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Some nonlinear operators for picture processing
Pattern Recognition VoL I1, pp. 341 342. Pergamon PressLtd. 1979.Printedin Great Britain. Pattern RecognitionSociety.
SOME
(KI31 3203,'79.'(1401 IH05S02.~UO
NONLINEAR
OPERATORS
FOR
PICTURE PROCESSING PAUL P. Rowe Department of Mathematical Sciences, North Dakota State University, Fargo, ND 58102, U.S.A. (Received 23 August 1978 ; received for publication 12 April 1979) Abstract -- This paper assumes that any non-negative function of two independent variables can be considered a picture. Also that some pictures might be processed to show fine detail better with an appropriate nonlinear operator or an operator that satisfies some other partial differential equation than the Laplacian. This paper discusses some of these operators.
Operator diffusion
Smoothing Enhancing Processing Non-negative functions Taylor's series Laplacian Grey level INTRODUCTION
DIFFUSION
If one starts with the linear diffusion equation t~g
[-92g
Nonlinear
and the gradient
Many operators for smoothing, enhancing and otherwise processing pictures have been discussed in the literature, Rosenfeld, I1~ Rosenfeld and Kak 121 and Schachter and Rosenfeld/31 These operators are very useful and much picture processing can be done with them. It is hoped that the operations introduced in this paper will also be useful for processing pictures and other non-negative functions of two independent variables. SIMPLE NONLINEAR
Picture
/(gf`] 2 (gf`] 2 X/\gx/ + \gy/ are rotation invariant. See Rosenfeld and Kak t2b (pp. 181, 182). Consider the nonlinear diffusion equation discussed by Ames t41 9g cq [-k' ` 9 0 ] k(gl~yy 9 t - 9 x L tg~ffxxJ + ~y • Expanded this becomes : 9g
Ot
92g]
[-c~2g
-- = k(0)[~2
+
929 7
~y~J
+
dk(g)F(gg`] 2 (~g`]27 ' d.q-L\dx/ + t~.s j'
since neither x nor y appear explicitly in k(g), the operator on the right side of this equation is also rotation invariant. An example of this would be
9t = k Lc3xz + O-y2j and the Taylor series
g(x,y,O) = ,q(x,y,r) - r ~9g( x, y,r)
9g
k g F "gg7
, g F ~gg7
r2 92g (x, and then substitutes the right side of the diffusion equation into the Taylor's series and neglects all higher order terms, the result is [92g
Now if this is substituted into the Taylor's series and higher order terms are neglected, the result is
c32g7
92g] f=g-kzlg( ,I-929 L~.~5.'+ 0 ~ j
~q(x,y,O)=g(x,Y,z)-krL~xZ + ~y2 f Now replace
g(x,y,O)by
[-6329f(x'Y)and6~2gTget
f = g - kr [O~5~z+ ~ j . See Rosenfeld and Kak ~21 (page 184). Now the Laplacian
Oz 9X 2
+
9Zg 9y 2
+ ctg'- ~ L\~-'c/[{gg ,]2 + (?~g)2]}~. This operator contains both the Laplacian and the gradient. Ofcourse, for some values ofct, f m a y t a k e o n non-integer values, but ahere are several ways of taking care of this, such as rounding. If ~t = 0, this reduces to the linear case. If one wants to see what happens when a picture is 341
342
PAUL P. ROWE
being processed but a certain grey level is to be left unchanged, then let k(g} = fl(g~ - g~)2. For example
k(g} = fl(g~ - 4~)z where fl and y are parameters that can bevaried at ones discretion. A more general method will be given later. For this case a2g
tent. Also if ct = 0, fl = 0, then this operator will not use the Laplacian. Schacter and Rosenfeld °~ suggested six other operators involving either the absolute value function or the maximum value function• These might also be useful here• For( example:
f= g - krt~[3-g]*i4-g]*(5-g) [- d2g 02g 1 * m a x t 0 ' ~ x 2 + O~-J
~
ag 2 2~0~-'(°~ -
L\~/
CONCLUSION
and for g = gi, f = gMORE GENERAL NONLINEAR EQUATIONS One could start out the equation
ag F x,y,t,g, ag a2g ag a2g'~ at = Ox' ax 2' ay' ~y2 } but this is too general for our purposes. For this work the most general considered is
ag =
I-a2g a2gI + kf2(o) [(aa~gx) 2 + (8g~21
kf,(g)L~x2 +
--
at
-
~J
SUMMARY
a o a2g ag a2g~ F x, y, t, a x ' ax 2' ay' 0-~-// k [ a20 a201
= fl(o) ~
[(0Ogx) 2
+ ~y2j + kf2(g)
(ag,~21 + \OYY/]"
Then
( g -
7 a29 / Fa2g + ~
+f2(g)
• Lt ) + t)Jj fffY
For example, one might use
f=g
I believe that some of the operators discussed in this paper would be useful in interactive computing where the effect of changing parameter values could be studied.
\~y/ j'
where
i:
It has been shown that some nonlinear operators for processing pictures exist. Many of these are rotation invariant. Many more could be obtained for specific purposes. It is generally assumed that a picture is a linear function of the independent variables. However, Andrews and Hunt m suggest that some are nonlinear. Therefore, it be that nonlinear operators should be used in somemaYcases.
- kr
{~13-vl,14-g[*
(5 --
g ) * ~a2g X2 + a j ~2
[(~X)2
+
BI3-gI*I4-oI*(5-g)
where Ihl means operator, g = 3, 4 increased, and g < have the effect of
(ajyy)2]} +
absolute value of h. With this or 5 is left unchanged, g > 5 is 3 is decreased if a, fl > 0. This will isolating those three grey-levels.
M a n y other variations of this are possible. Also if /7=0, ~t:#0, this operator will not involve the grad-
Since a picture, or more general non-negative surface of two independent variables, can in many cases be nonlinear, some nonlinear methods for picture enhancement, smoothing and general processing are developed in this paper. Most of them are rotation invariant and it is hoped that they will be useful in supplementing the linear operators that have proved useful in the past for picture processing. The nonlinear operators in this paper were dediffusion. The reason for doing this was that the linear operators previously developed for picture processing could be based on a linear diffusion model. REVERENCES
l. A. Rosenfeld, New York PictureProcessingbyComputer. (1969). 2. A. Rosenfeld and A. Kak, Digital Picture Processing. Academic Press, New York (1976). 3. B. Schachter and A. Rosenfeld, Some new methods of detecting step edges in digital pictures, Communs. ACM 21(2), 172-176 (1978). 4. W. Ames, Nonlinear Partial Differential Equations in Enoineerin#. Academic Press, New York (1965). 5. H. Andrews and B. Hunt, Digital Image Restoration. Prentice-Hall, Englewood Cliffs, New Jersey (1977).
About the Author - PAUL P. Rowe has been Associate Professor at the North Dakota State University in the Mathematical and Computer Science Department since 1969. He received his Ph.D. and M.S. from the Washington State University in 1960 and 1953 respectively and he received his B.S. from the Brigham Young University in 1950.
SOME
(KI31 3203,'79.'(1401 IH05S02.~UO
NONLINEAR
OPERATORS
FOR
PICTURE PROCESSING PAUL P. Rowe Department of Mathematical Sciences, North Dakota State University, Fargo, ND 58102, U.S.A. (Received 23 August 1978 ; received for publication 12 April 1979) Abstract -- This paper assumes that any non-negative function of two independent variables can be considered a picture. Also that some pictures might be processed to show fine detail better with an appropriate nonlinear operator or an operator that satisfies some other partial differential equation than the Laplacian. This paper discusses some of these operators.
Operator diffusion
Smoothing Enhancing Processing Non-negative functions Taylor's series Laplacian Grey level INTRODUCTION
DIFFUSION
If one starts with the linear diffusion equation t~g
[-92g
Nonlinear
and the gradient
Many operators for smoothing, enhancing and otherwise processing pictures have been discussed in the literature, Rosenfeld, I1~ Rosenfeld and Kak 121 and Schachter and Rosenfeld/31 These operators are very useful and much picture processing can be done with them. It is hoped that the operations introduced in this paper will also be useful for processing pictures and other non-negative functions of two independent variables. SIMPLE NONLINEAR
Picture
/(gf`] 2 (gf`] 2 X/\gx/ + \gy/ are rotation invariant. See Rosenfeld and Kak t2b (pp. 181, 182). Consider the nonlinear diffusion equation discussed by Ames t41 9g cq [-k' ` 9 0 ] k(gl~yy 9 t - 9 x L tg~ffxxJ + ~y • Expanded this becomes : 9g
Ot
92g]
[-c~2g
-- = k(0)[~2
+
929 7
~y~J
+
dk(g)F(gg`] 2 (~g`]27 ' d.q-L\dx/ + t~.s j'
since neither x nor y appear explicitly in k(g), the operator on the right side of this equation is also rotation invariant. An example of this would be
9t = k Lc3xz + O-y2j and the Taylor series
g(x,y,O) = ,q(x,y,r) - r ~9g( x, y,r)
9g
k g F "gg7
, g F ~gg7
r2 92g (x, and then substitutes the right side of the diffusion equation into the Taylor's series and neglects all higher order terms, the result is [92g
Now if this is substituted into the Taylor's series and higher order terms are neglected, the result is
c32g7
92g] f=g-kzlg( ,I-929 L~.~5.'+ 0 ~ j
~q(x,y,O)=g(x,Y,z)-krL~xZ + ~y2 f Now replace
g(x,y,O)by
[-6329f(x'Y)and6~2gTget
f = g - kr [O~5~z+ ~ j . See Rosenfeld and Kak ~21 (page 184). Now the Laplacian
Oz 9X 2
+
9Zg 9y 2
+ ctg'- ~ L\~-'c/[{gg ,]2 + (?~g)2]}~. This operator contains both the Laplacian and the gradient. Ofcourse, for some values ofct, f m a y t a k e o n non-integer values, but ahere are several ways of taking care of this, such as rounding. If ~t = 0, this reduces to the linear case. If one wants to see what happens when a picture is 341
342
PAUL P. ROWE
being processed but a certain grey level is to be left unchanged, then let k(g} = fl(g~ - g~)2. For example
k(g} = fl(g~ - 4~)z where fl and y are parameters that can bevaried at ones discretion. A more general method will be given later. For this case a2g
tent. Also if ct = 0, fl = 0, then this operator will not use the Laplacian. Schacter and Rosenfeld °~ suggested six other operators involving either the absolute value function or the maximum value function• These might also be useful here• For( example:
f= g - krt~[3-g]*i4-g]*(5-g) [- d2g 02g 1 * m a x t 0 ' ~ x 2 + O~-J
~
ag 2 2~0~-'(°~ -
L\~/
CONCLUSION
and for g = gi, f = gMORE GENERAL NONLINEAR EQUATIONS One could start out the equation
ag F x,y,t,g, ag a2g ag a2g'~ at = Ox' ax 2' ay' ~y2 } but this is too general for our purposes. For this work the most general considered is
ag =
I-a2g a2gI + kf2(o) [(aa~gx) 2 + (8g~21
kf,(g)L~x2 +
--
at
-
~J
SUMMARY
a o a2g ag a2g~ F x, y, t, a x ' ax 2' ay' 0-~-// k [ a20 a201
= fl(o) ~
[(0Ogx) 2
+ ~y2j + kf2(g)
(ag,~21 + \OYY/]"
Then
( g -
7 a29 / Fa2g + ~
+f2(g)
• Lt ) + t)Jj fffY
For example, one might use
f=g
I believe that some of the operators discussed in this paper would be useful in interactive computing where the effect of changing parameter values could be studied.
\~y/ j'
where
i:
It has been shown that some nonlinear operators for processing pictures exist. Many of these are rotation invariant. Many more could be obtained for specific purposes. It is generally assumed that a picture is a linear function of the independent variables. However, Andrews and Hunt m suggest that some are nonlinear. Therefore, it be that nonlinear operators should be used in somemaYcases.
- kr
{~13-vl,14-g[*
(5 --
g ) * ~a2g X2 + a j ~2
[(~X)2
+
BI3-gI*I4-oI*(5-g)
where Ihl means operator, g = 3, 4 increased, and g < have the effect of
(ajyy)2]} +
absolute value of h. With this or 5 is left unchanged, g > 5 is 3 is decreased if a, fl > 0. This will isolating those three grey-levels.
M a n y other variations of this are possible. Also if /7=0, ~t:#0, this operator will not involve the grad-
Since a picture, or more general non-negative surface of two independent variables, can in many cases be nonlinear, some nonlinear methods for picture enhancement, smoothing and general processing are developed in this paper. Most of them are rotation invariant and it is hoped that they will be useful in supplementing the linear operators that have proved useful in the past for picture processing. The nonlinear operators in this paper were dediffusion. The reason for doing this was that the linear operators previously developed for picture processing could be based on a linear diffusion model. REVERENCES
l. A. Rosenfeld, New York PictureProcessingbyComputer. (1969). 2. A. Rosenfeld and A. Kak, Digital Picture Processing. Academic Press, New York (1976). 3. B. Schachter and A. Rosenfeld, Some new methods of detecting step edges in digital pictures, Communs. ACM 21(2), 172-176 (1978). 4. W. Ames, Nonlinear Partial Differential Equations in Enoineerin#. Academic Press, New York (1965). 5. H. Andrews and B. Hunt, Digital Image Restoration. Prentice-Hall, Englewood Cliffs, New Jersey (1977).
About the Author - PAUL P. Rowe has been Associate Professor at the North Dakota State University in the Mathematical and Computer Science Department since 1969. He received his Ph.D. and M.S. from the Washington State University in 1960 and 1953 respectively and he received his B.S. from the Brigham Young University in 1950.