Based on Atkinson’s study (Atkinson 1997), Verhoeye and Wulf (2002)
presented a mathematical model to transform the SPM problem into one of assigning
classes to the subpixels using linear optimization techniques. Suppose that the coarse
resolution pixels are to be split into S
2 subpixels. The number of subpixels that have
to be assigned to class c has been derived from soft class proportion images. A
measure for spatial dependence SDV c,j will be computed for class c at each subpixel
j. Each subpixel has to be assigned a value of one or zero for each class, one
indicating an assignment to a particular class. Following this, the problem of
assigning each subpixel to a specific class emerges, which has the maximum value
of the spatial dependence.
17.4.1.2 Subpixel/Pixel Spatial Attraction Model (SPSAM)
Among all SPM methods, SPSAM has several advantages both in terms of its
simplicity and its explicit physical meanings (Mertens et al. 2006). For example,
spatial attraction, which calculates the spatial correlation between subpixels and their
surrounding pixels, is used in SPSAM as a simple tool to directly convey spatial
dependence. Without requiring prior knowledge on the spatial structure, which is
essential in some learning-based SPM methods, such as a two-point histogram,
indicator cokriging-based methods, and genetic algorithms, etc., SPSAM can obtain
satisfactory SPM results.
SPSAM was proposed based on the theory of spatial dependence. The assumptions of SPSAM are as follows: (1) The fraction values of neighboring pixels exert
the attraction toward subpixels within a central pixel, (2) a subpixel within the
central pixel can only be attracted by pixels surrounding the central one, and
(3) other pixels are assumed to be too distant to exert any attraction. Assuming
that closer pixels attract the subpixel more than the distant ones, SPSAM calculates
the attraction value of a neighboring pixel to a subpixel, based on their Euclidean
distance.
Figure 17.13 illustrates the labeled pixels and subpixels, the coordinate system,
and the distance between pixels and subpixels in SPSAM. A scale factor of S ¼ 4
means that the original central pixel P 11 contains 16 subpixels with labels: p 44 , p 45 ,
p 46 , p 47 , . . ., p 74 , p 75 , p 76 , p 77 . The distance between subpixel p ij within the central
pixel and the neighboring pixel P MN can be defined as:
d P MN , p ij
À
Á ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
i þ 0:5 À S M þ 0:5
ð
Þ
½
2 þ j þ 0:5 À S N þ 0:5
ð
Þ
½
2
q
i, j ¼ 0, 1, . . . , 8; M, N ¼ 0, 1, 2
ð
Þ
ð17:5Þ
SDV c, ij , the attraction values of subpixel p ij , which is assigned class c, by the
neighboring pixels, can be defined using Equation (17.4–2):
376
L. Lu
presented a mathematical model to transform the SPM problem into one of assigning
classes to the subpixels using linear optimization techniques. Suppose that the coarse
resolution pixels are to be split into S
2 subpixels. The number of subpixels that have
to be assigned to class c has been derived from soft class proportion images. A
measure for spatial dependence SDV c,j will be computed for class c at each subpixel
j. Each subpixel has to be assigned a value of one or zero for each class, one
indicating an assignment to a particular class. Following this, the problem of
assigning each subpixel to a specific class emerges, which has the maximum value
of the spatial dependence.
17.4.1.2 Subpixel/Pixel Spatial Attraction Model (SPSAM)
Among all SPM methods, SPSAM has several advantages both in terms of its
simplicity and its explicit physical meanings (Mertens et al. 2006). For example,
spatial attraction, which calculates the spatial correlation between subpixels and their
surrounding pixels, is used in SPSAM as a simple tool to directly convey spatial
dependence. Without requiring prior knowledge on the spatial structure, which is
essential in some learning-based SPM methods, such as a two-point histogram,
indicator cokriging-based methods, and genetic algorithms, etc., SPSAM can obtain
satisfactory SPM results.
SPSAM was proposed based on the theory of spatial dependence. The assumptions of SPSAM are as follows: (1) The fraction values of neighboring pixels exert
the attraction toward subpixels within a central pixel, (2) a subpixel within the
central pixel can only be attracted by pixels surrounding the central one, and
(3) other pixels are assumed to be too distant to exert any attraction. Assuming
that closer pixels attract the subpixel more than the distant ones, SPSAM calculates
the attraction value of a neighboring pixel to a subpixel, based on their Euclidean
distance.
Figure 17.13 illustrates the labeled pixels and subpixels, the coordinate system,
and the distance between pixels and subpixels in SPSAM. A scale factor of S ¼ 4
means that the original central pixel P 11 contains 16 subpixels with labels: p 44 , p 45 ,
p 46 , p 47 , . . ., p 74 , p 75 , p 76 , p 77 . The distance between subpixel p ij within the central
pixel and the neighboring pixel P MN can be defined as:
d P MN , p ij
À
Á ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
i þ 0:5 À S M þ 0:5
ð
Þ
½
2 þ j þ 0:5 À S N þ 0:5
ð
Þ
½
2
q
i, j ¼ 0, 1, . . . , 8; M, N ¼ 0, 1, 2
ð
Þ
ð17:5Þ
SDV c, ij , the attraction values of subpixel p ij , which is assigned class c, by the
neighboring pixels, can be defined using Equation (17.4–2):
376
L. Lu