Image Magnification Using Level Set Reconstruction
Bryan S. Morse and Duane Schwartzwald
Department of Computer Science, Brigham Young University
3361 TMCB, Provo, UT 84602
morse@ cs. byu. edu
Abstract
Image magnification is a common problem in imaging
applications, requiring interpolation to “ read between the
pixels”. Although many magnificatiodinterpolation algorithms
have been proposed in the literature, all methods
must suffer to some degree the effects of impefect
reconstruction false high frequency content introduced by
the underlying original sampling. Most often, these effects
manifest themselves as jagged contours in the image. This
paper presents a method for constrained smoothing of such
artifacts that attempts to produce smooth reconstructions
of the image’s level curves while still maintaining imagejdelity.
This is similar to other iterative reconstruction algorithms
and to Bayesian restoration techniques, but instead
. of assuming a smoothness prior for the underlying intensity
function it assumes smoothness of the level curves. Results
show that this technique can produce images whose
error properties are equivalent to the initial approximation
( interpolation) used while their contour smoothness is both
visually and quantitatively improved.
1. Introduction
Millions of digital images available today through the Internet
and other sources are frequently downloaded and integrated
into various types of media. While modern printers
and displays support fine detail, images available electronically
are rarely of such high resolution. This is especially
true for home computing, where limited bandwidth often
makes distribution of high resolution images impractical.
Typical screen resolution images are 72 to 100 dots per inch
( dpi) while even low cost printers are 1200 dpi or higher.
Although multiresolution formats have been developed for
distributing images at both screen displayable and printable
resolutions, use of such formats is still relatively uncommon
and requires that the high resolution data be available.
Thus, many image users are left today with low resolution
images displayed or printed on high resolution devicesstill
looking like low resolution images.
Bilinear
Original Image
Bicubic Level Set
Figure 1. “ Monarch” image with 3x magnification.
Compare the results of pixel replication
( sharpest edges, worst jaggies), bilinear
interpolation ( blurred edges, less jaggies),
bicubic interpolation ( sharper edges, worse
jaggies), and level set magnification ( sharp
as bicubic interpolation, smoother contours).
Standard interpolation methods treat the problem primarily
as either fitting a function or filtering ( or both) [ 9,28]. In
either case, the reconstruction is imperfect, and false highfrequency
components are introduced into the interpolated
image [ 7, 181. As a result, they still show artifacts of the
original discretization, as indeed all magnification methods
must to some degree. These artifacts demonstrate themselves
most commonly as alignment to the original pixels
( Figure l).’ This causes what should be smooth contours in
the image to be jagged the well known “ jaggies”.
Instead of approaching interpolation as “ fitting the function”,
this paper approaches it as “ fitting the visual geometry”:
reconstructing the geometry of the original image’s
‘ Color versions of this and other images in this paper may be found in
the conference proceedings CD ROM and on the author’s web site.
0 7695 1272 0/ 01 $ 10.00 0 2001 IEEE
1 333
level curves ( spatial curves of constant intensity). Using a
differential equation, this reconstruction smooths the image
contours from an initial approximation while maintaining
fidelity to the original lower resolution image. Thus, it directly
attacks one of the the most perceptible artifacts of
image reconstruction and causes the reconstructed image to
preserve smooth contours in the original.
2. Image Interpolation
Because interpolation attempts to approximate an intensity
function or surface from sampled data, most reconstruction
methods have their roots in either fitting functions to
sampled data or in sampling theory.
Functional interpolation treats an intensity surface as a
sampled two dimensional function and attempts to fit this
function to the samples, often using polynomials of various
degree ( linear, cubic, etc.) [ 23]. For two dimensional images,
these become bilinear and bicubic functions [ 8]: interpolation
in z then interpolation in y or vice versa. More sophisticated
interpolating functions and non exact fitting of
the data can be used ( e. g., [ 16]), but the idea is the samebest
fit functions. While these methods can do a good job of
approximating the image’s intensity surface, the metric for
evaluation is typically error in intensity, not visual appeal.
Filtering approaches often outperform function fitting
approaches by recognizing the frequency domain effects of
the original sampling [ 9, 281. These methods attempt to
undo the spectrum replication caused by sampling by approximating
the effects of a ( physically unrealizable) ideal
low pass filter [ 6. 15, 17, 18, 21, 221. Convolution by approximations
to a sinc function provide reasonable approximations
to such an ideal filter, but again, the objective is to
minimize pass through of the offending frequencies rather
than considering visual properties of the resulting image.
Edge directed interpolation algorithms [ 11 fit smooth
subpixel edges to the image and use these to prevent crossedge
interpolation. These methods create sharper edges,
and by fitting the edge contours spatially rather than functionally,
they also produce smoother edges. While these
methods in part address the effects of interpolation on visual
contours, they raise two questions: how do you define
the edges of interest, and what do you do elsewhere? Rather
than trying to extract specific image curves for smooth reconstruction,
the method presented here works by reconstructing
smooth approximations of all of the image levelset
contours simultaneously.
PDE based approaches to level set interpolation have
heen used previously in the literature [ 4, 5, 14, 191.
Caselles, et al. [ 5] showed that the only operators satisfying
certain requirements, particularly closure under interpolation,
are those involving second derivatives in the gradient
and tangent directions. One variation of this method
Figure 2. Isophote reconstruction errors introduced
by interpolation. When a black andwhite
edge ( a, magnified) is bicubicly interpolated
( b), it shows artifacts of the original
sampling. Individual level curves ( c, eight levels)
are jagged instead of smooth. When a
similar edge with more gradual transition ( d,
magnified) is interpolated ( e), the isophotes
( f, 20 levels) are also jagged.
( minimizing curvature in the gradient direction) can be used
to smoothly interpolate missing contours between known
curves or points. However, we want to interpolate the contours
themselves, not between them. Their second variation
( minimizing curvature in the gradient tangent direction,
which we use here) has been used for disocclusion [ 141
or inpainting [ 4] with remarkably successful results. However,
the solution for this equation is not unique [ 5], requiring
an additional constraint that minimizes total variation.
These approaches are also developed only for connected regions,
not filling in contours from isolated points.
3. Level Curve Interpolation
One of the most visually significant geometric properties
of images is their level curves or isophotes ( curves of
constant intensity). These curves are what give images their
perceptual contours. Although level curves don’t capture all
geometric information that one might want in analyzing image
content [ 1 I], reconstruction of the isophotes produces a
visually convincing reconstruction of the image.
Figure 2 shows an example of the effects of interpolation
on isophotes. If a simple black and white edge ( 24 is
interpolated bicubicly ( 2b), the result shows the underlying
pixel grid. The effects of this interpolation on the isophotes
can be seen by examining individual level curves in the reconstruction
( 2c).
Performing the same operations on a blurred edge ( intermediate
greylevels in the transition) shows similar results.
Even though each original level curve ( 2d) is straight, the
level curves of the resulting reconstruction ( 2e) are not ( 2f).
1 334
I
Original Isophotes Original Isophotes
pixel centers I pixel centers I
lsophote’curvature Srnootheh isophotes
visible as jagged artifacts
Figure 3. Constrained level set smoothing.
Smoothing the level curves as much as possible
while still maintaining level curve topology
and the values at the known pixels produces
a convincing image reconstruction.
These examples suggest an alternative approach to image
reconstruction: smooth fitting of level curves based on the
original image constraints ( Figure 3). This can be phrased
as a reconstruction/ optimization problem: find the set of
level curves that
0 Preserve level set topology,
Preserve intensities at known positions ( pixels), and
0 Are each as smooth as possible.
Notice that the problem, when stated in this way, follows
the general form of a constrained optimization problem and
uses the common “ smoothness” prior [ 3]. However, it is not
intensity surface smoothness but level set contour smoothness
that serves as the optimization prior.
Although there is no closed form solution for these constraints
( indeed, no unique solution exists without additional
constraints), this can be approached iteratively in a
fashion similar to gradient descent minimization:
I . Begin with an approximation of the magnification by
using existing interpolation algorithms, and
2. Iteratively minimize isophote curvature while preserving
fidelity to the original lower resolution image.
We may use any existing interpolation method as an initial
approximation, but as with all optimization methods,
the better the initial approximation, the better the result. In
particular, level set reconstruction smooths contours while
preserving edge sharpness, so it is only as sharp as the original
approximation used. We now turn to an explanation
of this second part: the iterative minimization of isophote
curvature constrained by the sampled image.
4. Level Set Manipulation
At first glance, manipulation of the level curves requires
explicitly finding and fitting each curve, much as individual
edges must be found and fitted in edge directed approaches
11. However, such explicit curve fitting is not
Increasing the
intensity here
movesthe .,
lewl CUM
, iDnetec nresiatys ihnegr teh e
moves the
level curve
Figure 4. Changes in pixel intensities move
level curves spatially according to ( 2).
necessary. Instead, we can directly manipulate the level
curve passing through each pixel respectively by manipulating
the intensities at that pixel.
Osher and Sethian [ 20,25] have demonstrated curve evolution
techniques for manipulating 1  dimensional curves in
2 dimensional domains by embedding the curve as a level
curve of a function $ : IR2 3 IR of two variables. Altering
this function alters its level curves and thus alters the specific
level curve that represents the curve of interest. The
relationship between changing the value of the function q5
and moving the curve in the direction of its normal is
where F is the speed of movement of the curve in its normal
direction and q5t is the change in the embedding function 4
with respect to time.
Research by Alvarez, Lions, and Morel [ 2], Sethian,
Malladi, and Kimmel [ 1 1, 12,13,25], and others has shown
that one can extend this to describe how changes in individual
pixel intensities alter their local level curves. The image
itself ( I) assumes the role of the embedding function 4:
It = FIIVIII ( 2)
By using the negative isophote curvature  K. as the speed
F, level curves contract at a rate proportional to their
curvature places of high curvature contract more quickly
than smoother parts of the curve ( Figure 4):
It =  K. llVI(( ( 3)
This can be used to perform edge preserving smoothing,
noise removal and other image enhancement [ 2, 11,12,13,
251, and shape evolution and description [ lo]. We use a
constrained form of this to reduce the artifacts of imperfect
reconstruction. Whereas other applications of level set
smoothing attempt to enhance the original image, we are
attempting to reconsfruct the original image ( warts and all).
Calculation of the isophote curvature IC similarly does
not require explicit representation of the level curve. It can
be calculated from local derivatives of the intensity 1271:
1 335
Substituting this into ( 3) and recognizing that JIVII( =
( I: + the desired flow is
5. Constrained Smoothing
We implement the differential equation in ( 5) using a
difference equation based on Euler’s method [ 23]. In this
form, the method produces the results expected of level set
smoothing: jagged edges are smoothed and thus the reconstruction
artifacts are diminished. However, standard levelset
smoothing does more than just smooth jagged edges it
also smoothes away features of objects, ultimately shortening
level curves until they disappear altogether. The result
is an appealing but oversmoothed image ( Figure 6a).
The method presented in this paper goes beyond simple
level set smoothing by imposing additional constraints
that preserve accuracy to the original image ( anchors), preserve
level set topology relative to these anchors, identify
and smooth jaggies rather than arbitrarily shorten all curves,
and to generally avoid oversmoothing. The combined effects
of these constraints are illustrated in Figure 6.
5.1. Image Anchors
Curvature flow provides two of our three goals: preservation
of isophote topology and isophote length/ curvature
minimization, but we must add an additional constraint to
preserve intensities at the original pixels:
( 6)
0 for an original pixel location
It = {  li\ lVIII otherwise
As the level set contours “ flo~” t, h ey are constrained by
the unchanging anchor pixels ( Figure 5). This is similar to
constrained curve evolution as presented in [ 25].
anchor
calculated
new contour
( not allowed)
contour
Figure 5. Anchor constraint. Original sampled
pixels retain their values during the contour
smoothing process. This preserves fidelity
to the original image.
5.2. Explicit Topology Constraint
If the level sets are moved too quickly ( too large a step
size for the numerical implementation of ( 6)), level curves
may move past their associated anchors. While methods
for ensuring the stability of level set smoothing have been
shown to preserve topology [ 26] and [ 24], such methods
require small step sizes that are often too slow for our intended
application. One may preserve the level set topology
of the initial reconstruction, especially relative to lowresolution
anchors, by introducting an explicit topology
contraint2 as folIows:
I. Calculate the desired next iteration value for all pixels
based on ( 6).
2. For each pixel that is increasing, limit its value to
less than the lowest next iteration value of the greatervalued
neighboring pixels.
3. Similarly, for each pixel that is decreasing, limit its
value to more than the largest next iteration value of
the lesser valued neighboring pixels.
The net effect of this is to ensure that all greater valued
neighbors stay larger and all lesser valued neighbors stay
smaller, thus preserving topology.
With this explicit topology constraint, coupled with a
step size reduction schedule, reasonable results can be calculated
for color images in as few as five iterations.
5.3. Inflection Constraint
As pointed out previously in Section 2, there is no unique
solution to this optimization [ 5]. This is because there
are constraints only on the smoothness of the curves themselves,
not between the curves. However, we don’t want to
arbitrarily smooth between the curves as in [ 4, 141 because
we want to maintain the sharpness of our original approximation,
By the same token, we don’t want to artificially
introduce discontinuities either. If allowed to continue constrained
only by original pixel anchors, level curve shortening
flow tends to act like an elastic band shrinking to fit
a set of nails hammered into a board: the minimum length
solution is piecewise linear ( Figure 6c). Indeed, if there is
only a single anchor bounded by a level curve, the curve
will shrink to this single point.
It must be remembered then that our goal is to smooth
contours, not simply shrink them. To smooth jagged contours
while preventing other forms of curve shortening, we
introduce a constraint that separates jagged contours from
curves without local inflections. This constraint simply requires
that if a pixel is increasing or decreasing in value, at
* It must be emphasized that this part of the algorithm is not necessary
unless one wants to use a more aggressive step size than stability requirements
would normally dictate.
I
1 336
b) Anchor constraints only
d) Anchor, topology, and inflection constraints
Figure 6. Effects of the constraints described in Sections 5.1 5.3. Unconstrained level set smoothing
oversmooths the image, removing jaggies but losing significant details ( a). Anchors help to constrain
the flow somewhat, but the curves may move past their anchors, producing results similar to unconstrained
smoothing ( b). Topological constraints allow the anchor pixels to serve their function, even
with an aggressive step size. However, small closed contours that surround a single pixel are still
allowed to collapse unconstrained. Likewise, contours between similar valued pixels are allowed
to shorten unconstrained until they produce piecewise linear segments, giving an almost polygonal
look to some of the contours ( c). These problems are addressed by the inflection constraint ( d).
least one of the neighboring pixels must change in the opposite
direction. ( See Figure 4.) Thus, Jagged contours may
be smoothed only by simultaneous pulling “ in” on convex
parts and “ out” on neighboring concave parts of the curve.
Already convex curves are not allowed to change.
separation. In particular, the anchor constraints guarantee
that any separation is sub pixel with respect to the original
image. Smoothing their level sets independently following
independent interpolation appears to produce color separation
no worse than the initial interpolation itself.
5.4. Color Images We have also experimented with a method based on Beltrurnijow
[ 1 11, which treats a color image not as three separate
2 dimepional manifolds but as a single 2 dimensional
manifold in a 5 dimensional space ( two spatial plus three
color dimensions). Our results using this method were,
however, not appreciably different from those produced by
treatment of the individual color planes independently.
Level set reconstruction can be extended to color images
by applying the constrained flow to the individual color
planes separately. Although this would appear to be susceptible
to separation as each plane flows independently, our
experience is that the constraints are sufficient to limit the
1 337
6. Results 7. Conclusions and Future Work I
Results ? f level set smoothing can be seen in Figures 7
10. In each case, the image enhanced by constrained levelset
smoothing preserves the sharpness of the original ( bicubicly
interpolated) approximation while reducing artifacts.
6.1. Quantitative Comparisons
A standard method for evaluating interpolated images is
to measure the error introduced by the interpolation as compared
to a “ perfect” interpolation. To measure this, we first
take a higher resolution image, reduce it by a factor f, then
enlarge it by the same factor f using reconstruction, and
compare the reduced then magnified image to the original.
Table 1 shows that the mean squared error for smoothed
bicubicly interpolated images is only modestly better than
for bicubic interpolation alone. This is not surprising,
though. The central idea of level set reconstruction is to
produce images that look better while staying true to the
data. While images smoothed with constrained level set
smoothing are not necessarily “ more accurate” than such
images without smoothing, their error is no worse and they
are more visually appealing.
To measure this other objective, producing images with
smoother contours, we also measured the mean absolute
level set contour curvature E{ IKI>. This might seem like
“ begging the question’ measuring the property we’re explicitly
trying to minimize but it is interesting to compare
the measurements of this quantity to visual impressions.
For example, bilinear reconstruction consistently produces
measurably smoother contours than bicubic reconstruction
while introducing more error agreeing with common wisdom
that bilinear reconstruction looks “ less jagged” but
“ more blurred”. Bicubic reconstruction, using a higherorder
fit to the original points, introduces less error but also
produces more jagged contours ( Figure 1). In each case,
bicubic reconstruction followed by level set smoothing produces
error results comparable to bicubic reconstruction but
produces smoother contours. Indeed, the contour smoothed
interpolated images consistently produce contours that are
significantly smoother than bicubic or bilinear reconstruction
without introducing additional intensity error.
For comparison, we also compared our results to a 3 x 3
box filtered original image. ( One must compare to a lowpass
filtered version of the original because of the low pass
filtering required prior to reduced sampling, unless one is
trying to sharpen as well as interpolate.) Bicubic interpolation
followed by contour smoothing consistently produces
results that are as smooth as the low pass filtered original.
Level set reconstruction, by focusing on visually significant
properties of interpolation artifacts, can significantly
improve the results of existing methods for image magnification.
The results have error characteristics comparable
to the initial interpolation method but with contour smoothness
comparable to the image prior to downsampling.
As with any iterative optimization technique, level set reconstruction
depends heavily on the initial approximation.
We have tested level set reconstruction using various initial
interpolation methods, and in all cases it significantly improved
the results. The effect of: level set reconstruction on
even better initial approximations should be explored.
Allebach [ 11 has noted that interpolation should consider
the original image samples as area based samples, not point
samples. We are pursuing variations of anchor constraints
that act as area average, not point, anchors.
Another area of continued exploration is to see whether it
is possible to achieve better results for larger magnifications
by applying a smaller magnification, performing level set
reconstruction, more, rnagnification, more level set reconstruction,
etc. We have tried this by comparing 4x magnification
followed by level set smoothing to a process consisting
of 2x magnification, smoothing, 2x magnification, and
smoothing again. The results are marginally improved, but
not significantly. However, it does suggest that this might
be a better way to perform larger magnifications.
Acknowledgments
This research was supported in part by funding from
Adobe Corporation. Our thanks go to Greg Gilley, Martin
Newell, Gregg Wilensky, and Peter Ullmann for their
helpful comments and support.
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