e XXXIX-B3, 2012
1 the number of nodes or
se data storage and
0 represent the contours
nt however, to maintain
ortant features such ag
st real world cases, an
1 little direction changes
à lot of direction change
2(a).
pproach, referred to as
'entation (Baxes, 1994)
ntour segments changes
hroughout the contour
to remove nodes from
segments and replaces
portions of the contour
lable number of contour
ient, depending on how
t are eliminated during
ed in the reconstruction
contour will not unduly
1 thereby nodes) where
le. The implementation
ut by first performing a
g line segment lengths
ain the contour details
image.
ning line segments are
are similar or identical,
to create a single line
ngth and overall angle
S a contour segment
/ more concise, while
gions of little change
| variable line segment
segments with similar
| removing redundant
s to compare the area
nd longest distance to
[hese parameters may
" deciding whether a
o another and thereby
lose nested contours it
des by removing the
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
middle contour and interpolate and/or diffuse between the other
two. However, this process may have the disadvantage of
smoothing and/or distorting particular high frequency (sudden
changes of brightness values within an image) information
necessary for a more accurate reconstruction of the original
image.
3. ENCODING 3D DATA POINTS
For easy extraction, processing and transmission over a digital
link, the contours data was encoded and saved in an ASCII file
(ie. txt). This file contains a two-row matrix where each
contour line defined in the matrix begins with a column that
defines the value of the contour line and the number of (x,y)
vertices in the contour line.
The remaining columns contain the data for the (x,y) pairs. The
x, y values are stored to the first decimal place and separated by
a point. In the coding shown below 120 is the grey-scale
intensity of a selected contour and the contour line is formed by
7 vertices. The x coordinate of the vertices are in the top row
(ie. 18.2 17.2...14.7) whereas the corresponding y coordinates
are placed in the bottom row (i.e. 110 109...93.2)
120 182 17.2 17 17.3 17.115.7 14.7
7 110109 108 99.5 97 95 032
Additional tests are presently being undertaken to ascertain the
use of binary files (as compared to ASCII .txt file) as a way to
further reduce memory requirements and speed of transmission
of contour data.
4. FROM 3D POINTS TO PIXELS
In this gridding process contours nodes are projected or mapped
on a uniformly spaced grid. Depending on the final resolution
required, this grid may be selected so as to create pixels in x and
y corresponding to the original input image. To determine the
pixel brightness which would exist at the intersections of a
regular grid using randomly spaced 3D node locations, several
interpolators may be used depending on the application and
accuracy requirements.
There exist several interpolation schemes available for this task.
Estimations of nearly all spatial interpolation methods can be
represented as weighted averages of sampled data (De Jong and
van der Meer, 2004). They all share the same general estimation
formula as shown in equation 1:
ñ
a ^ 4 , 1
£3) 8 A(x) an
Where Z is the estimated value of an attribute at the point of
interest xo, z is the observed value at the sampled point x;, 4; is
the weight assigned to the sampled point, and n represents the
number of sampled points used for the estimation (Webster and
Oliver, 2001).
In this work the interpolation process is an estimation process
which determines the pixel brightness which would exist on the
intersections of a regular grid using the randomly spaced nodes
of contours. Several local interpolators may be used depending
on the application and accuracy requirements. The method used
in this work is referred to as cubic splines. A brief explanation
follows.
The splines consist of polynomials with each polynomial of
degree n being local rather than global. The polynomials
describe pieces of a line or surface (i.e. they are fitted to a small
number of data points exactly) and are fitted together so that
they join smoothly (Burrough and McDonnell, 1998; Webster
and Oliver, 2001). The places where the pieces join are called
knots. The choice of knots is arbitrary and may have an
important impact on the estimation (Burrough and McDonnell,
1998). For degree n — 1, 2, or 3, a spline is called linear,
quadratic or cubic respectively.
In the ensuing tests, cubic splines were selected as they are very
useful for modeling arbitrary functions (Venables and Ripley,
2002) and are used extensively in computer graphics for free
form curves and surfaces representation (Akima, 1996).
5. TESTS AND RESULTS
Table 1 shows the results of a first test aimed at determining the
degree of accuracy and memory storage requirements to be
expected when reproducing a digital image from scattered
contour nodes. The figures are based on the Peter.tif (400?) in
Figure l(a) The R.M.S. (Spiegel and Stephens, 1999) is
random standard error of the differences of grey-scale values
between the original Peter.tif and the same image reconstructed
using the nodes for contour increments ranging between 2 and
8.
The figures in Table 1 show a linear degradation of the R.M.S.
as the contour interval is increased. The table also illustrates the
amount of memory required to store the contour nodes needed
for the reconstruction of Peter. As expected, as the contour
interval increases, the memory requirement decreases at the
expense of image quality.
Contour Accuracy Memory Maximum | Minimum
increments | R.M.S. required difference | difference
2 2.7 0.051 Mb 5 7
4 6.3 0.042 Mb 17 27
6 9.8 0.033 Mb 33 30
8 19.4 0.021 Mb 39 42
Table 1. Accuracy and memory requirements needed to
reconstruct the original image of the Peter using contour
increments between 2 and 8 grey-scale intensity values.
By way of comparison, the .txt file needed to store the x and y
coordinates of the nodes necessary to reproduce Peter (for
contour increment of 6 grey-scale values) used approximately
the same amount of memory of a JPEG compressed (0.029 Mb)
version of the same image for a compression ratio of 7:1, while
reproducing a more accurate image with improved visual details
(see Figure 3).
In Figure 3, the difference in visual quality between Peter
reconstructed with contour data (Figure 3(c)) and the
corresponding JPEG image (Figure 3(b)) is evident. Indeed, the
blocking effects, typical of a lossy JPEG protocol, were almost
completely eliminated.
In addition, the maximum and minimum differences of pixel
intensity values resulting from subtracting the JPEG version of
Peter from the original Peter.tif were respectively -36 and +44
grey-scale values with an R.M.S. equal to +/-35. By contrast,
the contour approach produced maximum and minimum
