International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
  
n 1/2 
1.(1 a: ) + (R Qs ar 4) 
* ne € "X a ( 
I=! 
NM 
where j is the band number. A generalized formula for the 
distance to the corners of a hyper-dimensional cube can be 
written as (i is the corner or component number): 
  
  
n /2 
D = ) 2 *(1-a)+(R 2 * 3. 5 
i x ( = aj ) NT. x a ( ) 
jl 
(001) e. 
NN. 
S. 
pum em E d 
(111) 
   
Band 3 
  
  
  
  
  
  
& 
(110) 
  
"3 255 
(010) 
Figure 6. nPDF Cube Model 
There are eight possible corners of a three-dimensional cube 
as is shown in Figure 6. Four of the corners can be selected 
as principal corners (1 through 4), the remaining corners (5 
through 8) are the complimentary to the four principal 
corners. For the hyper-dimensional cube model, "a" values 
for the equation (5) are as follows (j is the band number): 
D, : For all j values a0 
(jT 1245... a-0 
D» ifi j<36... a=1 
Ar jT. a0 
Dr fee as 
j= 1,4... a-0 
Dyed 
j72356... a=d 
The nPDF formula is: 
1310 
nPDF; - S * D; / QBIT * NB!2) (6) 
where: 
nPDF, = Component i of nPDF, 
1 Corner number, 
S = Desired scale for the nPDF axes, 
D. = Calculated distance for component i, 
i 
BIT = Number of bits of input data, 
NB = Number of bands used. 
Frequency plots of two nPDF components (hyper- 
dimensional distances) provide an excellent perspective of 
multidimensional data distribution. Depending on the 
spectral distribution of the classes of interest, the user can 
select corners which provide the maximum separation of the 
classes. A convenient scale for these nPDF components is 8 
bit in range, and thus a two-dimensional frequency plot 
requires a 256 by 256 array. 
The cube model has the advantage of being a conceptually 
simple way of describing corners in multidimensional space. 
However, it does tend to limit the choice of corners for four 
and higher dimensional data. Where this is a problem, the "a" 
values (see equation 1) are used to describe the corner 
location. Thus in Figure 6, corner Z2 is also labeled (001), 
which can be interpreted as a corner that has "a" value of zero 
for the first two bands, and that of one in the third band. 
Using this convention, the length of the list of "a" values 
depends on the number of input bands, and thus the corner 
corresponding to the origin in a four band image would be 
described as (0000). 
Prior to the classification process the spectral values for the 
entire scene are transformed into nPDF space. The software 
allows the user to view the distribution of the data and 
enhance the data by interactively stretching and rotating. 
This allows a rough visual identification of separable classes. 
For the supervised classification procedure the training field 
data are then plotted into nPDF space. Polygons can then be 
drawn around the obvious classes to delineate the spectral 
boundaries. The classification procedure uses the boundaries 
of these polygons to assign pixels to the appropriate class. 
5. RESULTS AND DISCUSSION 
Classification of the RDACS-3 dataset provided the highest 
overall accuracy (7696 overall accuracy for the overstory 
species classification and 94% overall accuracy for the other 
land cover classes such as agriculture fields). The AVIRIS 
datasets provided an overall accuracy of 69% for the 
overstory species and 83% for the other land cover classes. 
The Hyperion dataset provided 62% overall accuracy for the 
overstory species and 81% for the other classes (Figures 7 
and 8; tones of red were used for the forest species).