Linear Filtering of Images Based on Prop erties of Vision
V Ralph Algazi Gary E Ford and Hong Chen
CIPIC Center for Image Pro cessing and Integrated Computing
University of California Davis
Davis CA
EDICS IP Filtering
November
Abstract
The design of linear image lters based on prop erties of human visual p erception has b een
shown to require the minimization of criterion functions in b oth the spatial and frequency do
mains In this corresp ondence we extend this approachtocontinuous lters of in nite supp ort
For low pass lters this leads to the concept of an ideal low pass image lter whichprovides a
resp onse that is sup erior p erceptually to that of the classical ideal low pass lter
Intro duction
The use of hard cuto ideal low pass lters in the suppression of additive image noise is known to
pro duce ripples in the resp onse to sharp edges For high contrast edges human visual p erception
fairly simply determines acceptable lter b ehavior Ripples in the lter resp onse are visually masked
by the edge so that the contrast sensitivity of the visual system decreases at sharp transitions in
image intensity and increases somewhat exp onentially as a function of the spatial distance from
the transition
Algorithmic pro cedures using prop erties of human vision have b een describ ed for over years
The development of adaptive metho ds of image enhancement and restoration based on the use
Corresp onding author Email ford ece ucdavis edu Telephone Fax
of a masking function measure spatial detail to determine visual masking In active regions
of the image visual masking is high relative noise visibilityislow and the lter applied is allowed
to pass more noise until the sub jective visibili ty is equal to that in at areas
Whether the lter is adaptive or not the design of the linear lter to b e applied is a critical
issue Hentea and Algazi have demonstrated that the rst p erceptible image distortions due to
linear ltering o ccur at the ma jor edges and thus worst case design for visual app earance should
b e based on edge resp onse They develop ed a lter design approach based on the minimization of
aweighted sum of squared error criterion functions in b oth the spatial and frequency domains In
the spatial domain the weighting is by a visibili ty function representing the relative visibilityof
spatial details as a monotonically increasing function of the distance from an edge This visibili ty
function determined exp erimentally from the visibili ty of a short line p ositioned parallel to an
edge was also found exp erimentally to predict satisfactorily the visibility of ripples due to linear
lters
In the following weextendthework of Hentea and Algazi by considering the design and prop er
ties of one dimensional continuous lters of in nite supp ort two dimensional lters are generated
by D to D transformations We obtain a new formal result on the low pass lter of in nite
supp ort which is optimal for images It establishes the limiting p erformance that digital lters of
nite complexity can only approximate
Design of One Dimensional Filters for Images
The basic trade o in the design approach of Algazi and Hentea is maintaining image quality
while reducing unwanted artifacts or noise The image quality is measured by spatial domain
criterion function for the visibility of ripples in the vicinity of edges
Z
I w x u x u x dx
where u x is a unit step input pro ducing the lter resp onseu x u x h x where h x is the
point spread function of the lter denotes convolution and w x isaspatialweighting function
chosen to b e the visibility function
jxj
w x a
The frequency domain criterion function for the reduction of unwanted artifacts and noise is
Z
I W f jH f H f j df
d
where H f is the desired lter frequency resp onse and W f is the frequency domain weighting
d
function Hentea and Algazi minimized I under a constraintonI but wenow minimize the
equivalent criterion J I I where controls the relativeweights of the two criteria
with
To develop the optimality condition is expressed in the frequency domain using Parseval s
relation the transform of a zero mean step is used and calculus of variations is applied to the
criterion J resulting in the condition
H f
W f j f W f H f H f
d
j f
Substituting the spatial weighting function of with Fourier transform
b
W f f
b f
with b ln a b ecomes
H f f W f H f
d
f W f
b b H f
j f
b f b f j f
where This is a linear Fredholm integral equation of the second kind and a discussion
of the solution of this equation in terms of eigenfunctions of an equation of similar form arising
from a related approach to lter design is given in That approach is practically useful only
if the solution to the homogeneous equation related to is known in closed form or tabulated
which is not the case for our problem Thus in the design examples discussed b elow we apply a
series solution
LowPass Image Filter Design
For a low pass image lter the desired frequency resp onse is
jf j f
c
H f
d
jf j f
c
where f is the lter cuto frequency Wecho ose W f toweight the stopband resp onse only
c
jf j f
c
W f
jf j f
c
The conditions of and are applied to the integral equation The resulting equation
is solved with the Neumann series solution an iterative approach in which an approximation to
H f used in the convolution integral on the right hand side of generates the next approximate
solution Let the k th approximation to H f be H f then the k th approximation is
k
b b H f
k
H f H f j f
k u
b f b f j f
where
jf j f
c
H f
u
jf j f
c
f
The initial condition for the iterative solution is H f H f In the examples considered we
u
observed that the solutions converged to an accuracy of three decimal places in only eight iterations
convergence was not strongly a ected by the initial condition and there was no indication that
the solution was not unique A discussion of convergence for a series solution to a similar integral
equation is given in
As an example consider the design of a low pass lter with cuto frequency f normal
c
ized and Normalizing the viewing distance to six times the image height the resulting
angular increment is minutes of arc p er pixel and the appropriate value for a in w x is
b The ripples in the step resp onse of the resulting gure shown in Figure are
strongly suppressed The frequency resp onse of lter is shown in Figure
Figure ab out here
Figure ab out here
The plot showing the tradeo b etween I and I as a function of in Figure can b e used
to cho ose to meet sp eci cations on I or I Note the large decrease in the frequency domain
rejection that is required for a small improvement in spatial domain resp onse
Figure ab out here
Of indep endentinterest is the ideally bandlimited low pass image lter which results from
setting or The lter design in this case is equivalent to minimizing spatial domain
criterion I under the constraint that the stopband energy of I b e zero
The form of remains the same but H f and the initial condition H f b ecomes the
u
frequency resp onse of the classical ideal low pass lter
jf j f
c
H f
u
jf j f
c
The step resp onse of an example lter with cuto frequency f normalized and a
c
shown in Figure has a ripple resp onse that is b etter than that of the classical ideal low pass lter
The spatial error integral I for the ideal image low pass lter is less than that for the classical
ideal low pass lter The lter frequency resp onse for this example is shown in Figure
Figure ab out here
LowPass Filter Examples
To illustrate the prop erties of the low pass image lters discussed in the section ab ove we consider
two dimensional FIR approximations of the design examples and compare p erformance with that
of an equiripple lter on a noisy image To ensure ease of comparison of the halftone repro ductions
in a journal publication a test image with substantial distortion was chosen The results clearly
extend to images of lower contrast and smaller distortions High frequency noise was added to the
original image of Figure a by high pass ltering noise with a uniform distribution pro ducing the
noisy image shown in Figure b having p eak signal to noise ratio PSNR of dB
The noisy image was ltered by an equiripple low pass lter which approximates an ideal low
pass lter and a low pass image lter design with our approach eachhaving the same maximum
deviation in the stopband and cuto frequency f normalized frequency The lowpass
c
image lter is based on the one dimensional design with having frequency resp onse shown
in Figure The applied lters are two dimensional circularly symmetric obtained from one
dimensional lters by McClellan s transformation The equiripple lter resp onse in in Figure
c shows strong ripple resp onses at ma jor transitions and it pro duces a PSNR of dB The
low pass image lter resp onse in Figure d has suppressed the ripples and pro duced an improved
PSNR of dB Thus the low pass image lter is sup erior p erceptually and it provides b etter
noise reduction
Figure ab out here
The ripple resp onses of the low pass lters are compared in the checkerb oard images of Figure
Figures b through d show the resp onses of lters having a cuto frequency f normalized
c
frequency The resp onse of a classical ideal low pass lter is shown in Figure b showing very
strong ripple resp onse The resp onse of the ideal low pass image lter of Section with and
having the frequency resp onse of Figure is shown in Figure c where the ripples have decreased
substantially Finally in the resp onse to the low pass image lter from Section with frequency
resp onse shown in Figure for the ripples are di cult to p erceive
Figure ab out here
Discussion and Conclusions
Wehave reconsidered a metho d for the design of linear lters for image pro cessing based on prop er
ties of the human visual system whichinvolves the minimization of criterion functions in b oth the
spatial and frequency domains Wehave extended this work by obtaining new theoretical results
by considering continuous lters of in nite supp ort
An imp ortant limiting result for an ideal low pass image lter having in nite supp ort has b een
obtained This ideal low pass image lter is greatly sup erior p erceptually to the classical lowpass
lter and provides a design target for the imp ortant problems of image sampling and interp olation
Wehave found that interp olation lters designed with this approach provide b etter results than
bicubic lters that approximate classical ideal low pass lters
Acknowledgment
This work was supp orted by the University of California MICRO Program Grass Valley Group
Paci c Bell Lo ckheed and Hewlett Packard
References
B R Hunt Digital image pro cessing IEEE Proceedings vol pp
G L Anderson and A N Netravali Image restoration based on a sub jective criterion IEEE
Trans Syst Man and Cyber vol no pp
A Katsaggelos Iterative image restoration algorithms Optical Engineering vol pp
T A Hentea and V R Algazi Perceptual mo dels and the ltering of high contrast achromatic
images IEEE Trans Systs Man and Cyber vol no pp
V R Algazi and M Suk On the frequency weighted least square design of nite duration
lters IEEE Trans Circuits and Systems vol no pp
J H McClellan The design of two dimensional digital lters by transformations Proc th
Annual Princeton Conf Inform Sci and Syst pp
H Chen and G E Ford An FIR image interp olation lter design metho d based on prop erties
of human vision in Proc IEEE Intl Conf Image Proc vol I I I pp
List of Figures
Comparison of lter step resp onses
Low pass image lter frequency resp onse for
Design criteria I and I as a function of
Frequency resp onse for ideal low pass image lter
Low pass ltering a Original image b Noisy image c Noisy image pro cessed
with an equiripple low pass lter d Noisy image pro cessed with a low pass image
lter
Image ltering bylow pass lters a Original image b Filtered with classical
ideal low pass lter c Filtered with ideal low pass image lter d Filtered
with low pass image lter
x
ideal
u
x pixels
Figure Comparison of lter step resp onses
f
H
f normalized
Figure Low pass image lter frequency resp onse for
I
I
Criterion
Value
Figure Design criteria I and I as a function of
f
H
f normalized
Figure Frequency resp onse for ideal low pass image lter
Figure Low pass ltering a Original image b Noisy image c Noisy image pro cessed with
an equiripple low pass lter d Noisy image pro cessed with a low pass image lter
a b
c d
Figure Image ltering bylow pass lters a Original image b Filtered with classical ideal
low pass lter c Filtered with ideal low pass image lter d Filtered with low pass image
lter
a b
c d