A Robust Affine Matching Algorithm Using an Exponentially Decreasing Distance Function
Axel Pinz, Manfred Prantl, Harald Ganster
Institute for Computer Graphics
Technical University of Graz
Muenzgrabenstr. 11
A-8010 Graz, AUSTRIA
email: pinz@icg.tu-graz.ac.at
Abstract:
We describe a robust method for spatial registration, which relies on the coarse correspondence of structures extracted from images, avoiding the establishment of point correspondences. These structures (tokens) are points, chains, polygons and regions at the level of intermediate symbolic representation (ISR). The algorithm recovers conformal transformations (4 affine parameters), so that 2-dimensional scenes as well as planar structures in 3D scenes can be handled. The affine transformation between two different tokensets is found by minimization of an exponentially decreasing distance function. As long as the tokensets are kept sparse, the method is very robust against a broad variety of common disturbances (e.g. incomplete
segmentations, missing tokens, partial overlap). The performance of
the algorithm is demonstrated using simple 2D shapes,
medical, and remote sensing satellite images. The complexity of the algorithm is quadratic on the number of affine parameters. Keywords:
Affine Matching, Spatial Registration, Information
Fusion, Image Understanding Category:
I.2.10, I.5, I.4
1 Introduction
In Computer Vision, the establishment of correspondence between different sources of visual information is an
important issue. Affine matching has mainly been used for image to
model matching (e.g.[Beveridge et al., 1990], [Collins and Beveridge, 1993], see Fig. 1.a) and for image to image matching
(e.g. [Zabih and Woodfill, 1994], [Collins and Beveridge, 1993], [Flusser and Suk, 1994], see
Fig. 1.b), often with the purpose of spatial
registration of images [Brown, 1992]. Our motivation for this work
is driven by the idea of a general framework of "Information
Fusion in Image Understanding"
[Pinz and Bartl, 1992a], [Pinz and Bartl, 1992b], [Bartl et al., 1993]. In order to be able to deal
with multiple visual information on all levels of abstraction, proper
transformations of three different kinds are required:
1. spatial (coordinate) transformations,
2. radiometric transformations, and
3. transformations between
different levels of abstraction (signal, pixel, feature, and symbol
level [Luo and Kay, 1992]. Information that can be compared with respect to the above transformations is said to be in registration. While recent relevant work on fusion simply assumes Page 614
prior registration of the source images
(e.g.[Maître,1995, Clement et al., 1993], [Burt and Kolczynski, 1993], [Toet, 1989], we want to fuse visual information without the requirement of prior manual registration. In this paper we concentrate on the following case: - different source images, - different coordinate systems. Spatial registration can be
achieved by a conformal transformation (4 affine parameters), - the match is established at the ISR level (intermediate
symbolic representation [Brolio et al., 1989]), see Fig. 1.c.
Figure 1: Different types of matchingISR supports associative retrieval, spatial relations, and multi-level representations and is well suited for many image understanding
applications [Draper et al., 1993]. A so-called token (`image event')
represents one of the following `location' features: a point, a line,
a chain of points, a polygon, and a bitmap representing a region. For
each token, an arbitrary number of additional features can be
calculated (e.g. shape, spectral and texture parameters). Related
tokens are collected in a `tokenset'. Figure 2 gives
a simple example for an image of a purely 2-dimensional scene with 2D
shapes (`shapes' image Fig. 2.a). A region-based
segmentation of this image results in a tokenset of 11 bitmaps
visualized in Fig. 2.b. The process of segmenting an image
into a tokenset is covered in detail in [Pinz, 1994]. In the
context of this paper we want to emphasize that the intelligent use of
ISR (e.g. application of constraints, perceptual grouping, elimination
of irrelevant tokens) often leads to relatively sparse
tokensets, thus reducing the amount of data dramatically (as compared
to the pixels of the original image). This results in a substantial
reduction of computational complexity for the subsequent process of
matching of two different tokensets (e.g. 11 tokens versus 258000
pixels for Fig. 2).
Consider the situation sketched in Fig. 1.c: Starting
from two different images of the same scene, segmentation and feature
extraction processes are used to create two tokensets. Under certain
assumptions, the spatial relationship between the tokensets (and the
images) can be modeled by an affine transformation.
Figure 2: Region based segmentation of a 2D 'shapes' imageThis holds
for 2-dimensional scenes, as well as for planar structures in 3D
scenes (see [Collins and Beveridge, 1993], [Grimson et al., 1994], [Flusser and Suk, 1994] for more detail):  with and denoting locations in image 1 and image 2, respectively. A general affine transformation is defined by six independent parameters (a...f). However, a special case of the general affine transformation, the conformal transformation with four affine parameters is often sufficient for the given application, or the general affine problem can be converted
to a conformal one [Collins and Beveridge, 1993]. The conformal transformation can
be written as  consisting of a translation vector t = , a rotation angle , and a scaling factor s. While Grimson et.al. [Grimson et al., 1994] distinguish between two
different types of affine matching algorithms (hypothesize and
test and geometric hashing), Collins and Beveridge
[Collins and Beveridge, 1993] find four categories (key-feature algorithms, generalized Hough transformations, geometric hashing, and
constraint-based tree search). Our algorithm could be categorized as
a hypothesize and test approach which allows for many-to-many mappings between features.
A review of similar and related work in the area of object recognition, where the task is to determine the transformation between a model and an image, reveals many recent publications (e.g.[Basri and Jacobs, 1995, Rucklidge, 1995]), as well as fundamental work dating back to the mid 80s (e.g. [Ayache and Faugeras, 1986, Borgefors, 1988]). The general approach of Page 616
performing a search in transformation space for the best matching transformation, which is also followed in this paper, has been explored in the past, e.g. for the purpose of locating images of rigid 3D objects. The algorithm proposed by Cass [Cass, 1992] takes into account the inherent geometric uncertainty of data features when matching a model to image data. By assigning an uncertainty region to each data feature he restricts the space of possible transformations and arrives at an algorithm of polynomial complexity. However, his algorithm seems less suited for the matching of rather dissimilar patterns (like Fig. 11 or Fig. 14). Lowe, [Lowe, 1987] also matches 3D rigid model data with 2D images, exclusively using edges as primary features, which are subsequently segmented, perceptually grouped, and matched. His algorithm applies Newton's method and requires partial first order derivatives. Perhaps closest to our work we found the `Hierarchical Chamfer Matching' algorithm by Borgefors [Borgefors, 1988] using a quadratic distance function to measure the goodness of match. While she briefly mentions the general applicability of the method to many different kinds of features, she only reports on the matching of edges. In this context we want to point out that our approach concentrates on the capability of using any type of tokens (points, edges, lines, regions, as well as quite specialized and application oriented types, e.g. the blood vessels described in section 3.1.2, which are modeled as tubes), and thus could best be described as `affine matching of intermediate symbolic representations' or `affine matching at ISR level' (see Fig. 1.c). The paper is organized as follows: We start with a detailed
description of the affine matching algorithm, proceed with several
experiments on the recovery of simulated transformations and `real'
transformations between data sets originating from different sources, and finally discuss the performance of the algorithm.
2 The Affine Matching Algorithm
We are given two sets of tokens. One is the source and the other the
target tokenset for the affine matching algorithm. The aim now is to
find the conformal transformation to bring the source tokenset into
spatial registration with the target tokenset. The way we chose to
tackle this problem is to formulate it as the maximization of an
objective function as defined below.
2.1 The Objective Function
The target tokenset is first transformed into a binary image with the resolution of the image depending on the required accuracy. Region
tokens are not drawn with all their pixels set but just the border
pixels, as the border determines the position, scale and rotation. Now for each background pixel we calculate an inverse distance
function (see section 2.2) to its closest foreground pixel
(i.e. to the pixel belonging to the tokenset). In order to keep the
computational burden low, we do not transform the whole source
tokenset with the transformation in question, but just a number of
extracted sample points (described in section 2.4). The
objective function to be maximized is then defined as  Page 617
with being the number of sample points, the parameter vector for the conformal transformation and the coordinates of
the transformed sample points. Dist ( ) denotes the pixel value
at position ( ) of the distance image. There are, of course, some limitations on the allowable range for the translation, scale and rotation parameters. Otherwise a scale of zero would always be optimal in the sense that it would transform all
sample points onto one position and would result in a maximum value
for the objective function if that position is on the target tokenset. The limitations on the parameter range are a first coarse guess on the expected transformation values provided by the user and are
implemented as strong penalty terms in the evaluation of the objective function.
2.2 The Inverse Distance Function
The first distance function that might come into one's mind is the
Euclidean distance between a background pixel and its nearest
foreground pixel (or to be more precise: an inverse Euclidean distance having a maximum at the location of the foreground pixels and then falling off like a 45 degree ramp towards the
background). However, this will in general lead to an objective function which is not very robust and which can easily be fooled by points resulting from an incorrect transformation which do not hit the target tokenset but are just somewhat nearby. Fig. 3 depicts such an example. 
Figure 3: Due to a segmentation error a feature is missing in the target tokenset. Even though the sample points from the wrong feature do not match the target feature exactly, they achieve a higher score in the matching process simply because they are more numerous.
We want to have those points which hit the target tokenset exactly (within the accuracy of the image resolution) to obtain a higher weight than those which Page 618

Figure 4: An exponentially decreasing distance function gives stronger weights to points very close to the true location and thus makes the matching more robust. are just close to the true position. One can
achieve such an effect by choosing an inverse distance function that
falls off exponentially with the distance from the foreground pixels
(see Fig. 4). In such a case the number of transformed
points arising from an incorrect transformation and being close to the
true position must be much higher to result in a value for the
objective function that is larger than what we would have computed by using the correct transformation. Hence, such a choice of the distance function will make the whole matching procedure more robust against outliers. Figure 5.b-d show an intensity coded distance function corresponding to the tokens of Fig. 2.b.
2.3 Optimization Procedure
Now that we are able to compute an objective function for every
parameter vector of the conformal transformation, we have formulated
the problem of spatial registration as that of finding the global
optimum of a function. There exist many different methods for finding such an extremum in the literature. Most of them deal with cases where the global optimum is hid among many local extrema and considerable effort is spent to overcome the problem of getting stuck in one of those local extrema. However, in general, these methods require along time to converge and still do not guarantee to arrive at the global optimum. Provided we are dealing with tokensets that are not too dense (a
requirement we can often fulfill by choosing the proper parameters for the low-level segmentation tasks that produce the tokensets), we can, however, assume that the chosen objective function will not have that many local extrema. It will exhibit a structure that varies smoothly over large portions of the parameter space. These considerations lead us to an optimization procedure where we simply start local optimization from a couple of different seed points and take the
extremum with the highest score. Even though this procedure does not
assure to arrive at the global optimum, it performed well in
practice. Furthermore, it has the advantage that we can use fast and
reliable local optimization procedures like Powell's method (see
[Press et al., 1992].
2.4 Extraction of Sample Points
A tokenset can contain various types of tokens, including points,
lines and regions. Generally, the sample points are distributed
randomly over the tokenset. Page 619
Figure 5: Sample point selection and intensity coded inverse
distance function.
In some cases there are substructures
within the tokenset that are of higher significance than others and
should, therefore, get a higher share of the total number of sample
points. An example for such a structure are the vessels of
Fig. 9.b. There are many short vessels but just a few
long and significant ones. In order to achieve a selective distribution of sample points a weight (belief value) can be assigned to substructures of the tokenset. The actual attachment of the weight to individual tokens is up to the module generating the tokensets and is not the concern of the matching process. Tokens having a higher weight will be treated with
preference when the selection of sample points takes place. This
causes the matching algorithm to give priority to such substructures
in finding the correct match. Figure 5.a shows an
example for the sample point selection. Page 620
2.5 General Outline of the Algorithm
The general outline of the algorithm is depicted in
Fig. 6. The structure is simple and easy to implement. One
can use any local optimization technique often available from standard libraries. Figure 6: Schematic overview of the affine matching algorithm.
3 Experimental ResultsWe tested the algorithm on a variety of different image examples. Some of them were generated by simulating a conformal transformation,
whereas others stem from real multisource applications of medical
imaging and remote sensing. Page 621
3.1 Simulated TransformationsSimulated test data was used in order to provide the exact ground
truth for evaluating the accuracy of the algorithm and to incorporate some of the common disturbances generally found in many applications (like fragile segmentation, only partial overlap, etc.). 3.1.1 Shapes ImageThe first experiment was performed on a simple shapes image
(Fig. 2.a). The extracted tokenset (depicted by
Fig. 2.b) was disturbed by erasing some of the tokens and
adding artifacts corresponding to a noisy feature extraction process
(Fig. 7.a). The resulting tokenset was matched by our
algorithm onto a tokenset (Fig. 7.b) which in turn was
obtained by transforming the original tokens with a known conformal
transformation. Figure 8.a illustrates the initial
spatial relation of the two sets and Table 1 summarizes the used transformation parameters and the result computed
by the proposed method. By visually inspecting
Fig. 8.b one can confirm that the algorithm performed
well. Figure 7: Shape tokenset3.1.2 Medical Image
Our second experiment with known transformations was performed on an
image from a medical application. In contrast to the shape
experiment, the image itself and not the extracted tokenset was
transformed. An algorithm for the extraction of vessel structures modeling vessels as tubes of varying radius [Huang and Stockman, 1993] was applied onto both images to provide the tokensets. The vessel
extraction was done with slightly varying input parameters in order to simulate the fragility of low level segmentation
procedures. Figure 9 and 10 illustrate the
initial data set, the vessel segmentation results and the matched
tokensets. Again visual inspection and comparison of the data in
Table 2 reveals good performance of the affine matching
algorithm. Page 622
 Figure 8: Matching of shape tokensets
Table 1: Results for the shapes tokenset Table 2: Results for the simulated medical data setPage 623
Figure 9: Retinal images with superimposed vessel tokensets Figure 10: Matching of vessel tokens3.2 Real Data SetsThe real data sets consist of images that
exhibit a mutual transformation closely resembling (but not exactly
matching) a conformal transformation. Since the 'correct' conformal
transformation is unknown, we compare the result of our algorithm to
'measured' transformations. They are obtained by manual selection of
corresponding points in the two images and calculating the affine
parameters by least-squares adjustment. 3.2.1 Medical Data SetThe medical data set shows images of the human retina acquired by a
Scanning Laser Ophthalmoscope (SLO)[Webb et al., 1987]. The images were generated using different laser sources and examination methods (e.g. argon-blue laser, Page 624
fluorescein angiography). In order to detect
pathological changes and to perform the appropriate treatment
[Pinz et al., 1995a], [Pinz et al., 1995b] they have to be spatially registered. Similar to our experiment using a simulated transformation (section 3.1.2), the features used for registration are again the vessels
on the eye background. The reason for using them is that they are the only retinal features visible in all images acquired by the SLO. As with our previous experiments, Figure 11 and
Fig. 12 show the initial data, the tokensets and the
result of the matching process. Table 3 shows that the
results of the affine matching are very close to the manually
'measured' transformation. Figure 11: Retinal images with superimposed vessels Figure 12: Conformal matching of vessel tokensetsPage 625
Table 3: Results for the medical data set3.2.2 Remote Sensing Data SetFinally, we want to give an application example from satellite image
remote sensing. Figure 13 shows the near infrared
(approx. 0.8 ) channels of a Landsat TM image and a digitized
MKF6 image captured during the Russian AUSTROMIR mission. A land use
classification of these images was performed. We concentrate on the
forest regions represented as tokensets shown by
Fig. 14. As in the 'shapes' example described above, we use
the borders of the forest regions for affine matching. The original
images as well as the land use classifications were supplied by the
University for Bodenkultur and are described in more detail in
[Bartl and Pinz, 1992]. Figure 13: Two remote sensing images of an area in Lower Austria (Wr. Neustadt)
Figure 14: Forest regions resulting from land use classification Table 4: Results for the remote sensing data setFigure 15 shows the results for two different runs of the affine matching algorithm, the corresponding parameter settings are
given in Table 4. The Landsat tokenset is transformed to
the geometry of the MKF6 tokenset. In the case of the correctly
recovered conformal transformation (Fig. 15.a) we show an
overlay of both images and tokensets, while Fig. 15.b shows
an overlay of the correctly transformed Landsat image and the
incorrectly recovered Landsat tokenset over MKF6 image and tokenset. Page 627
Figure 15: Affine matching results
algorithm. There are several preconditions on the data which are
required for the algorithm to work reliably: -Rough estimates for the scaling s should be given. It is much better to select source and target tokenset in a way that s 1 (i.e. the source tokenset is scaled down), so that the majority of selected
sample points will be transformed to the interior area covered by the target tokenset. -A reasonable amount of overlap of the two
tokensets is required, otherwise the algorithm will only work in cases of very sparse tokensets with very dissimilar shapes. -Both tokensets have to be sparse. In the example shown by
Fig. 15.b, the information in the upper right corner of the
target tokenset (Fig. 14b) is too dense. If we allow for a
wide range of translations and scalings, the algorithm will be
'attracted' by this dense area, trying to map most of the source
tokenset onto it. -Up to a certain level of complexity, dense
tokensets can still be handled, if the number of sample points is
increased accordingly. In the experiments shown here, we used 300
sample points for the shapes and for the medical images, and 1500
sample points for the remote sensing data set. -As with any other affine matching approach, the algorithm will have difficulties with regular periodical structures (parallel lines, grids, etc.). If these conditions are fulfilled, the algorithm has already shown to perform well for a wide variety of visual data. Concerning the complexity of the proposed matching algorithm, it goes linear with the number of sample points and seed points, and
quadratically with the number of parameters of the affine transformation. Page 628
The concept of 'Affine Matching at the ISR-level' introduced in this paper seems to be a general and promising approach for many correspondence problems encountered in Computer Vision.
5 Implementation Details
The computing times given for the experiments were obtained on a
Silicon Graphics Power Challenge L (2 x R8000 75MHz processors)
running IRIX 6.01. The code was implemented using the KBVision
image processing package and no effort was spent on optimizing the
procedure for speed. As mentioned above the number of sample points
was 300 for the shapes and medical images and 1500 for the remote
sensing data set. We used 50 seed points for the optimization
procedure for all the experiments.
6 Conclusion and Outlook
In this paper, a general method for affine matching at the level of
intermediate symbolic representation (ISR) was introduced. The method is easy to implement and to parallelize. Since most applications have to deal with the extraction of some kind of significant
features, which can conveniently be represented at the ISR level, the method should be of common interest. It could be used in many
industrial, medical and remote sensing applications. Our algorithm relies on the coarse correspondence of tokens extracted from images. It does not require the establishment of point
correspondences. If the tokensets are kept sufficiently sparse, low
level features (e.g. edge elements) can directly be used without the
necessity of further processing (e.g. grouping edge elements into
straight lines, ribbons or corners). Since belief values guide the
probability that a certain token is selected for correspondence, and
many-to-many correspondences are possible, the method is very robust
against a broad variety of common disturbances (e.g. incomplete
segmentations, missing tokens, partial overlap). An extension of the
algorithm to recover a general 6 parameter affine transformation is
straightforward and computationally feasible. Multisource
visual information fusion (data fusion [Abidi and Gonzalez, 1992], consensus vision [Meer et al., 1990]) is a field of growing interest and
importance. The comparison and integration of information from several
sources without the necessity of prior (manual) spatial registration
is still an issue of ongoing research. In this context, the affine
matching algorithm constitutes just one module for spatial reasoning
in a complex framework for Information Fusion in Image Understanding
(proposed in [Pinz and Bartl, 1992a]).
Acknowledgments
This work was supported by the Austrian
'Fonds zur Foerderung der wissenschaftlichen Forschung' under
grant S7003. We thank Renate Bartl and Werner Schneider
(Univ. f. Bodenkultur Vienna) for putting the original Landsat and MKF6 images and the classifications of these images at our disposal. Several constructive comments by the anonymous referees helped to improve this paper and are gratefully acknowledged. Page 629
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