on

Holidays

and

UKB.

To

obtain

a

fair

comparison,

we

have

adjusted

the

threshold

of

the

color

SIFT

detector

so

that,

on

average,

it

approximately

produces

the

same

number

of

descriptors

per

image,

as

shown

below:

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Dynamic Replica Control Based on Fairly Assigned Variation of Data for Loosely Coupled Distributed Database Systems SUMMARY This paper proposes a decentralized and asynchronous replica control method based on a fair assignment of the variation in numerical data that has weak consistency for loosely coupled database systems managed or used by different organizations of human activity. Our method eliminates the asynchronous abort of already committed transactions even if replicas in all network partitions continue to process transactions when network partitioning occurs. A decentralized and asynchronous approach is needed because it is difficult to keep a number of loosely coupled systems in working order, and replica operations performed in a centralized and synchronous way can degrade the performance of transaction processing. We eliminate the transaction abort by fairly distributing the variation in numerical data to replicas according to their demands and updating the distributed variation using only asynchronously propagated update transactions without calculating the precise global state among reachable replicas. In addition, fairly assigning the variation of data to replicas equalizes the disadvantages of processing update transactions among replicas. Fairness control for assigning the data variation is performed by averaging the variation requested by the replicas. A simulation showed that our system can achieve extremely high performance for processing update transactions and fairness among replicas. The progress of world-wide computer networks has affected various fields of human activity, especially commercial activities, such as online shopping, trading, and banking. Applications used in commercial activities often handle numerical values, including the total stock of a particular product in a warehouse, production orders, and resources available to customers, such as rooms in a hotel. Such data is usually shared among many organizations all over the world. Those organizations usually place copies of the data in shared sites for better availability and scalability. As a result, a number of loosely coupled replicas exist around the world. It is difficult to keep loosely coupled replicated systems in working order for a long time. In addition, the type of user of these replicated database systems is changing from system experts to customers in commercial activities.. Control methods for replicated database systems can be classified into two types: strict and weak consistency. Strict consistency methods based on a quorum consensus and read-one-write-all [1] , [2] achieve serializability [3] but require a heavier overhead. On the other hand, weak con- sistency replica control methods [4] - [13] achieve high performance, availability, and scalability in return for permitting some inconsistency in the data. Several weak consistency replication methods [8] , [9] , [11] achieve extremely high availability and scalability for updating data anytime and anywhere using a decentralized and asynchronous approach. In such replication methods, any replica receiving an update transaction can initially process and then asynchronously propagate it among replicas. This type of data replication is called lazy-group replication [14] . In lazygroup replication, if an update transaction conflicts with already processed transactions, a conflict resolution procedure is invoked. This procedure depends on the semantics of a transaction. Replica control methods of this type have often been designed for specific applications. Conflict resolution can be defined using such application knowledge. When replicated database systems handle data using lazy-group replication even when network partitioning occurs, the asynchronous abort of already committed transactions can cause problems for customers and for the organizations managing or using replicas. Customers are required to issue requests to replicated database systems on the assumption that the asynchronous abort of already committed transactions can occur. For example, a customer sometimes chooses a distributor who can deliver a product earlier than others. In this case, the distributor must decide the earliest acceptable date for delivery. Assume that several replicas are separated from others but still continue to process update requests. The delivery date quoted to a customer might be extended because of the high number of requests processed by the separated replicas and the abort of some of the requests. When customers were notified about the longer delivery, they have to cancel their orders and place them to other distributors. Such behavior of replicated database systems confuses customers. From the viewpoint of organizations managing or using replicas (e.g., companies), the asynchronous abort of already committed transactions can be frequently imposed on some replicas. Because they usually compete with each other for their business, the probability of such an abort occurring in the result of customer trading can cause unfairness among replicas managed by the organizations.. To solve the above problems caused by the asynchronous abort of already committed transactions, we can use Data-Value Partitioning (DVP) [15] in a lazy-group replication method. DVP splits up the value of data for database systems. Each of them processes transactions usCopyright c 2005 The Institute of Electronics, Information and Communication Engineers ing the split value. However, because DVP needs to calculate the current value of data, it requires the periodical synchronous operation of replicas even though replicas can process transactions without interacting with others. This synchronous operation decreases the performance and scalability of transaction processing.. This paper proposes a decentralized and asynchronous replica control method based on a fair assignment of the variation in numerical data that has weak consistency. We call this the fairly assigned variation based replica control (FVRC) method. This method achieves decentralized and asynchronous replica control that eliminates the asynchronous abort of already committed transactions even if replicas in all network partitions continue to process transactions. We eliminate the abort of transactions by distributing the variation of numerical data to replicas and updating the distributed variation using only asynchronously propagated update transactions. Because this update of the distributed variation is performed asynchronously without synchronously calculating the precise global state among reachable replicas, our method can achieve high performance and scalability. It also enables fairness among replicas by reassigning the variation of data to replicas according to their demands so that the disadvantage of processing transactions can be balanced.. The remainder of this paper is organized as follows. Section 2 describes a decentralized and asynchronous replica control method based on a fairly assigned variation of data that has weak consistency. This section covers three issues. The first is the characteristics of data handled by the FVRC system and operations for it. The second is how a replica dynamically processes those operations in a decentralized and asynchronous way. The third is an algorithm for fairly assigning the variation of data among replicas. In Sec. 3, the evaluated results are discussed in terms of the performance of transaction processing and fairness among replicas achieved by our method. Section 4 discusses related work. Section 5 concludes the paper. We proposed fairly assigned variation-based replica control for a number of loosely coupled systems. This method achieves two important features in a decentralized and asynchronous way. First, it eliminates the asynchronous abort of already committed transactions even when network partitioning occurs. Second, it balances the demands of replicas to process update transactions so that successfully processed transactions have an equal effect on all organizations managing replicas. A decentralized and asynchronous approach is needed because it is difficult to keep a number of loosely coupled systems in working order, and replica operations performed in a centralized and synchronous way can degrade the performance of transaction processing.. We evaluated our method in terms of its transaction processing performance and fairness among replicas. The results showed that our system can achieve extremely high performance and fairness among replicas. Epsilon serializability [4] - [7] is a generalization of serializability [1] . It brings more concurrency by allowing a limited amount of inconsistency in transaction processing. To continue to process transactions even when a link or node failure occurs, systems using epsilon serializability need more reliability than the FVRC system. The FVRC system might restrict transaction semantics, but it is suitable for many applications used in the Internet because it enables weak coupling among replicas and allows unreliability of links and nodes.. The Escrow transactional method [3] , [23] is designed to offer non-blocking record updating by transactions that are long-lived. It provides high concurrency for transaction processing and allows distributed transactions in the presence of delayed messages and occasional line disconnection. An Escrow field allows fuzziness of the value of data. It improves the concurrency of transaction processing. However, this method does not simultaneously process update transactions in multiple partitioned networks. The idea of improving concurrency by allowing fuzziness is used in epsilon serializability as well as in the Escrow transactional method.. Data-Value Partitioning (DVP) [15] splits up the values of data and stores them as tokens on servers processing transactions. Ç etintemel et al. proposed token redistribution strategies for DVP and evaluated them using real wide-area message tracing [24] . They split the number into smaller portions, which were mainly to be consumed in only one direction, though they allowed the consumed number to be returned to a server. DVP can eliminate the abort of already committed transactions. On the other hand, our method achieves decentralized and asynchronous replica control to eliminate the abort of already committed transactions. It achieves high performance of transaction processing by eliminating synchronous interaction among replicas. In addition, the FVRC method provides symmetric operations in the positive and negative directions on numerical data and its premise is that numerical data is shared and scrambled by different organizations and companies for their business, which usually compete with each other. Our system enables fairness when those organizations and companies try to obtain a portion of the numerical data.. In our system, a replica provides the value of data using its copy and unconsumed local variation limits. As a result, the value of data read by a client includes some fuzziness but has the lower and upper bounds of the data. Yu and Vahdat proposed a technique for efficient error bounding [25] . A replica in our system provides the lower and upper bound of data by assuming that all local variation limits assigned to the other replicas are consumed. To reduce the error included in data handled by our system, we need tighter coupling among replicas. However, this is less suitable for use on the Internet.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure We present a practical algorithm for computing robust, multiscale curve and surface skeletons of 3D objects of genus zero. Based on a model which follows an advection principle, we assign to each point on the skeleton a part of the object surface, called the collapse. The size of the collapse is used as a uniform importance measure for the curve and surface skeleton, so that both can be simplified by imposing a single threshold on this intuitive measure. The simplified skeletons are connected by default, without special precautions, due to the monotonicity of the importance measure. The skeletons possess additional desirable properties: They are centered, robust to noise, hierarchical, and provide a natural skeleton-to-boundary mapping. We present a voxel-based algorithm that is straightforward to implement and simple to use. We illustrate our method on several realistic 3D objects. We have proposed the collapse measure, a novel importance measure that enables the robust computation of multiscale curve and surface skeletons of 3D objects. To our knowledge, this is the first truly global 3D importance measure that can be used to obtain both surface and/or curve skeleton hierarchies in a uniform manner. Underpinning the collapse measure is a physical process in which mass from the object boundary is advected or "collapses" onto the skeleton. This gives the measure an intuitive meaning, and allows us to reason about it and deduce several properties of interest. For the practical implementation of the proposed measure, we do not explicitly simulate the advection process, as this can be computationally unstable and complex. Instead, we propose a practical algorithm that is straightforward to implement, does not need any post-processing steps, and is robust, as it uses only integral operations. Our algorithm delivers good results on a wide range of real-world, complex objects. A limitation of our measure is that objects with tunnels cannot be handled. However, this is a known limitation for the analogous 2D feature-distance measure [5] . In detail, we can compute curve skeletons for such objects, but not simplify them, since we have no importance measure for their loop-parts.. In future work, we want to apply our skeletons in various applications, such as the simplification of complex 3D objects. Also, we consider an extension of the collapse measure that can handle objects with tunnels. The recursive nature of our approach makes it interesting to look at generalizations and applications beyond 3D, e.g. for time-dependent surfaces. Finally, on a more theoretical ground, we plan to investigate our conjecture (see Sec. III-B) in more depth to acquire more insight into the relation of shapes with their skeletal counterparts. Because curve and surface skeletonization methods differ in the precise skeleton definition that they use, the object representation they work on, and the applications they target, they are usually compared by their results based on various desirable properties as mentioned in Section I. In Section VI-A we discuss how our approach satisfies these desirable properties. In Section VI-B, we discuss how several key aspects of our approach relate to other approaches. We have implemented the algorithm from the previous section in C++ and have run it on a Pentium 4, 3GHZ with 1GB of RAM. As input we used several complex polygonal meshes from [42] , [43] , voxelized using binvox [44] for various resolutions. We used object resolutions ranging up to 512 3 voxels. For each object the resolution that we used is appended to the object's name. The Mobile object was created such that it incorporates several synthetic, nonnatural shapes. Various measurements are shown in Table I . Column "dim" indicates the dimension of the object. Columns "|Ω|" and "|∂Ω|" indicate the number of boundary and object voxels respectively. Columns "#paths" indicate the number of computed shortest paths, and "#comp." indicates how many times the connected components are computed ( Fig. 4, line 5 ), whereas "paths t" and "comp. t" show the total wall-clock time for both. The last column shows the total wall-clock time for the whole algorithm to complete. As can be seen, all objects are completed within 10 minutes. We use a simple caching scheme for shortest paths to prevent computing the same shortest path more than once as much as possible. The size of the cache is a user parameter and presents a tradeoff between speed and memory usage. In these measurements a cache size of 50 MB was used, which presents a speed-up factor of 3 to 4 approximately. For these resolutions, larger cache sizes do not improve speed significantly. Column "mem" shows the peak memory usage, which is below 800 MB for all the objects we considered. Fig. 7 shows the simplified skeletons S τ of three objects as computed by our algorithm, with the indicated values of τ . The importance measure is visualized using a rainbow color map, mapping 0 to blue (i.e. unimportant skeleton points) and 1 to red (i.e. central, important skeleton points). We observe that the non-generic cases in the Mobile object, such as the cylinder and extruded star, are handled well. The surface skeleton is mainly blue because its importance measure is significantly smaller than that of the curve skeleton, as explained in Section III-C. In order to highlight the variation of ρ on S \ C we can better use the equalized skeleton S ′ τ . Fig. 8 shows S ′ τ for the Bird object. It is interesting to see how S τ and S ′ τ progress differently when increasing τ . With S ′ τ , the surface and curve skeleton are simplified more simultaneously, especially near the periphery of the skeleton, which can be useful for some applications, as we will see in the next section. Fig. 9 shows the curve skeleton in isolation for various objects. Note that we modified the original Rockerarm object so that it is genus 0. We observe that the extracted curve skeletons reach into fine structures like the tentacle tips of the Octopus object and the Cow's tail, are centered with respect to the object surface, and exhibit very little wiggle noise. Cornea et al. [8] made a comparison between four curve skeleton methods: a basic thinning, distance field, geometric, and potential-field method. Among the objects tested was the Plane object. We observe that our approach delivers superior results on this object (compare Fig. 9 and [8, Fig.12] ). The price we pay is that our method is slower than the first three methods mentioned in [8] , although it is the same order as the potential-field method. More results of our skeletonization method are available online at [45] , including animations of S τ 's progression. Methods for computing medial axes and skeletons are commonly classified into four groups [8] : thinning or erosion [9] , geometric methods [5] , [10] , [11] , distance-field methods [7] , [12] - [14] , and general-field methods [15] , [16] . Such methods usually work either on a raster or voxel-based object representation, or on a boundary sampling. Thinning methods iteratively remove boundary voxels as long as connectivity is not violated. Removing voxels in order of a true distance-to-boundary fashion enforces the centeredness criterion [17] . Geometric methods often use the Voronoi-diagram of a boundary sampling and deliver a polygonal skeleton description [5] . These methods are quite complex to implement in 3D and the resulting surface skeletons can be difficult to handle. Other geometric methods compute skeletons using edge collapses [10] , from a mesh segmentation [18] , or by sphere sweeping [11] . Distance-field methods detect skeletons as singularities of various types of the distance transform of the object [3] , [13] and have been efficiently implemented using graphics hardware [14] , [19] , [20] . In general-field methods alternatives for the distance transform are used [15] , [16] . These methods are generally less sensitive to noise, due to the averaging effect that the field has.. A key aspect of any skeletonization method is its ability to produce robust, hierarchical skeletons. At the core of any such method is an importance measure for the skeleton points. Such measures have the desirable property that small values correspond to skeleton points that represent object details, whereas large values correspond to points that represent the object's main structure. Simplified skeletons can be obtained by pruning the skeleton using the importance measure [21] .. We distinguish between local and global measures. Local measures use only object properties from a small, limited vicinity of every object point, and are thus incapable of distinguishing between locally identical configurations, such as those sketched in 2D in Fig. 1 . Thresholding local measures usually delivers disconnected skeletons. Connectivity can be enforced, but this requires extra work [13] , [22] , [23] , and often makes the simplification non-continuous [21] , thus less intuitive. In contrast, global measures monotonically increase from the skeleton boundary to its center, and describe a skeleton point's importance to representing the whole object. Pruning global measures by simple thresholding always delivers connected skeletons.. In 2D, the feature-distance measure, also called collapsed boundary measure, is a simple, robust and efficient to compute global importance measure used both for polygonal and pixel object representations by several authors [5] - [7] . Fig. 1 sketches this measure: The importance of the skeleton point p equals the length of the smaller boundary curve delimited by points p a and p b , where p a , p b are the points at minimum distance of p: the so-called feature points. By this measure, the importance of q is regarded lower, which agrees with intuition. In 3D, the θ-SMA method uses a local measure based on the angle between the feature (or anchor) points [19] , [24] . Combining the angle with the distance-to-boundary value yields a more robust measure [22] . Siddiqi et al. propose a range of methods that use a local divergence-based measure to detect and classify skeleton points in 3D [13] , [25] , [26] . A similar approach using first-order moments is given by Rumpf and Telea [3] . Pizer et al. [27] give a comparison of methods that compute multiscale skeletons. The general-field method of Cornea et al. [16] delivers smooth curve skeleton hierarchies. However, only a limited number of hierarchy levels is generated and the different levels are not treated uniformly. Recently, Dey and Sun [28] presented a method that computes robust curve skeletons. Similar to Prohaska et al. [29] , they use shortest-geodesic length between feature points as an importance measure for the surface skeleton points. This is a global importance measure, used to find the middle of the surface skeleton. They obtain a curve skeleton by using the local divergence-measure of [13] on this geodesiclength measure.. Related to the curve skeleton is the Reeb graph: a 1D structure encoding geometrical and topological properties of 3D shapes [30] , [31] . It is constructed by following the evolution of the level sets of a real-valued function on the object boundary. The critical points of this function are represented by nodes in the Reeb graph, and the edges represent connections between critical points. The real-valued function, defined on the object boundary, should be chosen carefully to obtain a Reeb graph that is suitable for the application at hand. In contrast, our approach uses curve and surface skeleton definitions to obtain a real-valued function on the skeleton, the importance measure, which is then used to simplify it.. In this paper, we propose a global importance measure for computing multiscale surface and curve skeletons, by generalizing the 2D feature-distance measure to 3D. At the core of our measure is an advection principle. In the next section we present the conceptual model.

O O O O B I O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O


Reflected Brownian motion in the quadrant: Tail behavior of the stationary distribution Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a "direction of reflection" for each of the quadrant's two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and those conditions are restated here in a novel form; they are the only restrictions imposed on the process data in our study. Under those minimal assumptions, a large deviations principle (LDP) is known to be valid for the stationary distribution of Z. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x → ∞, and the asymptotic decay rate is the minimum value achieved by a certain function I(·) over the set B. Avram, Dai and Hasenbein [1] provided a complete and explicit solution for the large deviations rate function I(·). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(·) reduces to a relatively simple problem of least-cost travel between a point and a line. This paper continues earlier work by Avram, Dai and Hasenbein [1] , referred to hereafter as ADH. For the most part, we use the same notation and terminology as ADH, and to avoid wasteful duplication, readers will be referred to ADH for background information on various topics. The focus of our study is the class of two-dimensional diffusion processes called semimartingale reflected Brownian motions (SRBM's), which arise as approximations for open, two-station queueing networks. The state space for a process Z = {Z(t), t ≥ 0} in this class is the non-negative quadrant, which we denote by Q. The data of the process are a drift vector θ, a non-singular covariance matrix Γ, and two "directions of reflection" R 1 and R 2 (see Figure 1 ) that specify boundary behavior. In the interior of the quadrant, Z behaves as an ordinary Brownian motion with parameters θ and Γ, and roughly speaking, Z is pushed in direction R k when the boundary surface Z k = 0 is struck (k = 1, 2). To make this description more precise, let R be a 2 × 2 matrix whose first and second columns are R 1 and R 2 . Then Z can be represented in the form. where X is an ordinary Brownian motion with drift vector θ, covariance matrix Γ, and initial state X(0) ∈ Q, and Y is a two-dimensional process with components Y 1 and Y 2 that have the following properties:. • Y is continuous and non-decreasing with Y (0) = 0, and. • Y k only increases at times t for which Z k (t) = 0 (k = 1, 2).. This characterization of Z should be viewed as an informal paraphrase of the precise mathematical definition recounted in Section 4 of ADH. Readers can also find there a compact summary of the well developed foundational theory for SRBM in the quadrant. Building on that foundational theory, it will be shown in Appendix A of this paper that two simple restrictions on the process data are necessary and sufficient for the following to hold: first, there exists a diffusion process Z of the form described above (when it exists, Z is also unique in distribution); and second, Z is positive recurrent, meaning that it admits a stationary distribution. The two simple restrictions are the following:. R is a P -matrix;. In general, an n × n matrix is said to be a P -matrix if all of its principal minors are positive; in our two-dimensional case, (1.2) means that both the diagonal elements of R and its determinant are strictly positive.. To repeat, conditions (1.2) and (1. 3) together assure the existence of a positive recurrent process Z of the form described above, and that process Z is unique in distribution. Moreover, if either (1.2) or (1.3) fails to hold, then either the existence or the positive recurrence of Z must also fail. As we shall explain in Appendix A, (1.2) and (1.3) constitute a compact re-expression of necessary and sufficient conditions for the existence, uniqueness and positive recurrence of Z that have been known since the mid-1990's. The development in Appendix A shows that (1.2) is necessary for existence of a positive recurrent process Z, regardless of what θ may be. Condition (1.3) has long been known as a necessary and sufficient condition for positive recurrence of certain multidimensional SRBM's that arise in queueing theory, cf. Harrison and Williams [6] . A forthcoming paper by Dai and Harrison [3] shows that (1.3) is necessary for positive recurrence in general (that is, for SRBMs in the n-dimensional orthant, without special assumptions on R) but is not sufficient for positive recurrence in a general setting.. ADH enunciated a large deviations principle (LDP) for the stationary distribution of Z as Conjecture 4.1 of their paper. The precise statement of the LDP is somewhat technical, and it need not be repeated here, but the following loose paraphrase will be useful for expositional purposes: for sufficiently regular sets B (including convex sets having nonempty interiors), the stationary probability of xB decays exponentially as x → ∞, and moreover, the asymptotic decay rate is governed by the solution of a certain variational problem. The relevant variational problem involves a target point v as a parameter, and its optimal value I(v) can be animated as the minimum cost to travel from the origin to v. The asymptotic decay rate referred to immediately above is the minimum value achieved by I(·) over the set B. Following standard terminology, we call I(v) the large deviations rate function. As we shall explain in Appendix B, the LDP is actually valid whenever (1.2) and (1.3) hold; this had not been proved for all relevant cases when the ADH paper was written, but the gaps have now been filled. ADH developed a complete and explicit solution for I(v), in addition to proving some partial results for higher-dimensional problems. In this paper we re-express the ADH solution in a simplified form, and offer a more extensive explanation of its structure.. The remainder of the paper is organized as follows. Section 2 addresses some relatively simple variational problems in the plane: the main goal is to find a least-cost path from a point to a line, and from a line to a point, where the instantaneous cost rate is a quadratic function of the velocity vector. (Such quadratic functions arise in large deviations theory for Brownian motion.) In Section 3 those results are applied in a more particular setting, preparing the way for later developments. Section 4 is devoted to a variational problem in the upper half-plane, with "oblique reflection" at the boundary; we show that it reduces to the simpler problems solved in Section 3. In Section 5, assembling the component parts developed earlier, we present our simplified restatement of the ADH solution for the large deviations rate function, which is compactly summarized in Proposition 5.3. Section 6 contains some concluding remarks about related earlier work, especially the random walk theory developed by V. Malyshev and his co-workers, and also summarizes the contributions of this paper relative to ADH. The justification of (1.2) and (1.3) as necessary and sufficient conditions for existence, uniqueness and positive recurrence of the process Z is presented in Appendix A, and Appendix B reviews antecedent work on the LDP whose rate function is our central focus. Because our development starts with preparatory propositions having no obvious connection with the problem of ultimate interest, readers may wish to look first at Section 5, then quickly scan the earlier sections in reverse order to see the overall structure of the argument before getting into details. For general v ∈ U , we have defined K(v) as the minimum cost achievable with a simple path x ∈ S(0, v). One has from Propositions 4.2 and 4.3 that (4.24). where H(·) is given by the three equivalent expressions (3.15)-(3.17), and f (·) is the basic cost functional (2.2); denoting by r and ξ the polar radius and polar angle of v, and recalling from Section 3 that the unit vector µ has polar angle π + α, readers can easily verify the alternative formula The contributions of this paper lie in three areas. First, by identifying the alternative conditions (1.2)-(1.3) for existence and positive recurrence of Z, and by updating the ADH Figure 14 : Three cones whose union is the wedge W references on associated large deviations theory, we provide a clean and definitive setting for study of the rate function I(v). Second, by making the standard change of variable v → cM v described in Section 5, and by treating the angles α i and β i in Figure 13 as our basic problem data, we are able to express I(v) in a highly simplified form (see Proposition 5.3). Finally, and perhaps most importantly, our analysis reduces the calculation of I(·) to a relatively simple variational problem whose fundamental role has not been recognized earlier, namely, the problem of least-cost travel between a point and a line (see Section 3). The three-cones structure portrayed in Figure 14 for the large deviations rate function is identical to a structural finding by Malyshev [10, 11, 12] for random walks in the quadrant. That work, which uses transform relationships and complex variable methods to study the stationary distribution and its tail behavior, is recapitulated in the English-language book by Fayolle, Iasnogorodski and Malyshev [5] ; see Kurkova and Suhov [8] for further explication of Malyshev's approach and methods. Random walks with boundaries are more complicated in virtually every regard than the SRBM's analyzed here, and not surprisingly, Malyshev's asymptotic theory is less explicit than the ADH analysis on which we have expanded.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Design preserving garment transfer We present a fully automatic method for design-preserving transfer of garments between characters with different body shapes. For real-life garments, such transfer is performed through a knowledge intensive and time consuming process, known as pattern grading. Our first contribution is to reformulate the criteria used in professional pattern-grading as a set of geometric requirements, respectively expressing shape or design preservation, proportionality, and fit. We then propose a fully automatic garment transfer algorithm which satisfies all of these criteria while ensuring the physical plausibility of the result. Specifically, we formulate garment transfer as a constrained optimization problem and solve it efficiently through iterative quadratic minimization. As demonstrated by our results, our method is able to automatically generate design-preserving versions of existing garments for target characters whose proportions and body shape significantly differ from those of the source. The method correctly handles the transfer of multiple layers of garment. Lastly, when source 2D patterns are available, we output graded patterns suitable for manufacturing the transferred garments. Our fully automatic design-preserving transfer method leads to significant time savings for both computer artists and fashion designers. Modeling of complex real or virtual garments is a highly time consuming and knowledge intensive task. For instance, it took an experienced computer artist three hours to generate the woman's outfit in Figure 1 (left), a process that included creating 2D patterns, placing them on the mannequin, running cloth simulation and then iteratively adjusting patterns and parameters, repeating the process until the desired effect was achieved. If we wished to design a similar looking outfit for a young girl who has very different proportions, see Figure 1 (left), this tedious process would need to be repeated practically from scratch. Consequently, dressing a large number of virtual characters in similarly designed outfits, Figure 1 (right), is currently prohibitively time consuming, even if state-of-the-art garment modeling interfaces such as [Umetani et al. 2011 ] are used.. In real-life, professional pattern makers adapt garments to people with different proportions and body-types through a complex process known as grading or pattern grading. Manual grading aims to preserve the design or look of the garment while adjusting it to fit the proportions of the new wearer [Moore et al. 2001] . This process requires significant specialized expertise and relies on a set of precomputed rule tables for standard body sizes. Applying a similar process to virtual garments can be especially challenging since the characters to be dressed often have non-standard body proportions for which no rule-tables exist. While skinning-type garment transfer techniques [Cordier et al. 2003; Wang et al. 2005 ] adjust the garment to fit the target character's proportions reasonably well, they often fail to preserve the original design, requiring manual editing to recover it [Meng et al. 2012] .. Instead, we focus on automatic design preserving garment grading or transfer, introducing a method that produces garments that fit the proportions of the new wearer while retaining the original design as much as possible. As emphasized in the literature, garment transfer is an ill-posed problem. Formalizing what makes a garment look the same on characters of different body shapes, although the input and output garment surfaces obviously differ, is in itself a challenge. Therefore, our first contribution, presented in Section 3, is to analyze professional pattern-grading criteria and reexpress them in terms of geometric requirements for garment-mesh transfer. We then introduce a garment transfer algorithm that satisfies these requirements, as our main contribution. We formulate garment transfer as a constrained optimization problem which optimizes design preservation subject to a number of geometric constraints arising from proportionality, plausibility and other considerations elaborated upon in Section 3. To efficiently generate the graded garments we solve this problem using an iterative process, where each iteration consists of solving a quadratic minimization problem.. As demonstrated by the results, our method enables fully automatic, realistic, design preserving transfer of garments between characters with vastly different body shapes and proportions. Consequently, it significantly simplifies the modeling of new dressed characterspre-existing wardrobes can be instantly transferred from one character to another without the need for any specific user skill. Our method also allows efficient generation of large groups of similarlydressed characters for the film industry -e.g. warriors, workers, dancers, etc, without the usual restriction to clone-like models.. Plausible virtual garments have a low distortion mapping to a set of 2D patterns, enabling texture mapping and actual manufacturing. Our design preservation criteria, as discussed in Section 3 explicitly aim to preserve plausibility. Thus given the 2D patterns for the source garments, our method directly generates the 2D patterns for the transferred one (Figure 1 , middle). By automating garment transfer, our work greatly simplifies the generation of dressed characters, currently one of the more tedious and difficult task for computer artists. It enables transfer of both virtual and real-life garments between characters with very different body shape and proportions automatically adapting source patterns to fit the transferred garment. Our technique lets non-expert users to adapt pre-existing wardrobes to their characters of choice, removing the tailoring knowledge barrier. It is directly applicable to clothing crowds of arbitrary characters, as it requires no per-output tweaks. The two key components of our approach are a geometric formulation, and subsequent efficient optimization, of the set of criteria used in manual pattern-grading.. Our method has three main limitations. We rely on purely geometric considerations in the setting of the tight regions. Reverse engineering designer intent can be an interesting topic for future research. Similarly we ignore the impact of fine cloth mechanics on the shape of real-life or simulated output garments. However as demonstrated by the figures and the accompanying video our outputs retain the source design following simulation. Lastly, the method is sensitive to the quality of the cross-parameterization between the source and target models.. Possible improvements of the method include further speedup through the use of tailored solvers, such as [Ascher and Boxerman 2003] and pattern regularization consistent with tailoring approaches. It would be very interesting to apply our method to resizing of other 3D shapes, comparing the results with state of the art [Kraevoy et al. 2008; Bokeloh et al. 2011 ].. Figure 12: Transfer of garments to progressively more different models (front and side views): a stockier shorter female, a pregnant female, and an exaggerated male character. The texture on the transferred dresses was generated using the automatically computed patterns. Discontinuities (e.g. the line under the chest) reflect seam locations. Figure 13 : Top: A dress (a) transferred to a fatter model using skinning [Wang et al. 2005 ] (b) shows visible artifacts along silhouettes. On this dress (c,d) our automatic method (yellow) generates slightly better results than the user assisted approach of Meng et al. [2012] (blue) (arrows point to undesirable artifacts). Bottom: When transferring a dress (Figure 12 , top) to a more different pregnant model, using the indicated user-specified profiles (a) Meng et al. [2012] (blue) generate severe artifacts, resulting in a dress which is not physically plausible (floating around the belly). In contrast, our automatically generated output (yellow) reflects the input design.. 36:10 • R. Brouet et al. Previous research relevant to our work can be roughly split into three areas: general garment modeling approaches, grading and transfer methods for garments, as well as resizing and transfer methods for other domains.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O B O B I O O O


Languages of logic and their applications Concerning the logical description languages, in the past 40-50 years many authors have introduced a number of structurally very different first-order languages. Some of these languages follow the structure of a given future model, other ones have been prepared for the description of an arbitrary model. Other variations of the first-order languages do not follow the whole structure of any model: they have been prepared only for the relations definable over the universe in order to be able to prove the generalizations of a number of difficult logical results. The semantics of the first-order languages is based on the interpretation of their extralogical symbols by a suitable model. In some cases, in the interpretation all possible models can be in focus, but there are cases when the models over a special universe are regarded. The naming problem of the universe element of the model arises at this stage. The efforts for solving this problem lead to different approaches. Here, we present the most important language definitions and some characteristic semantics. We investigate the different approaches and conclude that they do not indicate essential differences. In fact, they have been only motivated by seeking for an easier way to achieve the just fixed target. Moreover, we try to point out the suitability connections of languages and semantics definitions.

O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O