SIMULATION OF A PANCHROMATIC SAND BY SPECTRAL LINEAR COMBINATION OF MULTISPECTRAL BANDS

Nelson D.A.Mascareiihas; Gerald Jean F.Baoon; L e i l a IMaria 6. Fonseca

I n s t i t u t o N(aciona1 de Pesquisas Espaciais - INPE - SCT Caixa Postal !515 - 12201 - Sao Jose dos Campos, SP, B r a s i l

The s i m u l a t i o n o f one o r more bands o f a mu1 t i s p e c t r a l sensor by c o m b i n a t i o n o f o t h e r bands r e p r e s e n t s an a t t r a c t i v e p o s s i b i l i t y . I n t h i s work an a n a l y s i s o f t h e s i m u l a t i o n o f t h e s p a t i a l l y degraded SPOT panchromat ic band by l i n e a r c o m b i n a t i o n o f t h e m u l t i s p e c t r a l bands has been p e r f o r v e d u z i v g l e a s t squares methods. The r e s u l t s ivc1ildC: 1 ) i? t h e o r e t i c a l s t u d y o f t h e s i g n a l t o n o i s e r a t i o o f t h e s i m u l a t e d band as compal-ed w i t h t h e m u l t i s p e c t r a l bands, and 2) e x p e r i m e n t a l work showing a n u m e r i c a l l y and v i s u a l l y reasonab le e s t i m a t e o f t h e panchromat ic band.

Image S i m u l a t i o n ; L e a s t Squares Method; SPOT S a t e l l i t e .

1. INTRODUCTION

The s i m u l a t i o n o f s a t e l l i t e imagery i s a u s e f u l t o o l as an image p r o c e s s i n g t e c h n i q u e . One c o u l d ment ion a t l e a s t two a p p l i c a t i o n s i n t h e remote sens ing area. a) The l a r g e s i z e s o f remote sens ing images impose a severe burden on t h e communicat ions l i n k between t h e s a t e l l i t e and t h e ground. There fore , t h e s i m u l a t i o n o f an image by ground p r o c e s s i n g c o u l d o f f e r t h e p o t e n t i a l o f a decrease on t h e d a t a r a t e . b ) I n sensor f e a s i b i l i t y s t u d i e s i t i s common p r a c t i c e t o deve lop s i m u l a t i o n procedures b e f o r e t h e a c t u a l sensor i s b u i l t , so t h a t many t e c h n i c a l aspects can be p r e d i c t e d i n advance.

The main i d e a o f t h i s paper i s t o p r e s e n t a s i m u l a t i o n t e c h n i q u e f o r a sensor band based on l i n e a r combina t ion o f o t h e r bands. T h i s i s p o s s i b l e i f t h e r e i s a s u b s t a n t i a l s p e c t r a l o v e r l a p b e h e e n t h e s i m u l a t e d band and t h e bands t h a t a r e l i n e a r l y combined. As an example o f t h i s procedure, we p r e s e n t a s tudy o f t h e s inu la , ion o f a degraded, 20 12 r e s o l u t i o n SPOT

panchromat ic band (P) , by l i n e a r combina t ion o f t h e two m u l t i s p e c t r a l bands (XS1 and XS2) t h a t s p e c t r a l l y o v e r l a p t h e panchromat ic band. The use o f t h e t h i r d m u l t i s p e c t r a l band ( X S 3 ) was d i s c a r d e d because o f t h e n e g l i g i b l e s p e c t r a l o v e r l a p between P and XS3.

2. THEORETICAL BACKGROUND

L e t f ( X ) be t h e s p e c t r a l c h a r a c t e r i s t i c o f t h e scene, gi(X), i=1 ,2 be t h e s p e c t r a l c h a r a c t e r i s t i c s o f t h e two m u l t i s p e c t r a l bands and gp(X) t h e s p e c t r a l c h a r a c t e r i s t i c o f t h e panchromat ic band.

The observed v a l u e a t each p i x e l o f t h e m u l t i s p e c t r a l b'ands can be a p p r o x i m a t e l y g i v e n by :

bi = 1 f ( A ) gi ( A ) ~ A i = 1,2 (1 1

L i k e w i s e , f o r t h e panchromat ic band

We want t o e s t i m a t e p by 6, where 6 would be g i v e n by a f i c t i t i o u s band t h a t i s a l i n e a r c o m b i n a t i o n o f t h e two m u l t i s p e c t r a l bands, i e :

( 3 )

I f t h e c o e f f i c i e n t s and<, can be determined, t h e e s t i m a t e o f p a t each p i x e l would be g i v e n by p = 5, b l + E2 b2 ( 6 )

By comparing eqs. ( 2 ) and ( 5 ) i t i s c l e a r t h a t we have t o approximate xhe s p e c t r a l c u r v e g (A) by t h e l i n e a r c o m b i n a t i o n a,gl(X) + E2g2(!). A r e s o n a b l e c r i t e r i o n t o adopt i s based on t h e l e a s t squares procedure, t h a t i s , we want t o m i n i m i ze

( 7 ) f J [ g p ( h ) - ( [ x l g , ( X ) + a , g 2 ( X ) ) I 2 dX

by t h e a p p r o p r i a t e c h o i c e o f = land @ z . I n o r d e r t o do t h i s we i d i s c r e t i z e t h e c u r v e s g,(x), gl(A)

32 1 CH2971-0/91/0000-0321$01.00 0 1991 IEEE

and g 2 ( X ) over n points and a l i n e a r regression model i s obtained g ( A , ) = a lg l (x , ) + a,g2(Al) + r l

g (A,) = a l g l ( A , ) + a 2 g 2 ( x Z ) -t r 2 P

p .

The sca la rs r . , i = 1 , 2 ... n take i n t o account the e r ror in the approximation. By defining

I-rn J n x 1

We obtain the l inear model gp= 5 a t 1 (10)

I t i s well known t h a t under the hyphotesis t h a t the rank of the matrix G i s given by the number of columns ( 2 in t h i s c a s e ) , t h e l e a s t squares solut ion of t h i s overdetermined system i s given

(11 1 T -1

- ? = ($5) 5 4p = G+ Yp

where G+ i s the Moore-Penrose pseudoinverse o f 5 (Lewi s-and Ode1 1, 1971 ) . For reasons t h a t wi l l become c l e a r in the next sect ion, one may w a n t t o impose an additional cons t ra in t on the values of U, and a2 , namely t h a t a1 t a, = 1 . I n matrix form, t h i s can be expressed as

- _ A U = t , where A = ( I 1 ) and t = 1 ( 1 2 )

This l i n e a r equal i ty constrained l e a s t squares problem has the solut ion given by (Lewis and Odell, 1971):

T - l T T T -1 - - & : = - + ( G _ - G ) A (5 (G;) A (t-5:) (13)

where - CL i s given by equation ( 1 1 ) .

3. SIGNAL TO NOISE RATIO CALCULATION

Let us assume t h a t the l i n e a r cons t ra in t given by eq. ( 1 2 ) i s imposed. I n p rac t ice , It was ver i f ied t h a t the components a1 and a, are pos i t ive so no inequal i ty cons t ra in ts have t o be imposed.

I n order t o ca lcu la te the signal t o noise r a t i o of the simulated band and compare with the signal t o noise r a t i o of the bands b l a n d b,, l e t us assume t h a t each of these bands i s degraded by addi t ive noise, i e

b , = s1 + n b, = s 2 f n 2 (14)

Assume f o r s implici ty (without any loss of genera l i ty ) the following conditions: E(s , ) = E ( s 2 ) = E ( n l ) = E ( n , ) = 0 (15)

E(n: = E (n; ) = 0; (16)

E(S: =E ( S ; = 0' ( 1 7 )

Furthermore, l e t us a l so assume uncorrelated noise processes on the two bands, i e

E ( n , n , ) = 0 (18)

For each mult ispectral band

S

0;' SNRb = -2-

5 2 n (19)

For the simulated band given by - . . p = a,s, t i 2 s 2 + i 1 n l + G2n2 (20)

w S P

U2 we have S N R i j = __

"P ( 2 1 )

We need an additional assumption which i s f requent ly ver i f ied in prac t ice , namely, high cor re la t ion between mult ispectral bands, i e

E 1 (22) psls2

Then, i t can gas i ly be shown t h a t

But 1 - 2 & 1 3 2 < 1 s ince &,and G 2 > 0. Therefore S N R p > SNRb (24)

This r e s u l t shows t h a t an improved signal t o noise r a t i o i s obtained in t h e simulated band . Let us remark t h a t t h e key assumptions leading t o t h i s der ivat ion are: 1 ) l i n e a r constrained est imators ( & + & = 1 ) and 2) high cor re la t ion between the combined bands b l and b,. I t should be observed t h a t in the experimental r e s u l t s we obtained a cor re la t ion coef f icent of .89 between b , and b,.

4. SIMULATION USING SPOT IMAGES

The use of only XS1 and XS2 SPOT mult ispectral bands t o simulate a s p a t i a l l y degraded SPOT panchromatic band i s j u s t i f i e d by the f a c t t h a t XS3 has l i t t l e overlap with the P-band ( s e e Figure 1 ) .

Figure 1 - Typical Spectral S e n s i t i v i t y of HVR Instruments (source: CNES; SPOT-Image, 1986.

The spectral curves of XS1, XS2 and P were d iscre t ized with 10 points ( i e , n=10), leading t o the following numerical r e s u l t s f o r the unconstrained solut ion

322

- r 0.3426 1

~ r 0.4937 -1

= L 0.397 J

a = L 0.5046 _I

and the constrained solut ion

- r 0.3426 1

~ r 0.4937 -1

= L 0.397 J

a = L 0.5046 _I

and the constrained solut ion

Figure 2 shows the d iscre t ized XS1 and XS2 curves. Figure 3 displays the d iscre t ized P curve, the unconstrained estimated P curve and the l inear constrained estimated P curve.

r-.

Figure 2 - Discretized XS1 and XS2 curves

Figure 3 - Discretized P , ? and p curves

5. EXPERIMENTAL RESULTS 5.1. Spa t ia l Simulation o f 2Om - Resolution SPOT

Panchromatic Band

Since bands XS1 and XS2 have a sampling r a t e of 20m, we degraded the 10m P band, according t o the following two s teps .

a ) Step 1 : Spat ia l F i l t e r i n g of the 10m Resolution Panchromatic Band

The P-band was s p a t i a l l y degraded by applying twice the low pass l i n e a r f i l t e r with f i n i t e impulse response given by

3000 I 1 6 9 337 169

This impulse response was obtained following the procedure suggested by (Banon, 1990) and considering the at tenuat ion f a c t o r Y of the modulation t r a n s f e r function at half the sampling r a t e ( t h a t i s , a t Nyquist frequency) given in Table 1 .

______-__

row 0.16 .~

Table 1 - Attenuation Factor \

The value f o r P-band a r e those of (CNES; SPOT Image, 1986); the values f o r XSP-band are not those of the previous reference b u t have been reevaluated so t h a t the visual impression of the simulated panchromatic band be s imilar t o the mult ispectral bands from t h e point of view of spa t ia l resolut ion.

b ) Step 2: Resampling of the F i l te red 10m Resolution Panchromatic Band

The f i l t e r e d 10m resolut ion panchromatic band of s tep 1 was resampled a t 0.5 spa t ia l r a t e , i e , keeping only the f i r s t pixel of each pa i r of consecutive pixels and keeping only the f i r s t l i n e of each pa i r of consecutive l ines . Figure 4 shows the r e s u l t of t h e spa t ia l simulation of the 20m resolut ion panchromatic band.

I t should be noted t h a t the f i l t e r i n g and resampling procedures could a l so be done using mul t i ra te d i g i t a l signal processing techniques, according t o (Fonseca; Mascarenhas, 1988).

5.2. Spectral Simulation of 2Om - Resolution

5.2.1. Unconstrained Simulation

The spec t ra l ly simulated image has lower mean and lower variance than t h e s p a t i a l l y simulated band. The reason f o r t h i s l i e s in the f a c t t h a t t h e spectral curves do not incorporate o f f s e t and gain fac tors t h a t a re used in ground processing.

5.2.2. Constrained Simul a t ion

I t was observed t h a t the histogram of the spec t ra l ly simulated image under a l inear cons t ra in t i s c loser t o the histogram of the s p a t i a l l y simulated image than the previous case; however, mean and variance of spec t ra l ly simulated image a r e s t i l l lower (although c l o s e r ) than the s p a t i a l l y simulated image.

5.2.3. Gain and Offset Adjustment

The adopted procedure i s t o introduce the gain and o f f s e t adjustments so t h a t the mean value and the standard deviation of the estimated band ( e i t h e r through the unconstrained or the constrained est imat ion) a re equal t o the s p a t i a l l y simulated 20m resolut ion band, i e , in t h e unconstrained case: E ' = such t h a t E ( n ' ) = E ( P O )

where po i s the s p a t i a l l y simulated panchromatic band. Analogous expressions a re va l id f o r the constrained estimators. I n order t o avoid increasing r o u n d off e r r o r s oa, and a& where combined by applying t h e transformation d i r e c t l y on bl and b Z . I t was ver i f ied t h a t the histogram of the gain and o f f s e t compensated estimator without cons t ra in t approximates be t te r the histogram of the s p a t i a l l y degraded panchromatic band than the constrained estimator.

5.2.4. Mean Square Error Between Jmages

The m.s.e. between the s p a t i a l l y simulated band (P, and the spec t ra l ly simulated bands without cons t ra in t ( P 2 s ) , w i t h cons t ra in t (P;c ) , w i t h o u t cons t ra in t with gain a n d o f f s e t compensation ( P Z s G ) and w i t h cons t ra in t with gain of o f f s e t

Panchromatic Band

bl + G, b, ) t 6

a ( p ' ) = ( p ' )

323

compensat ion (P, 1 were computed, a c c o r d i n g t o t h e e x p r e s s i o n CG

112 2 5 6 2 5 6

LEWIS, T.O.; ODELL, P.L. E s t i m a t i o n i n L i n e a r Models, P r e n t i c e H a l l , Englewood C l i f f s , N.J., 1971.

These r e s u l t s a r e d i s p l a y e d on T a b l e 2.

P= 1 10.75 I 4.87 I 2.65 I 2.68 I T a b l e 2 - Mean Square E r r o r Between

S p a t i a l l y and S p e c t r a l l y S i m u l a t e d Bands

One can observe t h a t : a ) P, approx imates P, b e t t e r t h a n PzS; b ) t h e r e s s l t s f o r P, and P, a r e approxrmate ly t h e same. It s h h q d be obGGrved t h a t t h e comparison between t h e s p e c t r a l l y s i m u l a t e d band and t h e s p a t i a l l y degraded band r e q u i r e d a smal 1 t r a n s 1 a t i o n a l r e g i s t r a t i o n procedure t h a t was per fo rmed manua l ly . It i s expected t h a t even s m a l l e r mean square e r r o r s c o u l d be o b t a i n e d i f no r e g i s t r a t i o n were necessary.

5.2.5. Visual results

F i g u r e 4 d i s p l a y s 256 x 256 images o f P, P, and Pz . F i g u r e 5 d i s p l a y s P,, Pzs , P,CG a n a t h e a b s o l u t e v a l u e o f t h e d i f f e r e n 5 e image (Pn - P z s G ) , t h r o u g h a l i n e a r c o n s t r a s t s t r e t c h t h a t i n c r e a s e d t h e maximum v a l u e o f t h e a b s o l u t e v a l u e o f t h e d i f f e r e n c e image f r o m a p p r o x i m a t e l y 16 t o 85 f o r v i s u a l purposes. F i g u r e 6 d i s p l a y s t h e SPOT m u l t i s p e c t r a l bands XSl(bl) and XSZ(b2).

It i s observed t h a t Pz, has l o w e r c o n s t r a s t t h a n P2 due t o unknown g a i n and o f f s e t f a c t o r s . Pqc approx imates P, b e t t e r t h a n P z ~ , which i s i n accordance w i t h T a b l e 2. Fur thermore , PZSG and PZCG a r e v e r y c l o s e t o P2 i n v i s u a l te rms.

6. CONCLUSIONS

Through l e a s t squares c r i t e r i a t h e p o s s i b i l i t y o f s i m u l a t i n g one band b y l i n e a r c o m b i n a t i o n o f bands t h a t s p e c t r a l l y o v e r l a p t h e d e s i r e d band has been demonstrated. N u m e r i c a l l y and v i s u a l l y a c c e p t a b l e r e s u l t s were ob ta ined.

BIBLIOGRAPHICAL REFERENCES

BANON, G.J.F. Simulaczo de Imagens de B a i x a ResolucZo (Low R e s o l u t i o n Image S i m u l a t i o n ) , SBA: C o n t r o l e e Automaczo, v o l . 2, n.2, M a r c h / A p r i l 1990, pp. 180-192 ( I n Por tuguese) .

CNES; SPOT I M A G E Guide des U t i l - i s a t e u r s des Ejonn6es SPOT, Vol . 1 Manuel de Reference, l e r e E d i t i o n , Toulouse, 1986.

FONSECA, L.M.G.; MASCARENHAS; N.D.A.; Methods f o r Combined I n t e r p o l a t i o n - R e s t o r a t i o n Through a FIR F i l t e r Design Technique, 1 6 t h I n t e r n a t i o n a l Congress o f t h e I n t e r n a t i o n a l S o c i e t y f o r Photogrammetry and Remote Sensing, Kyoto, Japan, J u l y 1988 i n I n t e r n a t i o n a l A r c h i v e s o f Photogrammetry and Remote Sensing, 27, p a r t B3, Commission 111, pp. 196-205.

“SG ’ “ C G F i g u r e 5 - Images o f P, ,

S G ‘ IP, - P,

F i g u r e 6 - Images o f XS1 and XS2

and

324