Raised cosine function for image restoration

Signal Processing 5 (1983) 61-73 North-Holland Publishing Company

SHORT

61

COMMUNICATION

RAISED COSINE FUNCTION FOR IMAGE RESTORATION H.B. K E K R E Computer Centre, Indian Institute of Technology, Bombay 400076, India S.C. S A H A S R A B U D H E

Electrical Engineering Department, Indian Institute of Technology, Bombay 400076, India N.C. GOYAL

Computer Centre, Indian Institute of Technology, Bombay 400076, India Received 12 November 1981 Revised 5 April 1982

Abstract. In this paper, a new set of raised cosine functions is proposed. These functions have all the useful properties of the spline functions with the additional advantage of continuous infinite derivatives as against only a finite number of derivatives in case of the spline functions. Because of this property they exhibit a smoother behaviour. The property of smoothness, coupled with convolution, makes the raised cosine functions readily applicable to the image restoragtion problem, where the degradation is through a shift invariant blurring function. The results confirm the superior behaviour of these functions in comparison to spline functions. Zusammenfassung. In diesem Beitrag wird eine neue Klasse von Funktionen vorgeschlagen, die auf Potenzen von Kosinusfunktionen basieren. Diese Funktionen besitzen alle niitzlichen Eigenschaften der Splinefunktionen. Im Gegensatz zu den Spline-funktionen, die nur endlich viele stetige Ableitungen besitzen, haben die hier behandelten Funktionen jedoch eine unbegrenzte Zahl stetiger Ableitungen und deshalb einen glatteren Verlauf. Durch diese Eigenschaft sind die hier behandelten Funktionen besonders geeignet ffir die Bildaufbereitung mit Hilfe der Faltung, wenn die Verzerrungen des Bildes durch ein verschiebungsinvariantes Verunsch~irfungsfilter erfolgt sind. Das bessere Verhalten der hier behandelten Funktionen wird durch experimentellen Vergleich mit Splinefunktionen bestfitigt.

R6sum6. On propose dans cet article un nouvel ensemble de fonctions cosinus 61ev6es. Ces fonctions poss6dent toutes les propri6t6s utiles des fonctions spline avec l'avantage additionnelle des d6riv6es continues infinies contre un nombre fini de d6riv6es dans le cas des fonctions splines. A cause de cette propri6t6, elles ont un comportement plus adouci. La propri6t6 d'adoucissement coupl6e avec la convolution rend ces functions cosinus 61ev6es directement applicables au probl6me de restoration d'images of1 la d6gradation est due b, une fonction de flou invariant par translation. Les r6sultats confirment le meilleur comportement de ces fonctions en comparaison avec les splines. Keywords. Image restoration, raised cosine functions, spline functions, approximation, filtering, shift invariant degradation.

1. Introduction

characteristics

of t h e s y s t e m . I m a g e

resoration

c a n b e d e f i n e d as r e c o n s t r u c t i o n of a n o r i g i n a l There seems to be no imaging system which can

( o b j e c t ) i m a g e b y i n v e r s i o n of t h e d e g r a d a t i o n

g i v e p e r f e c t q u a l i t y i m a g e of an o b j e c t . T h u s , t h e

phenomenon

images obtained through an imaging system may

large digital computers have recently opened the

be

way for high resolution image processing by digital

degraded

due

to

physical

0165-1684/83/0000-0000/$03.00

limitations

and

0 1983 North-Holland

it h a s u n d e r g o n e .

The

advent

of

62

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal/ Raised cosinefunctions--Restoration

techniques. With increasing availability of digital input/output imaging devices it is convenient to use digital computers for image processing. The advantages are its flexibility, accuracy and noise immunity etc. Digital data transmission is becoming increasingly popular so that even if the original image is not discrete, there are genuine bonuses to be gained in sampling and conversion to discrete form. In general, the degrading system could be very complex. However, in many cases of practical importance, such as camera motion, atmospheric turbulance and blurring due to lenses etc. may be represented by a shift invariant point spread function (PSF). If the imaging system is assumed to be linear and the PSF to be separable the degraded image can be described by the superposition integral of the form t"

g(x, Y)= ld - -

(3o

and forming local basis. In this paper, a new set of basis functions is presented. These functions have all the useful properties of spline functions. However, these functions have an infinite order of continuous derivatives as against only ( m - 1) continuous derivatives for the spline of degree m. Hence, they exhibit a smoother behaviour as compared to the spline functions. The proposed raised cosine functions are used for restoration of degraded image by exploiting their convolation property. The same degraded image is used for restoration through corresponding spline function. The resultant restored images obtained through raised cosine function are found to be better than those obtained through the corresponding spline functions. In all the examples considered the mean square error obtained by using raised cosine functions is very much less as compared to the spline functions.

hl(x-~) (3o

2. Raised cosine function over a finite interval

I"

oo

=l

h2(y-/3)

The first order raised cosine function over a finite interval, - T <~x <~T, is defined as

co co

Cl(x)=

~-~

l+cos~x

,

- T < - x <-

co

where f(x, y) is the original two-dimensional image, h(x, y)=ha(x)" hz(y) is the point spread function representing the imaging system, and g (x, y) is the degraded image. For the simulation of degradation and restoration by means of a digital computer, the continuous image model must be discretized in both amplitude and spatial position. Thus, the continuous image field and the point spread function, must be transformed into arrays of numbers. Generally, this transformation produces some error, i.e., the inverse transform of these arrays is not exactly the original field. To increase accuracy, the spline functions can be used for the restoration of images [1]. Spline functions are a class of piecewise polynomial functions having limited support Signal P r o c e s s i n g

0,

elsewhere,

Ca(x) = ~-~ u ( x + T ) - c o s ~ ( x + T ) + 1 [u(x - T ) - c o s T7r( x - T ) + ] 2T

=fl(x + T ) + - f l ( x - T)+

(3)

where fl(x) = ~-~ u ( x ) - c o s - ~ x

(4)

and {~ u (x) =

forx ~>0, for x < 0.

(5)

63

ll.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal / Raised cosine functions--Restoration

Also

fl(X)+=lfl(X) t0

f o r x ~>0, for x < 0.

(6)

The Fourier transform of Cl(x) is given by a 2 [sin w T ] , ~-[C~(x)] = ~ 2 _ w 2 t ~ J

(7)

o~ "n"l T =

The raised cosine function Cl(x), along with its Fourier transform ~ [ C l(x)] are plotted and shown in Fig. l(a). For comparison, the first degree spline function S l(x) with its Fourier transform ~ [ S l(x)] is also plotted along with it in Fig. l(b). From the above figures it can be noticed that the spectrum of raised cosine function is similar to that of a band limited function because most of its energy is concentrated in its main love and the side lobes have negligible energy contents. The energy concentration in the main lobe of raised cosine function is more in comparison to that in the case of spline functions: the main lobe energy in J~[Cl(x)] total energy in ..~[Cl(x)]

0 0.5 1.0 --~.x

-8rr

-6rr

2.1. Family of raised cosine functions The raised cosine functions of higher order can be generated through the repeated convolution of Cl(x) similar to the spline function family, i.e.,

C2(x) = Cl(x) • Cl(x), C3(x) = C2(x) • Cl(x),

C"~(x) = Cm-l(x), C~(x)

(8)

where * denotes the convolution process. It is obvious that all raised cosine functions defined in eq. (8) have their maxima at x = 0 and spread of C ' ( x ) is from -roT to +roT. Now, using eq. (3)

0.999364,

the main lobe energy in ~[Sl(x)] = 0.997055. total energy in ~[S l(x)]

-1.0-0.5

It is clear from the above figures that for the raised cosine function the energy outside the main love is 0.064% whereas for the first degree spline function it is 0 . 3 % . Thus, for raised cosine function practically all the energy is concentrated in the main lobe and a negligibly small energy is outside. The raised cosine functions are expected to show better performance as compared to spline functions, as most of the practical signals are band limited.

-4rr

-2"a

2n

0

4n-

6~r

8~"

~cO

Fig. l(a). First order raised cosine function and its Fourier transform.

-1.0-0.5

0

0.5 --e.x

1.0

Fig.

-8n'

-6"ff

-411"

-2n

0

2TT

4"n

6"rr

l(b). First degree spline function and its Fourier transform. Vol. 5, No. 1, January 1983

64

H.B. Kekre, S.C. 8ahasrabudhe, N.C. Goyal / Raised cosine functions--Restoration

1

in eq. (8) gives

C2(x) = [fl(X q'-T)+ - f l ( x

1

x

- T)+]

+ ~9( ~ ) +x

× [fl(x + T)+ - f l ( x - T)+] =

1 x 2

f3(x)+ ~-~~ [~ (~)+ IN(X)--~COS ~T(~) +] . ( ~x) + smTr

(9)

f2(x + 2 T)+ - 2/2(x)+ +f2(x - 2 T)+

where (10)

f2(x)+=fl(x)+ *fl(x)+.

In general

C'(x)

f,(x)+ =

~

a [l(x)3[.ix)+8cos=(X) ] t6 \ T ] +

3 =

x

T

+

2

( )

y. (_1) j m + 1 i=o j f,,[x +(m-2/)T]+

cos

(11) where ( 7 ) denotes the/th binomial coefficient of

105

.

x

mth degree expansion. Here, (12)

f,.(x)+ = f.,-l(x)+ * fl(x)+.

The function fro(x)+ can easily be computed through its Laplace transform. The Laplace transform of f l ( x ) + is given by

,[i ,] ~[fl(x)+]=FI(S)=2-T

1

s

S2+Ot 2

a 2

"IT

2T s(se+a2) '

(13)

ct = - ~ .

Hence, the Laplace transform off,. (x)+ is given by

2 Fro(s) = 2 T s

m

+o~ 2

(14)

"

By taking the inverse Laplace transform of Fro(s) the expressions for the first four functions are given below.

1[

(x)]

fl(X)+=~-~ u(x)-cos~" ~ + , /2(x)+ = ~1 [ ( T )

+

(15a)

Substituting eqs. (15) in eq. (11) any member of the family of raised cosine function can be obtained. The expression for the first four members of the family are given in the Appendix. The first four members of the family of the raised cosine functions are plotted along with the corresponding spline functions for comparison in Fig. 2. Though the spread of mth order raised cosine function is from - r o T to m T but it can be observed that the value of the function is negligibly small after a certain period. Thus the higher order functions do not loose their local support capability and space (time) limited characteristics. For the sake of convenience the basis functions are given double subscripts nomenclature. The superscript m denotes the order of the function and a subscript i indicates the point xi on abscissa at which the basis function attains its maximum. Thus the expression for raised cosine function of order m, which attains its maxima at xi can be given by applying shift invariant property to eq. (11), as

[ u(x)+ ~cos 1 .n.(x) ]

+

C ? ( x ) = Y'. (-11 j

i=O

-3sin

2rr

SignalProcessing

rr

+

,

(15b)

×f,,,[x - x i + ( m - 2]) T]+.

(16)

65

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal / Raised cosine ]:unctions--Restoration RAISED SPLINE

COSINE FUNCTION - FUNCTION -1.0

/

0.8

-~ 8

\ -2

-1 0 1 (o) FIRST ORDER RAISED COSINE FUNCTION ~ FIRST DEGREE SPLINE FUNCTION.

-i

0

i

2

(b) SECOND ORDER RAISED COSINE FUNCTION ,{, CUBIC SPLINE FUNCTION.

2 (c) THIRD ORDER RAISED FIFTH DEGREE.

-~.

-3

-2

COSINE

-1

(d) FOURTH ORDER RAISED OF SEVENTH DEGREE.

0 COSINE

FUNCTION

1

OF

SPLINE

2

FUNCTION (f( SPLINE

FUNCTION

Fig. 2. Raised cosine functions and spline functions. Thus the properties of raised cosine function can be summmarized as follows. (i) The functions are shift invariant, i.e., C ? (x ) = C ? k (X -- Xk ).

(17)

(ii) The functions have limited support and thus they form the local basis. (iii) All the functions are strictly positive. (iv) Since they are generated through the successive convolution of C l ( x ) , it follows that Cm(x) * C"(x) = Cm+"(x).

Thus any member of the raised cosine family can be obtained through convolution in the sense that the set of raised cosine functions is closed with respect to the convolution operation.

(v) The raised cosine functions have infinite number of continuous derivatives. (vi) From the Fourier transform of raised cosine functions it can be seen that they are of band limited nature for all practical purposes. The properties (i) to (iv) are common to those of spline function. These are very useful properties for the purpose of image processing as the property of shift invariant function results in a circulant matrix while the limited support property leads to a banded matrix which can efficiently be solved on computer [9]. The property of nonnegativity matches with the non-negative nature of the intensity of images. The additional properties increase the usefulness of the functions for data interpolation and restoration of the band limited signals. Vol. 5. No. 1, January 1983

66

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal / Raised cosine functions--Restoration

3. Restoration through raised cosine functions

From the eq, (1) the deterministic part of the degraded image in case of a linear shift invariant imaging system in one dimension can be described by P oo

g(x) = ]

h(x -c~)f(c~) da

(18)

d - - oo

where g (x), f(x) and h (x) are the degraded image, original image and point spread function, respectively. Using raised cosine functions as the basis, by sampling uniformly, the object and point spread function can be represented by

function of order m. The continuous image may then be obtained through interpolation by raised cosine functions of order m [10].

4. Illustrations

4.1. Rectangular pulse image In order to evaluate the relative performance of the family of raised cosine functions over that of the spline functions, a rectangular pulse image f(x) of constant amplitude A and width w, was blurred by convolving it with a polynomial point spread function of the form

[ x;l

K

f ( x ) = Y~ fkC~(x),

(19)

h(x)=b 1 -

, -p<~x<~p

(25)

k=l L

h(x)= Z htCT(x)

(20)

/=1

where Cm(x) and C"(x) are raised cosine functions of order m and n, respectively, centered at the origin. The corresponding interpolations coefficients are given by fk and h~. Substituting these coefficients into eq. (18), gives

g(x ) = Y. Z fkhlC'~ (x ) * C7 (x ). k l

where b is the normalizing coefficient to make the area under the curve equal to unity. The expression for the blurred image is as follows:

A

g(x)=-~p [X + p - y ]

(21)

Exploiting the convolution property of raised cosine functions as in eq. (8) it is seen that g(x) is represented by raised cosine function of order m. Eq. (21) can now be written as

O~x~y'p,

g(x) = 0,

L

p \p

/p

y - p ~
g(x)=A,

g(x) =

A

y+p<~x<~y+w-p, [y + w - p + x ]

P=K+L+I

gpCp +" (x - p T )

Z p=l K

L

= ~., Z fkhtCk~++t"[x-(k+l) T] k=l

(23)

1=1

Comparing the coefficients of raised cosine function gives K

gp = ~, fkhp-k, p = 1, 2 . . . . . P.

(24)

k=l

Thus g is the result of convolution of f and h. By deconvoluting g through h one can evaluate the vector f, which are the coefficients of raised cosine Signal Processing

y+w-p~x~y+w+p, g(x)=0

elsewhere

(26)

where y is the point on the abscissa from where the original image originates. The original and the blurred images along with the point spread function are shown in Fig. 3.

67

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal / Raised cosine functions--Restoration

F-~

POINT--SPREAD FUNCTION~-! ~ " \,

ORIGINAL, / B L U R R E D FUNCTION-~/*- FUNCTION

,Cx)

/,,

/;

/I\

,/,.

gCx)

-3

i . , .~, ii . . . . O 5

i 10

.

.

.

N

I ,.X,

o

:\

3

i , , , , i .... 15 20

.

i 25

. . . .

II i

~

30

,

,

X-l-

35

Fig. 3. Original and blurred pulse image with point spread function.

The blurred image is now approximated by the raised cosine functions of order m. Three values of m (2, 3 and 4) are considered. The coefficients so obtained were deconvolved with those of raised cosine function of order n, (n = 1), obtained by approximating the point spread function. This resulted into the coefficients of raised cosine function of order (m - n) corresponding to the original image. The continuous image was obtained by interpolation through the raised cosine function of order (m - n). The schematic block diagram of the various operations involved is shown in Fig. 4. The restored images obtained from different order of raised cosine functions are shown in Fig. 5. The original image is also, shown for comparison. Since the original image is known the mean squared errors between the original function

and reconstructed functions are calculated in each case at knots and for overall span. The procedure is repeated by replacing the raised cosine functions by spline functions of various degrees. For approximation of blurred image, the spline functions ot degree 3, 5 and 7 are used and the point spread function is approximated by linear function (i.e., spline of degree one). The reconstructed images are shown in Fig. 6. The mean square error at knots and for overall span have been calculated in each of the above cases and are shown in Table 1. The results shown in Figs. 5 and 6 indicate that the ripples in the restored images are of negligible amplitude in case of the raised cosine functions, while they are substantial in case of the spline functions restoration. Thus, the raised cosine func-

ORIGINAL

~CONVOLUT~)g

, IMAGEpoINT

_IOEORADED

IMAGE

SPREAD I FUNCTION (a

DEGRADED IMAGE [

J

IMAGE

IAPPROXIMATIONBY

.J m t h ORDER " RAISED COSINE FUNCTION

•SPREAD POINT

I APPROX I MAT ION .J BY n th ORDER

FUNCTION

"L RAISEDFUNCTCOSINEoN

(b)

BLURRING

I JDECONVOLUTION

J ~

IMAGE

ICOEFFIOENTSOF

_] I

(m-n)th

"[ C] OORSDI NE ER FUNCTONJ RAISED

1 'MAGi__

RESTORATION

Fig. 4. The schematic block diagram for image blurring and its restoration. Vol. 5, No. 1, January1983

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal / Raised cosine functions--Restoration

68

RESTORED

,, L ,

,

,

,

.

0

.

i

~ ,

,

i

(O)

I

= i

~ , J . . . .

15

10

5

RAISED

J .

COSINE

.

.

.

25

20 FUNCTION

OF

.

.

.

30

ORDER

35

TWO

..... I ....................... 1 .....

O

5

(b)

L.

.

.

.

.

10

15

RAISED

COSINE

,

3

5 (c)

.

,

L

.

.

.

.

20

25

FUNCTION

L

.

.

.

.

i

OF

,

,

30

ORDER

,

,

i

,

35

THREE

,

,

.

.

.

.

J

10 15 20 25 30 RAISED COSINE FUNCTION OF ORDER FOUR

35

Fig. 5. Restored pulse image using raised cosine functions.

,

ORIGINAL . . . . . . RESTORED

',

,, O

'

'

- -

i

i

i

.

,

I

i

.

.

10 (a)

i

•

,

~ v J

5

L

0

.

.

.

.

i

SPLINE

.

5 (C)

.

.

i

.

i

.

.

.

.

L

OF

.

.

.

.

i

15 FUNCTION

,

=

,

i

,

.

,

,

i

.

.

.

.

,

.

.

OF

L / ~

.

.

.

=

30

,

i

.

.

.

.

25

DEGREE

,

,

Processing

J

35

.

i

30~

35

FIVE•

.

20 DEGREE

t

.

.

25 SEVEN•

.

.

.

v

.

.

30

Fig. 6. Restored pulse image using spline functions. Signal

'

THREE.

20 OF

i

25

DEGREE

15 FUNCTION

10 SPLINE

.

20

FUNCTION

10

(b)

i

15

SPLINE

.

0

,

35

69

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal / Raised cosine (unctions~Restoration

Table 1 Percentage mean square error between the original and restored (pulse) images obtained through the various raised cosine functions and spline functions S. No.

Approximating functions for blurred image Raised cosine function

1. 2. 3.

Spline function

percentage mean square order at knots

overall span

percentage mean square error degree at knots

overall span

two three four

0.66605 0.48466 0.59166

three five seven

1.01451 0.72370 0.85115

0.0173 0.000104 0.00213

tions seem to be comparatively better than the splines for the purpose of restoration. The percentage m e a n square error (m.s.e.) figures confirm the superior performance of the raised cosine functions.

0.2557 0.0242 0.0151

....

jiii',iiiii'i',i,,,

4.2. Portrait of girl

The procedure explained in Fig. 4 is extended to a real life portrait of a girl, the original photograph of which is shown in Fig. 7. This is a coarsely discretized image and is quantized only in 16 grey levels. The size of image is kept as 128 x 64 so as to incorporate it into a single page of line printer page. The original image (shown in Fig. 7) is the photograph of the computer line printer output obtained by the character over printing of 16 grey levels. The degraded image shown in Fig. 8 is :~i~i~i~::i: ,!K;~!!'i!"'L-':'.~ii~!i L::~:.

,iiiiii!iiiiiiiiii!i!i!iiiiiiiiii?ii!i "iF;!Lr!?!;.'i'~;'":

i~i"i~ii~;*iL~i|- ~ L~'~i:-':,

lll~ilill~iih!iiiiiiii;~' i!;i~i~ii:~,,.:"i

i= ~ii[.."{i.~

-i :: i

~

"=

ii!l~|~4~11i~ii!K!Yi !~P!I{'.'lllBi[? • t|iTi"!!ili!ii '~i!i?iliY. ~?'iii:~t: _ff ~.-!-~.~ i! i!i i-ii

~-[!

."

:~: :

18

Fig. 7. Original girl image.

,

dill, ....

!lllliiiili!!!iii!ii:iiiiii!!i!iifiiiiii!!!!!!i! !!$iiiii!!iiHgf[i~fliiiHii!![iglgliiiil' 19

Fig. 8. Blurred girl image. obtained by convolving the point spread function of Fig. 3 in vertical direction with the original image of Fig. 7. The restored images are obtained by approximating the degraded image successively through the raised cosine functions of the order 2, 3 and 4 respectively. The point spread function is approximated by raised cosine function of order one. To compare the performance of the raised cosine function with the spline functions, the point spread function is approximated by first degree spline function and the restoration is done by approximating the degraded image through the cubic spline, spline function of degree 5 and spline function of degree 7 respectively. The percentage mean square error in each case are obtained and tabulated in Table 2. The various restored images obtained through the above procedure are shown in Figs. 9 to 14. Vo[. 5, N o . 1, J a n u a r y 1983

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal I Raised cosine functions--Restoration

70 Table 2

Percentage mean square error between the original girl image (128 x 64) and restored girl images obtained through the various raised cosine functions and the spline functions S. No.

Approximating functions for the blurred girl image (Fig. 8) Raised cosine function

1. 2. 3.

Spline function

order

percentage m.s.e,

fig. no,

degree

percentage m.s.e,

fig. no.

two three four

0.059 0.1056 × 10 -3 0.0080

9 11 13

three five seven

3.8945 0.09945 2.617

10 12 14

......

i+,i

i~'*!+/~ ++'':+

,+i+

[,ii, ,l!!i+il!iii "

i!"i "+M+i +;:ill7+'i~!'i

+, !!l+ll,!i:

tlI. N7

! ii'+++lliiiiiii+!ii!+ ++'"'+ + +,+,.++,

~?.=-.~i~iiu+"_+!

20

Fig. 9. Restored girl image: blurred image is approximated by raised cosine function of order two, MSE = 0.059%.

l+++++,++++++

Fig. 11. Restored girl image: blurred image is approximated by raised cosine function of order three, M S E = 1.056 × 10-4%.

:::::::::

++fTltll~iiil!iiliiilll+~

:

+,,+,+.i,,,,+,.+,l,,,+++,,:,,,++,++i++++r, "~i~lll~l'l:+:i3llll~l~Sil!!i ? rT~N ~ *);';" :~"+i!l-~lili'.'litl!ll~.¢lil;l~illi~:C: ~+

"llliti : : t"Piih~ :

.i;-:i+ • [

• iit]+'~+~:!iii!!:!i

li

"+.+,lfgmu~l+ili+iH:;+l+f:t'g;i{q+~ll+lliiit+t" 7+;÷~++llllllliiiiiiilci~l++H~

]H~J(~lll~i:~llTltiiil:.'Jeillg+ !.:

i !i ill+ ~i-;,. ;C~ ,:iiljll~i..711"i

l!illlllllll!+lT+l+!l:i~:it+l~l+ii+tt+!+l+l *::¢::llf/t: "'-:-~+;': ..... :l .;l).il

:

el!ill:"

ii++~;Nli++i++iii~;:+ ~.+-+-+-i~i,+!; iPlll.i++ ~.~*d i*'.+| +Ii+iL+Ii ++l ++. i.lllli.=-h ilk. illlmi.~l. !~:.!ii'~i'ii~ll

i! " ":

": +i.lq+'.-"+isil i''

-777+: 7 +$++7~7 7+: :

+:

7+~-~77777, 7'+ii{+...l+

Fig. 10. Restored girl image: blurred image is approximated by spline function of degree three, M ~ E = 3.8945%. Signal Processing

l~:~

++++!++~++h, ++~hl+;!i~fi' ~-i; ~!i+ii~s7+~+7++ +++++)++++!!++++++7+?++++~J++: ) + : ,: :7 ++"l 7 +7]+7' '+:7+'

Fig. 12. Restored girl image: blurred image is approximated by spline function of degree five, MSE = 0.09945 %.

H.B. Kekre, S.C. Sahasrabudhe, N.C. Goyal I Raised cosine functions--Restoration

~:~;.

~

-I

~

'

,.!

~i

':

iiit,~

tt

i ,~r!i{i~.tl i i f i ...... . .,. . . ..... ,I .. '

[li::v:..

,

~• i

:7

,,i~i!~7. iiiil,)?

t.

.

:

~71~.:k~.,,~i~:,iiI:i7 in

Fig. 13. Restored girl image: blurred image is approximated by raised cosine function of order four, MSE = 0.008%.

71

Where degradation is produced through the convolution of the blurring function with the original object function. This set has all the useful properties of the spline functions. In addition, the raised cosine functions have all the infinite continuous derivatives as against only ( m - 1) derivatives of spline function of degree m. Because of this they exhibit a smoother behaviour as compared to spline functions. Since the blurring phenomena is essentially a smoothing process the raised cosine functions are better suited to represent the blurred image. It has been shown by the illustrations considered that more accurate reconstruction of the original object is obtained through the raised cosine function.

References

:'~'.~:~m'nt~..m.'~,,.'"~'"~'~ i~!~:.".F,~' :.-!iii=,;!i.'.'llllrl~ "r''ili~ii~

~

~!.,"'('i~' ~

:~li!:~'"

i id.lll~li~.i!i~i~!!-'..-~.~.-~.:~d "~:: ~ , . : , u l # ' . ' " , , ~. i~i.l ~l O ,.#!!i.'l, I IdI l,BI Il t h .................. : .: = . l,lrl~.~.llr !~

i'-I,iI11!!IIIIIi|i : "~ ~' i~ili~l[ ]lli:l iI!|tiiidi~liii[ !! ! ~ : ~ : : : ~ ':: :##~'~[II: s~." ![| lil]'i

ti ~H~ii'i|iji:.\-~,~

:i ! "I'::!ilIIHH ifltlt|l;.l~! i:::'" iliil.qtt, i !i "

i...1_~!:

1

"i':

! " :. ~:..I.~H :::: :::;~" HT:'III " f:.'-:i'i"~c :ii!i~'lt! i

i ili!J)Tii ;: iiiiiiiiiii', iil

~c~='-:~iiY'i i iIiii77] iili~ ii, l i!i'=~ II

Fig. 14. Restored girl image', blurred image is approximated by spline function of degree seven, MSE = 2.617%.

The appearance of images and the magnitude of mean square errors indicate the superiority of raised cosine functions over the spline functions.

5. Conclusion A new set of basis functions, the family of raised cosine functions, is introduced for the purpose of image restoration. This is a closed set, with respect to convolution. The property of convolution can be exploited successfully for image restoration.

[1] M.J. Peyrovian and A.A. Sawchuck, "Image restoration by spline functions", Applied Optics, Vol. 17, No. 4, 15 Feb. 1978, pp. 660-666. [2] H.S. Hau and H.C. Andrews, "Cubic spline for image interpolation and digital filtering", IEEE Trans. Acoust. Speech Signal Process, Vol. ASSP-26, No. 6, Dec. 1978, pp. 508-517. [3] L. Levi, "Motion blurring with decaying detector response", Applied Optics, Vol. 10, No. 1, Jan. 1971, pp. 38-41. [4] A.W. Lohmann and D.P. Paris, "Influence of Longitudinal vibrations on image quality", Applied Optics, Vol. 4, No. 4, Apr. 1965, pp. 393-397. [5] A.A. Sawchuck, "Space variant image motion degradation and restoration", Proc. IEEE, Vol. 60, No. 7, July 1972, pp. 854-861. [6] S.C. Sahasrabudhe and A.D. Kulkarni, "Shift variant image degradation and restoration using singular value decomposition", Compt. Graphic and Image Processing, Vol. 9, 1979, pp. 203-212. [7] V.N. Mahajan, "Degradation of an image due to gaussian motion," Applied Optics, Vol. 17, No. 20, 1978, pp. 3329-3334. [8] H.C. Andrews and B.R. Hunt, Digital image restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977, Ch. 4, pp. 61-89. [9] H.B. Kekre, S.C. Sahasrabudhe and N.C. Goyal, "Fast algorithm for the solution of banded linear systems", Int. J. Comput. Elect. Engng., Vol. 8, No. 1, Jan-Mar. 1981, pp. 21-27. rio] H.B. Kekre, S.C. Sahasrabudhe and N.C. Goyal, "Image data interpolation through spline functions", JIETE, Vol. 27, No. 4, April 1981, pp. 125-130. Vol. 5, N o . 1, J a n u a r y 1983

H.B. Kekre,S.C.Sahasrabudhe,N.C. Goyal/ Raisedcosinefunctions--Restoration

72

Appendix

With the help of eqs. (16) and (12) the expressions for the first four members of the raised cosine function family are obtained and given below. C~(x) = ~-~ 1 [u(x)-cos ~(x rr +T)+] - ~ 1 [u(x)-cos ~(x rr -T)+] ,

(A. la)

]_3sinrr(X+2T~ ] C~(x)=-d-1f [ ( x- T- T ) +[u(x)+~1 cos rr(X+2T~ ~--T--;+J \ T ;+J x 3 . x 1 [(7)+b~x,+ ~1 cos o (~) x +] - ~---~ 2T s,n zr(~)+] 1 [(x-2T) [ 1 ,rr(X-2T~ ] 3 rc(X-2T~ ] +4-T Y + u(x)+~cos \ T ]+j -~--~ sin \--~---]+], 1 rr(X+3T~ ] + 9 ( x + 3 T ~ ~co~ , w , + ~ ~,--~,+s'n rr(X+3T~ , T ,+

~x, ~1[ 1 ( -x ?- Ti b) , x ,

[

32 u(x)-cos

77"

,,(x,~,~]] ~[l(x+~[u(x)_l \Y]+JJ-~l_2 \ T ] +

9 /x+T\

(A.lb)

{x+T'~

~ COS

¢x'"~]

rrk---~----]+j

3 [u(x)_cosn.(x+T~ ]] \ - - - ~ ] +JJ

+8--7 k - - ~ ) + sin 7rk - - - ~ ] +-7- ~ 3 [I(x--T~2 [

1

7T(x--T~ l + 9 ( x - T ~

7.r(X-r~

+--8T \ T ]+ u(x)-~ cos \ T ]+J 8~'\ T /+sin \ T ]+ 32[u(x)_cos x - T 1 [l(x-3T~ 2 ~" (~)+]]-~L2\~'/+[1-~c°s + -8~ -9 (~T3T) + sin ~ t[x-3T'~ -W-)+-7 1 [1/x +4 T \3 F C4(x ) = .i_~

1

3 [

1

u(x)-cos 7,

rc(x-3T~ ] \~]+j

(X~T3T) ]] +

[I +4 T "~+] -~-~ 3 (x \ ~ '+/ +4 T ~2 sin ,rr(X - +~ - T)

( - - T - ) + i u ( x ) + ~ cos z r k ~ )

42/x+4T\ F

rr(x+4T~ ]

41

105 sin n.(x+4T~ ] \-g-'/÷j

; t--V-)÷["(x)+~ cos \--g--;÷j + ~ 1 [l(xTT

4(x+2T~

rr \

T '/+

3 1 x 3

8~

)

1

x+2T

3 (~T__)

_ 3_3_(x~2 sin (~)+

co, (~)+] 8o,~,+

x ;42 (T)+[u(x)+~ co, ,n.(T) +] + ~105 s , r . ~(7)+] SignalProcessing

sinzr(x+2T~

[u (x) + ~--~cos ,~(x+2r~ ] + 105 sin ~(x+2T~ ] \ T '/+J ~ \ T '/+J 1

(A.lc)

,

÷

H.B. Kekre,S.C.Sahasrabudhe,N.C. Goyal/ Raisedcosinefunctions--Restoration 1 rl/x-2r\'r 1 ,~(x-2r~ ] 3 (x-Zr'~ ~ (,-2r'~ 4T[g~)+[u(x)+-~cos \----~/+j-~-~\~----/+sinTr\ T /+ 42/x-2T\

I-

+ I _1_ 6 _ ~ [ ~ ( x ? T )

--3+[U(X)I-NCON

42(x-gT'~ [u \~/+L

(x-2T~ ]

41 1

105

73

. 7r(X-2T~ ] r /+J

7r(X -4 T '~ ] 3 [ (x - 4 T '~2 rr(x - 4 T '~ ] \ T ] + J - 8 - ~ L \ ~ - - - - / + sin \----T---/+J

41 (x-4T'~ ] 105 . [x-4r'~ ] (x) + g-4 c°s 7r \ ~ T - - / + J + ~ sm rr I ~ ) +] "

Vol. 5, No. 1, January 1983