24 3D Object Classification, Visual Search …
369
R
u
i , I
=
r
1
i , r
2
i , . . .
.
Vector d i is constructed in the form:
d i =
r
1
i , r
2
i , r
1
i , r
2
i
.
3. Descriptor matrix D formed from vectors d i :
D =
d i 1
d i 2
. . . |d i N
, where∀i j , i j+1 , d
1
i j
≤ d
1
i j+1
.
Let us make some comments about the algorithm described above:
• Because of the sorting procedure in step 3 of Algorithm 1, it becomes impossible to
reconstruct the model from its descriptive representation, because there are exactly
N ! ways to order the element set, which means that one descriptive representation
of the form of Eq. 24.1 corresponds to several possible polygonal models.
• By entering into consideration, the distance not only to the nearest intersection
point with the model, as is done in [19], but also to the next intersection points
with the models surface, it is possible to achieve that polygonal models that have
the same external surface shape, but different internal structure will have different
descriptors.
• To find out the lengths of vectors r
i , it is possible to use a computationally efficient
ray procedure with a plane from the work [20].
Next, we will construct an algorithm that allows us to calculate the similarity
measure between two descriptive representations described above. This measure
similarity is a function of the following type:
μ(D 1 , D 2 ): R
4×N
× R
4×N
→ [0, 1].
In order to design a search algorithm based on the similarity of descriptive representations in the future, it will be sufficient to use the metric ρ in the descriptor space
defined earlier in Sect. 24.3.3, let us define the distance function of the following
type:
ρ
M p,s , M k,l
= 1 − μ
D N
M p,s
, D N
M k,l
,
where M p,s , M k,l are two polygonal models and D N
M p,s
, D N
M k,l
are their
descriptive representations in the form of Eq. 24.1 with dimensions shape 4 × N . For
convenience of notation, two different descriptive representations will be written as:
D 1 =
d
11
d
12
. . . |d
1N
;
D 2 =
d
21
d
22
. . . |d
2N
.
369
R
u
i , I
=
r
1
i , r
2
i , . . .
.
Vector d i is constructed in the form:
d i =
r
1
i , r
2
i , r
1
i , r
2
i
.
3. Descriptor matrix D formed from vectors d i :
D =
d i 1
d i 2
. . . |d i N
, where∀i j , i j+1 , d
1
i j
≤ d
1
i j+1
.
Let us make some comments about the algorithm described above:
• Because of the sorting procedure in step 3 of Algorithm 1, it becomes impossible to
reconstruct the model from its descriptive representation, because there are exactly
N ! ways to order the element set, which means that one descriptive representation
of the form of Eq. 24.1 corresponds to several possible polygonal models.
• By entering into consideration, the distance not only to the nearest intersection
point with the model, as is done in [19], but also to the next intersection points
with the models surface, it is possible to achieve that polygonal models that have
the same external surface shape, but different internal structure will have different
descriptors.
• To find out the lengths of vectors r
i , it is possible to use a computationally efficient
ray procedure with a plane from the work [20].
Next, we will construct an algorithm that allows us to calculate the similarity
measure between two descriptive representations described above. This measure
similarity is a function of the following type:
μ(D 1 , D 2 ): R
4×N
× R
4×N
→ [0, 1].
In order to design a search algorithm based on the similarity of descriptive representations in the future, it will be sufficient to use the metric ρ in the descriptor space
defined earlier in Sect. 24.3.3, let us define the distance function of the following
type:
ρ
M p,s , M k,l
= 1 − μ
D N
M p,s
, D N
M k,l
,
where M p,s , M k,l are two polygonal models and D N
M p,s
, D N
M k,l
are their
descriptive representations in the form of Eq. 24.1 with dimensions shape 4 × N . For
convenience of notation, two different descriptive representations will be written as:
D 1 =
d
11
d
12
. . . |d
1N
;
D 2 =
d
21
d
22
. . . |d
2N
.