### Table 7.4: Accuracy for face recognition using invariants and signatures. Invariant Type

in Dedication

### Table 1n3a Top matches between an0ene invariant signatures

in Image Indexing and Retrieval Using Image-Derived, Geometrically and Illumination Invariant Features

"... In PAGE 4: ... Finallyn2c we show the results of applying illumination invariants to further verify a correct match. Table1 shows both the en0bectiveness and limitan2d tions of using an0ene invariantsn2c when dealing with obn2d jects under perspective transformations. An0ene invarin2d ant signatures of the airplane images were stored in a database n28performed on0fine using Eq.... In PAGE 4: ... 4n29 were used to determine the similarity between each pair of signatures. Each row in Table1 refers to a Rank n28using an0ene invariantsn29... In PAGE 4: ... Entries printed in boldface are the expected n28correctn29 matches. It is clear from Table1 that the an0ene invariant works well in cases where the object is far from the camera relative to its size. Query images B and D are consistent with this scenario.... ..."

### Table 1: Signature for stack operations

2000

"... In PAGE 5: ... Signatures are well-known from the definition of abstract data types. For example, in the description of a stack we have sorts STACK, INT and BOOL, and operators push, pop, and empty, as shown in Table1 . A term of this signature is push(empty, 8).... In PAGE 17: ...Aggregation Aggregation reduces sets of points to points (Table 10). Operation Signature Semantics min, max ! [1D] min( (U)); max( (U)) avg ! [1Dnum] 1 jintvls(U)j P T2intvls(U) sup(T)+inf (T) 2 avg[center] points ! [2D] 1 n P p2U !p avg[center] line ! [2D] 1 kUk P c2sc(U) !c kck avg[center] region ! [2D] 1 M R U (x; y) dU where M = R U dU single ! if 9u : U = fug then u else ? Table1 0: Aggregate Operations In one-dimensional space, where total orders are available, closed sets have minimum and maximum values, and functions (min and max) are provided that extract these. For open and half-open intervals, we choose to let these functions return infimum and supremum values, i.... In PAGE 18: ....2.5 Numeric Properties of Sets For sets of points some well known numeric properties exist (Table 11). Operation Signature Semantics no components ! int [1D] jintvls(U)j no components points ! int jUj no components line ! int jblocks(U)j no components region ! int jfaces(U )j size[duration] ! real [1Dcont] P T2intvls(U) sup (T ) inf (T ) size[length] line ! real kUk size[area] region ! real R U dU perimeter region ! real flength(@U) Table1 1: Numeric Operations For example, the number of components (no components) is the number of disjoint max- imal connected subsets, i.e.... In PAGE 19: ... The time domain inherits arithmetics from the domain of real numbers, to which it is isomorphic. Operation Signature Semantics distance ! real [1Dcont] ju vj ! real [1Dcont] minfju vj jv 2 V g ! real [1Dcont] minfju vj ju 2 U; v 2 V g ! real [2D] dist(u; v) = p(u:x v:x)2 + (u:y v:y)2 ! real [2D] minfdist(u; v)jv 2 V g ! real [2D] minfdist(u; v)ju 2 U; v 2 V g direction point point ! real see below Table1 2: Distance and Direction Operations The direction between points is sometimes of interest. A direction function is thus included that returns the angle of the line from the first to the second point, measured in degrees (0 angle lt; 360).... In PAGE 19: ...ention them because they have to be included in the scope of operations to be lifted, i.e., the kernel algebra. Operation Signature Semantics and, or bool bool ! bool as usual (with strict evaluation) not bool ! bool Table1 3: Boolean Operations 4.2.... In PAGE 20: ... For values of intime types, the two trivial projection operations inst and val are offered, yielding the two components. Operation Signature Semantics deftime moving( ) ! periods dom( ) rangevalues moving( ) ! range( ) [1D] rng( ) locations moving(point) ! points isolated(rng( )) moving(points) ! points isolated(S rng( )) trajectory moving(point) ! line rng( ) n flocations( ) moving(points) ! line S rng( ) n flocations( ) traversed moving(line) ! region ((S rng( )) ) moving(region) ! region S rng( ) routes moving(line) ! line (S rng( )) n ftraversed( ) inst intime( ) ! instant t where u = (t; v) val intime( ) ! v where u = (t; v) Table1 4: Operations for Projection of Temporal Values into Domain and Range All the infinite point sets that result from domain and range projections are represented in collapsed form by the corresponding point set types. For example, a set of instants is represented as a periods value, and an infinite set of regions is represented by the union of the points of the regions, which is represented in turn as a region value.... In PAGE 22: ...o the given domain or range values, e.g., get the part of the moving point when it was within the region, or determine the value of the moving real at time t or within time interval [t1; t2]. Operation Signature Semantics atinstant moving( ) instant ! intime( ) (t; (t) quot; ) atperiods moving( ) periods ! moving( ) f(t; y) 2 jt 2 T g initial moving( ) ! intime( ) limt!inf(dom( )) (t) final moving( ) ! intime( ) limt!sup(dom( )) (t) present moving( ) instant ! bool (t) 6 = ? present moving( ) periods ! bool fatperiods( ; T ) 6 = ; at moving( ) ! moving( ) [1D] f(t; y) 2 jy = bg at moving( ) range( ) ! moving( ) [1D] f(t; y) 2 jy 2 Bg at moving( ) point ! mpoint [2D] f(t; y) 2 jy = ug at moving( ) ! moving(min( ; ))[2D] f(t; y) 2 jy 2 Ug atmin moving( ) ! moving( ) [1D] f(t; y) 2 jy = min(rng( ))g atmax moving( ) ! moving( ) [1D] f(t; y) 2 jy = max(rng( ))g passes moving( ) ! bool fat( ; x) 6= ; Table1 5: Interaction of Temporal Values With Values in Domain and Range In Table 15, the first group of operations concerns interaction with time domain values, the second interaction with range values. Operations atinstant and atperiods restrict a moving entity to a given instant, resulting in a pair (instant, value), or to a given set of time intervals, respectively.... In PAGE 22: ...o the given domain or range values, e.g., get the part of the moving point when it was within the region, or determine the value of the moving real at time t or within time interval [t1; t2]. Operation Signature Semantics atinstant moving( ) instant ! intime( ) (t; (t) quot; ) atperiods moving( ) periods ! moving( ) f(t; y) 2 jt 2 T g initial moving( ) ! intime( ) limt!inf(dom( )) (t) final moving( ) ! intime( ) limt!sup(dom( )) (t) present moving( ) instant ! bool (t) 6 = ? present moving( ) periods ! bool fatperiods( ; T ) 6 = ; at moving( ) ! moving( ) [1D] f(t; y) 2 jy = bg at moving( ) range( ) ! moving( ) [1D] f(t; y) 2 jy 2 Bg at moving( ) point ! mpoint [2D] f(t; y) 2 jy = ug at moving( ) ! moving(min( ; ))[2D] f(t; y) 2 jy 2 Ug atmin moving( ) ! moving( ) [1D] f(t; y) 2 jy = min(rng( ))g atmax moving( ) ! moving( ) [1D] f(t; y) 2 jy = max(rng( ))g passes moving( ) ! bool fat( ; x) 6= ; Table 15: Interaction of Temporal Values With Values in Domain and Range In Table1 5, the first group of operations concerns interaction with time domain values, the second interaction with range values. Operations atinstant and atperiods restrict a moving entity to a given instant, resulting in a pair (instant, value), or to a given set of time intervals, respectively.... In PAGE 23: ....3.3 The Elusive when Operation We now consider (speculate about) an extremely powerful yet conceptually quite simple oper- ation called when, whose signature is shown in Table 16. The idea is that we can restrict a Operation Signature Semantics Syntax when moving( ) ( ! bool) ! moving( ) f(t; y) 2 j p(y)g arg1 op[arg2] Table1 6: The when Operation time dependent value to the periods when its range value fulfils some property specified as a predicate. If we had such an operator, we could express a query such as Restrict a moving region mr to the times when its area was greater 1000 as: mr when[FUN (r:region) area(r) gt; 1000] Here the result would be of type mregion again.... In PAGE 26: ...Signature Semantics derivative mreal ! mreal 0 where 0(t) = lim !0( (t + ) (t))= speed mpoint ! mreal 0 where 0(t) = lim !0 fdistance( (t + ); (t))= turn mpoint ! mreal 0 where 0(t) = lim !0 fdirection( (t + ); (t))= velocity mpoint ! mpoint 0 where 0(t) = lim !0( (t + ) (t))= Table1 7: Derivative Operations between two points, and the vector difference (viewing points as 2D vectors). This leads to three different derivative operations, which we call speed, turn, and velocity, respectively.... In PAGE 27: ... 4.2 Table1 8: Operations on Sets of Database Objects The syntax for applying this operator is arg1 op[arg2; arg3]. The semantics can be defined formally as follows.... ..."

Cited by 114

### Table 1. Invariants of arrangements

2001

"... In PAGE 24: ... Otherwise, we merely show the underlying matroid of A, which keeps track of the incidence relations. Table1 contains the list of objects and invariants that we associate with A. Some immediate numerical information is extracted from the intersection lattice, L(A).... ..."

Cited by 12

### Table 1 Numerical results for surface curves shown in Fig. 2.

"... In PAGE 6: ...8 GHZ Pentium 4 with 1GB RAM. Timing results for our method (2) are given in Table1 . For a fast orthogonal projection of a point onto a surface (7) we refer the reader to the literature (cf.... In PAGE 6: ... Since we have just implemented a simple un-optimized algorithm for the solution of (7) we do not give timing results for this part of the algorithm. Table1 contains the following informa- tion, which we explain at hand of the Moebius strip example: given 7 input points on we apply 4 subdivision steps which results in 97 output points on . In subdivision step 1; 2; 3; 4 the optimization (2) has been performed 4; 3; 2; 2 times.... ..."

### Table 1. Invariants of arrangements De ning polynomial

2001

"... In PAGE 24: ... Otherwise, we merely show the underlying matroid of A, which keeps track of the incidence relations. Table1 contains the list of objects and invariants that we associate with A. Some immediate numerical information is extracted from the intersection lattice, L(A).... ..."

Cited by 12

### Table 1. Invariants of arrangements De ning polynomial

2001

"... In PAGE 22: ... Otherwise, we merely show the underlying matroid of A, which keeps track of the incidence relations. Table1 contains the list of objects and invariants that we associate with A. Some immediate numerical information is extracted from the intersection lat- tice, L(A).... ..."

Cited by 12

### Table 4: PCS Signatures for Curved Bus Lines

"... In PAGE 5: ...Table 1-3 list the PCS signatures and standard deviations of four straight wires. Table4 lists the cases have curves. According to these tables, opens and shorts have deviations over 20%.... ..."

### Table 4. Prediction of time-invariant covariates

2005

"... In PAGE 7: ... set S10=L10 T=S10 T=L10 S10=Lall T=Lall Sall=L10 T=Sall Sall=Lall training 99:53% 98:74% 98:59% 99:53% 98:59% 99:53% 98:74% 99:53% testing 91:16% 91:32% 92:13% 91:16% 92:11% 26:46% 91:32% 26:46% Table 3. CCRs after applying the time-dependent predictive model The results of using time-invariant predictive model for SDB is presented in Table4 . Table 4 shows that it is possible to calculate the predictive matrices at the training stage and then apply then successfully on the testing stage, thus achieving good recognition performance.... ..."

Cited by 3

### Table 3: Summary of admissible operations and invariant-preserving actions in response to the manipulation of object pairs. Kind Set

"... In PAGE 11: ... Table 2 indicates the invariant-maintaining actions to be applied to the corre- sponding relations whenever objects are created or deleted and details under what circumstances the operations are admissible. Table3 depicts how changes in participation of object pairs influence individual objects.2 A programming language supporting first-class relationships and relationship invariants must comply with the semantics defined by the invariant-preserving model.... In PAGE 11: ...2 A programming language supporting first-class relationships and relationship invariants must comply with the semantics defined by the invariant-preserving model. Such a language must monitor that only the invariant-preserving actions identified in Table 2 and Table3 can be applied. This in turn guarantees the properties (4), (5), and (6).... ..."