METHODS USED IN INCREASED RESOLUTION
PROCESSING
Polygon based interpolation and robust log-polar based registration.
Stéfan van der Walt, Ben Herbst
University of Stellenbos
h, South Afri
a
stefansun.a
.za
Keywords: geometry, interpolation, registration, image-pro
essing, astronomy, log-polar transform
Abstra
t: A polygon-based interpolation algorithm is presented for use in sta
king RAW CCD images. The
algorithm improves on linear interpolation in this s
enario by
losely des
ribing the underlying
geometry. 25 frames are sta
ked in a
omparison. When sta
king images, it is required that these
images are a

urately aligned. We present a novel implementation of the log-polar transform that
over
omes its prohibitively expensive
omputation, resulting in fast, robust image registration.
This is demonstrated by registering and sta
king CCD frames of stars taken by a teles
ope.
1 Introdu
tion
In this paper we des
ribe two te
hniques whi
h
have proved useful in the
onstru
tion of reso-
lution rening algorithms. The rst is polygon-
based interpolation. When sta
king RAW CCD
readouts, the frames are aligned and interpola-
tion is applied to obtain values on a
ommon grid.
Unlike bi-linear interpolation, that
an
ause blur
and is not well suited to the geometry of the prob-
lem, geometri
interpolation is designed to take
not only the positions but also the shape of pixels
into a

ount.
The se
ond te
hnique is image registration or
alignment using the log-polar transform (LPT).
While the LPT has been proposed for this pur-
pose before, the
omputational
ost is prohibitive.
We show how the LPT
an be used in a way that
requires mu
h fewer
omputations, without
om-
promising robustness or a

ura
y.
2 Polygon-based interpolation
Given an image, or grid of pixel values, we
would like to
al
ulate the value at an arbitrary
point in the grid. Bi-linear interpolation esen-
tially uses a weighted average of the
losest four
pixels, based on their distan
es from the target
position [11℄. If pixels are seen as innitely small
points in spa
e, this method yields a

urate re-
sults.
When, however, we
onsider that pixels have
nite areas, this method is only reasonably a
-
urate when the target pixel is horizontally and
verti
ally aligned with the original grid. As soon
as rotation or skew
omes into play, linear inter-
polation no longer resembles the underlying ge-
ometry, as is the
ase when sta
king raw CCD
frames. The CCD sensor
omprises a number of
light-sensitive
apa
itors, arranged in a grid. For
the purpose of this arti
le we will assume that
the shape of these
apa
itors is square, but the
method des
ribed allows for any geometry, also
in the arrangement on the sensor itself (permit-
ted that there are no holes, as in the
ase of the
olour-masked CCD).
Given a target pixel position, (r, s), and a
transformed (for example rotated and translated)
sour
e pixel at (m,n) with value Wm,n, we would
like to
al
ulate the
ontribution of the sour
e to
the target. We
onstru
t a quadrilateral polygon
at the target position with verti
es
[
xt
yt
= s s + 1 s + 1 s
r r r + 1 r + 1
]

as well as at the referen
e position with verti
es
[
xr
yr
= n n + 1 n + 1 nm m m + 1 m + 1
]
.
The verti
es xr and yr are then inversely
transformed to align with the target grid. The
sour
e pixel value is weighted with the area of
the polygon interse
tion between sour
e and tar-
get pixels (see Fig. 1). This method has the ad-
vantage that it
an be adapted a

urately for
any kind of spatial transformation, although it
may require adding more verti
es to support non-
linear transformations.
Using this te
hnique, 25 frames provided by
the NASA Pathnder mission were sta
ked [10℄.
The frames were aligned using lo
alised fea-
tures [7℄, with trivial outlier reje
tion. A high-
resolution grid was spe
ied after whi
h the poly-
gon interse
tions were
al
ulated using the Liang-
Barsky algorithm [16℄. The results are shown in
Fig. . Note that this is not a super-resolution al-
gorithm (although the interpolation
an
ertainly
be
ombined with su
h a statisti
al estimation
pro
ess), but simply in
reased resolution sta
k-
ing.
3 Registration
Registration algorithms
an be divided into
two broad
lasses: those that operate in the
spatial and frequen
y (i.e. Fourier) domains,
respe
tively. In the spatial domain, there are
sparse methods in
luding lo
al des
riptors, that
depend on some form of feature extra
tion, and
dense methods that operate dire
tly on image val-
ues su
h as opti
al ow and
orrelation. The
two
lasses generally dier in that the spatial
methods are lo
alised, whereas the frequen
y do-
main methods [12, 3, 5, 4℄ operate globally. At-
tempts have been made to bridge this gap, by
using wavelet and other transforms to lo
ate
information-
arrying energy [2℄. These have been
met with varying su

ess.
Ea
h registration method has its own parti
u-
lar advantages and disadvantages. Fourier meth-
ods, for example, are fast but ina

urate, suf-
fer from resampling and o

lusion ee
ts [13, p.
1425℄, and only operate globally. Iterative regis-
tration, on the other hand, is highly a

urate but
extremely slow, and prone to misregistration due
to lo
al minima in the minimisation spa
e.
These problems led to the development of
methods based on lo
alised interest points [1, 7,
14, 15℄, su
h as the s
ale-invariant feature trans-
form (SIFT) [9℄, the fast Speeded Up Robust Fea-
tures (SURF) [6℄ and others [8℄. All these meth-
ods depend on unique lo
alised features, whi
h
are available in many images. There are, how-
ever,
ases where it is very di
ult to distinguish
one feature from another without examining its
spatial
ontext.
As an example, we will use frames re
orded
by a CCD mounted on a teles
ope pointing at a
deep-spa
e obje
t. It is very di
ult to nd fea-
tures to tra
k in these images, be
ause the stars
(all potential features) are virtually identi
al and
rotationally invariant. Sin
e lo
al features fail,
and global methods are slow and unreliable, we
would like to nd an algorithm that
an bridge
the gap.
We will pro
eed to show that the log-polar
transform (LPT) is an ideal
andidate. While
previously its use has been limited due to its high
omputational
ost, we develop ways of redu
ing
those
osts and making the LPT behave more like
lo
al features.
4 The log polar transform
The log-polar transform (LPT) spatially
warps an image onto new axes, angle (θ) and log-
distan
e (L). Pixel
oordinates (x, y) are written
in terms of their oset from the
entre, (xc, yc),
as
x¯ = x− xc
y¯ = y − yc.
For ea
h pixel, the angle is dened by
θ =
{
arctan
(y¯
x¯
)
x¯ 6= 0
0 x¯ = 0
with a distan
e of
L = logb
(√
x¯2 + y¯2
)
.
The base, b, whi
h determines the width of the
transform output, is
hosen to be
b = eln(d)/w = d 1w ,
where d is the distan
e from (xc, yc) to the
orner
of the image, and w is the width or height of the
input image, whi
hever is largest.
When warping images, it is not possible to use
the forward transform. Sin
e we use dis
rete
o-
ordinates (integer x and y values), more than one