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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
each pixel as a member of class. The probabilities are then con-
verted to a posteriori probabilities of class membership, which
are assessed as the probability density of a case for a class rela-
tive to the sum of the densities. For each pixel, the a posteriori
probabilities sum to 1.0. At each resolution, the highest a poste-
riori probability and its related class are output for each pixel.
With this approach, the feature layers obtained from coarse
resolutions do not have to map back to the finest resolution
specified and the computation cost is lower than that in the first
strategy.
The third approach is a top-town filtering approach. This multi-
resolution procedure starts with the coarsest resolution image.
The finer resolution images are used only when necessary. The
posterior probabilities of the training data are used to determine
the threshold at each resolution level. For pixels with maximum
posterior probabilities greater than the threshold are assigned to
their related classes and are excluded (masked) for subsequent
processing. All pixels with maximum posterior probabilities
less than the threshold are regarded as mixed pixels, or pixels
that do not have identical signatures and cannot be identified at
this resolution level. Their posterior probabilities are used as
prior probabilities at next resolution level.
More efficient classification algorithms can be used. This in-
cludes the Fuzzy-set and neural net classifiers (Foody, 1999;
2000), approximating the model by Markov, conditional inde-
pendence or fractal structure obtained from multi-scale analysis
(Solberg et al. 1996; Li et al. 2000).
2.4 Error analysis
At the same spatial resolutions, classification errors can be
influenced by many factors, such as scheme selection, spatial
autocorrelation among and within classes and classification
techniques, to name a few. Traditionally classification error
analysis is based on the use of an error matrix or contingency
table to derive descriptive statistics such as overall accuracy,
kappa coefficients, etc. The error models for area classification
maps across scales are largely unsolved and little work has been
done, although it is becoming more important with the
increasing availability of multi-scale data (Tate and Atkinson
2001). The emphasis in this framework is to explore how
spatial autocorrelation of classification errors among and within
classes changes with spatial resolutions. The variance loss from
fine resolution to coarse resolution will be calculated. The
relationship between these errors and spatial autocorrelation
parameters obtained from the multi-scale analysis will be
analyzed for each class.
3. CASE STUDY
The multi-resolution framework was tested using simulated
multi-resolution images derived from IKONOS data for a por-
tion of the western part of Kingston City, Ontario, Canada. The
IKONOS image has a spatial resolution of 4 m with four spec-
tral bands. Ten land cover classes were used, including resi-
dential roof, industrial/commercial roof, Road, lawn, conifer
tree, deciduous tree, bare/cleared land, water, new crop, and
wetland.
4 m IKONOS image was aggregated progressively into four
nominal resolution levels (8 m, 12 m, 16 m, and 20 m) by an
averaging method. Both single-resolution and multi-resolution
classification were conducted. The single-resolution classifica-
1189
tion was used as a benchmark for evaluating various multi-
resolution approaches.
Initial exploratory data analysis was carried out as three trials.
The first trial used histograms to determine the types of distri-
bution exhibited by each band. The second trial included the
mean and standard deviation to assess distribution properties.
The final trial established if each band offers or similar or dif-
ferent information, i.e. are they correlated?
A set of sampling data was selected for each class. The semi-
variogram was used in the case study to decide the spatial auto-
correlation level in each class. Each value of a pixel was com-
pared to its neighbors at varying distances (lags) and the differ-
ences were calculated. The semivariance was then calculated by
averaging the summation of squared differences for pixels. In
most cases the semivariance tends to increase with spatial inter-
vals. After reaching a maximum value, the semi-variogram flat-
tens (called the sill). The lag at which the sill is reached is
called the range. The range indicates the extent to which val-
ues sampled from a spatial process are similar (spatially re-
lated). Detailed theoretical and mathematical exploration of
variograms can be found in Cressie (1991), and Woodcock et
al. (1988a; 1988b).
The non-directional or isotropic semi-variogram was calculated
and plotted for to assess the degree of spatial autocorrelation in
respective bands in the case study. The ranges were determined
by visual examination and through a comparison of piecewise
slopes. Table 1 lists the ranges for each class.
Classes Range (m)
Residential roof 16
Commercial/industrial roof 28
Road 8
Lawn 14
Conifer trees 6
Deciduous trees 12
Cleared/bare land 30
Water 12
New crop 24
Wetland 8
Tablel. The ranges obtained from semi-variogram
The shape and range of each semi-variogram were useful for
determining suitable sizes for training data, sampling interval,
resolution or window sizes used for spatial feature extraction.
Based on the discussion in previous section, when image reso-
lution is close to or coarser than the range of a class, an L-
resolution scene model is generally most appropriate for that
class. Otherwise, spatial features that incorporate tex-
ture/contexture information should be generated at H-
resolution.
Training data were selected by visually identifying and manu-
ally digitizing blocks of pixels. As a general rule, the length and
width of small blocks for each class were close to the range ob-
tained from the semi-variogram, so that each block was big
enough to represent the spectral and spatial properties of each
class. Thus, the heterogeneity or autocorrelation within each
class was included in the training data. The distance between
any two blocks was greater than or equal to the range of the
semi-variogram, so the pixels in one block were correlated, but
not spatially autocorrelated with those in another block.