SME
Remote Sensing · Problem type
Minimum-distance classifier
Use this for “A pixel reflects 0.10 in red and 0.40 in near infrared. Which class mean is nearest: water, vegetation or soil?”
Chart
Distance
in
feature
space:
d
=
√
Δx² + Δy²
from
the
pixel
(0.1,
0.4)
to
each
class
mean.
Water:
√(
(0.1 − 0.05)
2
+
(0.4 − 0.03)
2
)
=
0.373
Vegetation:
√(
(0.1 − 0.06)
2
+
(0.4 − 0.45)
2
)
=
0.064
Soil:
√(
(0.1 − 0.2)
2
+
(0.4 − 0.28)
2
)
=
0.156
Nearest: vegetation (0.064), so the pixel is classed vegetation.
This is an example. Type your own values to start a new problem.
ρred
Pixel’s red reflectance
example
ρNIR
Pixel’s near-infrared reflectance
example
dW
Distance to water
worked out (not typed)
dV
Distance to vegetation
worked out (not typed)
dS
Distance to soil
worked out (not typed)
Formulas
dW
=
√(
(ρred − 0.05)
2
+
(ρNIR − 0.03)
2
)
0.3734
=
√(
(0.1 − 0.05)
2
+
(0.4 − 0.03)
2
)
dV
=
√(
(ρred − 0.06)
2
+
(ρNIR − 0.45)
2
)
0.06403
=
√(
(0.1 − 0.06)
2
+
(0.4 − 0.45)
2
)
dS
=
√(
(ρred − 0.2)
2
+
(ρNIR − 0.28)
2
)
0.1562
=
√(
(0.1 − 0.2)
2
+
(0.4 − 0.28)
2
)
Assumptions
•
Class means (red, near infrared), from training areas: water (0.05, 0.03), vegetation (0.06, 0.45), soil (0.20, 0.28).
•
Distance is the straight line (Euclidean) in the plane of the two bands, the feature space.
•
The pixel goes to the class whose mean is nearest; the caption names it.
Step-by-step
WE KNOW
ρred = 0.1, ρNIR = 0.4
FIND
distance to water (dW), distance to vegetation (dV), distance to soil (dS)
Step 1 · Find distance to water (dW)
Use dW = √((ρred − 0.05)² + (ρNIR − 0.03)²)
Pythagoras in feature space, from the pixel to the water mean (0.05, 0.03).
dW
=
√(
(0.1 − 0.05)
2
+
(0.4 − 0.03)
2
)
(0.1 − 0.05)
2
=
0.0025
(0.4 − 0.03)
2
=
0.1369
dW = 0.3734
Step 2 · Find distance to vegetation (dV)
Use dV = √((ρred − 0.06)² + (ρNIR − 0.45)²)
Pythagoras in feature space, from the pixel to the vegetation mean (0.06, 0.45).
dV
=
√(
(0.1 − 0.06)
2
+
(0.4 − 0.45)
2
)
(0.1 − 0.06)
2
=
0.0016
(0.4 − 0.45)
2
=
0.0025
dV = 0.06403
Step 3 · Find distance to soil (dS)
Use dS = √((ρred − 0.2)² + (ρNIR − 0.28)²)
Pythagoras in feature space, from the pixel to the soil mean (0.2, 0.28).
dS
=
√(
(0.1 − 0.2)
2
+
(0.4 − 0.28)
2
)
(0.1 − 0.2)
2
=
0.01
(0.4 − 0.28)
2
=
0.0144
dS = 0.1562
Check
Put the numbers back in. Both sides should match.
0.3734
=
√(
(0.1 − 0.05)
2
+
(0.4 − 0.03)
2
)
✓
0.06403
=
√(
(0.1 − 0.06)
2
+
(0.4 − 0.45)
2
)
✓
0.1562
=
√(
(0.1 − 0.2)
2
+
(0.4 − 0.28)
2
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.