A problem arises in making color composite images because there are 210 different possible color presentations of TM three-band images. A method is given for reducing that 210 to a single choice, decided by the statistics of a scene or subscene, and taking into full account any correlations that exist between different bands. Instead of using total variance as the measure for information content of the band triplets, the ellipsoid of maximum volume is selected which discourages selection of bands with high correlation. The band triplet is obtained by computing and ranking in order the determinants of each 3 x 3 principal submatrix of the original matrix M. After selection of the best triplet, the assignment of colors is made by using the actual variances (the diagonal elements of M): green (maximum variance), red (second largest variance), blue (smallest variance)

Sheffield, C. A.

English

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https://ntrs.nasa.gov/search.jsp?R=19830019023 2020-03-21T02:49:52+00:00Z
ORIGINAL PAGE 10
--	 OF FGQj QUAM Y
^ de available under WASA sponsorship
	 E83-103 0 15wl^ dlcin the interest of early and	 i	 a
semination of Earth Resources Survey 	 ^1
program
and wit	 a
	
h	 libility 	7I.FCT l rir;	 COMB i NAT i ON5 T.l fMAPPER WA.
for any use m,^^, , , reol-	 ^,	 ? .	 J
by	 ,	 'I
Charles Sheffield,
	
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ORIGINAL PAGE 1^	
Earth Satel lite Corporation,	 RECEIVED
7222 47th Street, 	 ^^
	
OF POOR QUALITY	 s NASA ST! FACiLITyChevy Chase, Md. 20815. 	 ecG ACCESS DEPT, ^ I
ackground. Since the human eye employs three primary colors, and the
Thematic Mapper returns seven bands of data, one obvious problem that arises
in making color composite images is the choice of bands. The choice in
non-trivial, since three bands can be selected from seven in 35 ways. Also,
any band can be assigned any color. This gives a total of 210 different
possible color presentations of TM three-band images.	 In this note we present
a way of reducing that 210 to a single cholca, decided uniquely by the
statistics of a scene or subscene, and taking full
	
account of any correlations
s	 w
that exist between different bands.
We should remark here that one well-known and widely used approach to
this problem of choice is through the use of principal component images.
However, such methods offer a new problem as great as the one that they solve.
For although the first three principal components contain in a statistical
sense as much information as can be presented using three colors, the
resulting scene is completely data dependent.
	 It Is thus difficult for an
interpreter to apply any previous experience of color-surface relationships to
the analysis of v, principal
	
components	 image.
Definition Sf ±b& method.	 Consider the 7 x 7 variance-covariance matrix M for
the scene or subscene,
	 ignoring for the moment the fact that the thermal band
Is of	 inherently
	
lower resolution than the rest.
	 Any triplet of bands will
	 be
represented within this 7 x 7 matrix by a 3 x 3 submatrix.
(E83-10303)	 SELECTING BAND COMBINAUUNS	 N83 -27294
WITH THEMATIC MAPPER DATA	 (Earth Satellite 1.
Corp.)	 12	 p HC A02 /MF A01	 CS.:L 20P
U nclas
G3/43	 00303
oRjGjf4oR Qu ^^pF p0 Sheffield	 2.
. .I
	 Vida,
iq
Considering now tha 3-dimunsioni4i subspace spanned by any particular band
triplet, the associated variance-covariance matrix defines an b;lipsoid within
the subspace. Further, the sum of the squared principal axes of this
ellipsoid represents the total variance accounted for by these three bands
(see Figure 1). One could plausibly (but as we shall see, wrongly) argue that
the best three bands are those with the largest sum of squared principal axes,
and hence accounting for the largest total variance. This is, after ail,
exactly the argument applied in employing principal component images. Since
the trace of a matrix is invariant under rotational transformations, and since
the sum of squared principal axes is equal to that trace, the band triplet
that accounts for the most possible variance can be found from the original
variance-covariance matrix simply by selecting the three bands with the
largest diagonal elements. There is no need to examine all 35 band
combinations.
To see what is wrong with this approach, consider an extreme case where
there happens to be perfect correlation between a pair of bands. For
convenience, suppose that those bands are 1 and 2, and suppose that the
variance of band 1 (and therefore of 2) is larger than that of any other band.
The 7 x 7 matrix M then has the form:
a	 a
a	 a	 r	 _
b
C
1	 '
where a>b,c,...
i
Sheffield 3.
O'RONAV' 
PAlar. 1$
oV pQ©!R 
Q"^"" `t
The rotation matrix that will diagonalize the upper left 2 x 2 submatrix
then has the form:
I	 ^, I
f T l
_ !J
	
l	 d
T
and thus after rotation the upper left 2 x 2 submatrix will have the form:
2a	 0
0	 0
As expected, one eigenvalue Is zero; but the other is the sum of 'the variances
from the original bands 1 and 2. Since a is assumed to be large, both bands 1
and 2 will be included in the triplet that accounts for maximum variance --
despite the fact that if either one of them is used, adding the other
contributes no new information.
The problem lies in the use of total variance as the measure for the
information content of the band triplets. This is equivalent to use of the
sum of squares of ellipsoid principal axes, and there is no penalty associated
with a very small principal axis provided that it occurs in association with a
large axis (see Figures 2 and 3), as was the case for the above example.
We propose the use of a new measure for the information content of the
triplet, and one that avoids the undesirable property demonstrated above. We
will select the ellipsoid of maximum vglu 1e. This discourages selection of
pairs of bands with high correlation, since in such cases one eigenvalue will
be close to zero and the corresponding ellipsoid volume will be small.
Since the ellipsoid volume is simply 4/3.7fabc, where a, b, and c are the
principal axes of the ellipsoid, the volume of the ellipsoid associated with a
i
a
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i
l
,.
Sheffield 4.
vftNAV  PAGE IS
POOR QUALITY
particular band triplet is a constant multiple of the square root of the
product of the elgenvalues for the 3 x 3 variance-covarlance matrix of that
triplet. However, undor rotational transformation the product of the
eigenvalues is equal to the determinant of the original 3 x 3 Y,ubmatrtx. Thus
we can select the band triplet that provides the ellipsoid of maximum volume
simply by computing and ranking in order the determinants of each 3 x 3
principal submatrix of the original matrix M. The band triplets associated
with these determinants will then be ranked in order of decreasing overall
information content. Given the original matrix M, the total computation to
achieve this ranking is trivial. It requires a few hundred multiplications,
followed by a sort of a list of 35 items. A BASIC program to perform this is
given as an Appendix to this note.
This procedure gives the best triplet, but the assigment of colors is
still to be made. Now we can make use of the actual variances (the diagonal
elements of M). Since the eye is most sensitive to green, next to red, and
least to blue, we will assign green to the band triplet member of maximum
variance (i.e. most variation within the image), red to the triplet member of
second largest variance, and blue to the triplet member of smallest variance.
The definition of bands for production of a color image is now complete.
Examples &nd comments. The procedure has been applied to a number of scenes
of very different ground cover, including Washington D.C., Death Valley, and
Cement, Oklahoma. The results for Washington and for Death Valley are given in
Tables 1 and 2, together with the associated variance-covariance matrices. The
following comments apply to all scenes ytudled to date.
1) The band combination 1,4,5 (in the order blue, red, green) is usually,
but not always, the selected triplet. In cases where it does not rank first,
Slief',Itield	 S.
it ranks second or third.
2) The natural color combination 1,2,3 and the standard false color
combination 2,3,4, both place far down In the rankings. In the case of
Washington, the natural color combination is 29th (lower than anything except
some thermal band combinations, which are low for another reason to be
discussed shortly); the 2,3,4 combination was ranked in 16th place. For Death
Valley, the 1,2,3 natural color combination ranked 32nd, and the 203,4
combination ,Just above It, at 31st. This is presumably a consequence of the
very high correlations between the first four bands.
3) Triplets that rank high always Include eli•her band a or band 6 (note:
the bands here are ordered by incrensing wavelength, so the thermal band is
band 7). This emphasizes the great Importance of these new bands on general
information-bearing grounds.
4) The triplet selected Is not always or even usually the triplet wi't'h
the greatest individual variances, though large variances are naturally
preferred somewhat in the selection process.
Other sons i darat i on.9 e ns( aom	 .
1) The statistical analysis performed here used P tapes (all that we had
available) in which the original histograms had already been modified by the
gains and offsets. It would be preferable to work with data that have had no
gains or offsets applied, I.e., with A tapes prior to any radiometric
correction. If band selection of this type becomes common, it would be nice to
have A tapes generally available from the EROS Data Center.
2) The thermal band is of lower resolution than the rest, thus it would
not be appropriate to give it the same weight in the selection process. Now
should one therefore de-weight It? One argument runs as follows: The maximum
Information that a scene can contain Is given by the number of pixels, since
VShaftield 6.
ORIGINAL PAGE IS
OF Pool; QUALITY
in the ultimate case there would be no correlation between pixels, and each
i
would carry independent information about some feature of the surface. in such
a case, the amount of Information that the thermal band can contribute is only
t
1/16th that of the other bands, because there are 16 times fewer pixels in
that band. Therefore one should deweight the thermal channel by a factor of
16. Such deweighting was performed in the experiments reported here. However,
we should also note that this made no difference at all to the preferred band
triplets, since even without deweighting we found no case where a triplet
involving the thermal channel was in the top five.
3) It is obvious when one looks at images created from the triplet 1,4,5
that for some applications this combination will be much inferior to others,
n
such as natural color and standard false color. This restates the old truth,
one man's noise is another man's signal. However, the preferred triplets have
another advantage: they provide images of unusual clarity, with far less
residual striping than is seen in, for example, the natural color images.
4) Although combinations such as 1,4,5 produce images that are at first
sight unfamiliar and unusual, the assigned colors are not scene-dependent.
Thus In contrast to the scene dependent colors of principal component or ratio
Images, the interpreter quickly learns to associate colors wi^;h particular
ground condition. We therefore believe that there are definite advantages to
seeking color composites from the original bands, rather than through band
ratios or band combirations. 	 `t
P4_	 ^
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ORIGINAL PAGE IS
OF POOR QUALITY
:r -r M
-1 x = 1
(ellipsoid surface of
constant probab:.lity)
Figure 2. High correlation.
bands 1 and 2.
A
if
	
Sheffield 7.
i;
Figure 1: The variance-covariance ellipsoid, principal axes 4-Al
I	 /
{
1
!1
Figure 3. Low correlation, bands_
1 and 2, but lower
individual variances.
2
large, +'aZ
 smaller than in Fig.2.
small '-A . X` 	 than in Fig. 2._larger
E.
Based on ellipsoid volumes, Fig. 2 case is preferred over Fig. 1 case, s
although the former accounts for a greater total variance.
t
4
V"
APPENDIX: THE BEST-BAND PROGRAM. 	 ORIGINAL PAGE IS
	 Sheffield 8.
OF POOR QUALITY
20 PRINT "SELECTION OF BEST THREE BANDS BASED ON ELLIPSOID VOLUME"
30 PRINT "DEATH VALLEY WITH REDUCED THERMAL VARIANCE"
40 DIM R(36),Q(36)
50 DIM U(36)2V(36)
60 DIM M(818)
70 REMARK: M is the variance-covariance matrix for the scene or subscene.
80 REMARK: The arrays R,Q,U and V are storage arrays used in the program.
90 REMARK: Note that the program assumes that band 7 is the thermal data,
and band 6 is the 2.2 micrometre data.
100 REMARK: The instructions 190 to 230 (except for 220, which sets a count)
reduce the variance of the thermal channel to allow for the lower spatial
resolution of the thermal channel pixel.
190 FOR I = 1 TO 6
200 M(I,7) = M(I,7) / 4
210 NEXT
X20 C = 1
230 M(7,7) = M(7,7) / 16
240 PRINT "RANK	 DETERMINANT	 COMBINATION"
250 FOR I = 1 TO 5
260 FOR J = I + 1 TO 6
270 FOR K=J+1 TO 
280 D1 = M(I,I) * (M(J,J) * M(K,K) - M(J,K)	 2)
290 D2 = M(I,J) * (M(J,K) * M(I,K) - M(I,J) * M(K,K))
300 D3 = M(I,K) * (M(I,J) * M(J,K) - M(I,K) * M(J,J))
310 DT = D1 + D2 + D3
315 REMARK: The next instruction makes the determinant an integer; this is
not necessary, it is done for convenience of output only.
320 DT = INT (DT)
330 N = 100 * I + 10 * J + K
340 R(C) = DT:Q(C) = N
350C =C+ 1
360 NEXT
370 NEXT
380 NEXT
385 REMARK: The next piece of code sorts the determinant into descending order.
390 FOR I = 1 TO 35
400N =0
410 FOR J = 1 TO 35
420 IF R(I) I	 R(J) THEN 440
430N =N +1
440 NEXT
450 U(N)	 = R(I):V(N)
	 = Q(I)
460 NEXT
470 FOR I = 1 TO 35
480 PRINT I,U(I),V(I)
490 NEXT
500 PR# 0
510 END
d1
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ORIGINAL PAGIZ IS
'OF' POOR QUALI'ry
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Sheffield 10.
SELECTION OF BEST THREE BANDS BASED ON ELLIPSOID VOLUME
THIS IS THE WASHINGTON SCENE WITH REDUCED VARIANCE ON THE THERMAL CHANNEL
RANK DETERMINANT COMBINATION
1 433858 145
2 205811 345
3 138551 146
4 124784 245
5 101638 456
6 71723 156
7 62960 346	 ORIGINAI. pAGE.: Eg8
9
49759
39992
135	 OF POOR QUALITY134
10 39609 246
11 36060 356
12 22847 125
13 21953 256
14 16732 124
15 11646 235
16 9709 234
17 7967 136
18 5094 457
19 4752 157
20 3634 126
21 3606 147
22 2294 467
23 2194 357
24 1945 347
25 1616 236
26 1386 567
27 1348 257
28 1130 247
29 727 123
30 688 167
31 276 367
32 215 137
33 175 267
34 84 127
35 43 237
TABLE l.b Ranked results for Washington D.C. scene.
M U1 t+ M N CO
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ORIGINAL PAGE 65.
OF, POOR QUALITY
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Table 2.b Ranked results for Death Valley scene.
° Sheffield
	
12.
SELECTION OF BEST THREE BANDS BASED ON ELLIPSOID VOLUME
DEATH VALLEY WITH REDUCED THERMAL VARIANCE
ar
RANK DETERMINANT COMBINE} ^rION
1 1462581 145
2 859695 156 it
3 684248 135
4 601687 146 onIelm' PAGE? 13
5 432952 345 OF POOR QUALITY
6 346425 157
7 328331 356
8 319827 245
9 275534 456
10 263989 136
11 219239 256
12 204146 125
13 167450 346
14 137060 357
15 127643 246
16 121117 167
17 107494 457
18 103781 235
19 89506 257
20 76827 126
21 75913 134
22 49163 367
23 40621 467
24 39230 236
25 37614 147
26 31621 267
27 21579 137
28 21322 124
29 20256 567
30 9168 347
31 8118 234
32 7895 123
33 7197 247
34 5037 127
35 2407 237


Selecting band combinations with thematic mapper data

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