FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES 1
On the Completeness of
Coding with Image Features
Wolfgang Förstner
http://www.ipb.uni-bonn.de/foerstner/
Timo Dickscheid
http://www.ipb.uni-bonn.de/timodickscheid/
Falko Schindler
http://www.ipb.uni-bonn.de/falkoschindler/
Department of Photogrammetry
Inst. of Geodesy and Geoinformation
University of Bonn
Bonn, Germany
Abstract
We present a scheme for measuring completeness of local feature extraction in terms
of image coding. Completeness is here considered as good coverage of relevant image
information by the features. As each feature requires a certain number of bits which are
representative for a certain subregion of the image, we interpret the coverage as a sparse
coding scheme. The measure is therefore based on a comparison of two densities over the
image domain: An entropy density pH(x) based on local image statistics, and a feature
coding density pc(x) which is directly computed from each particular set of local fea-
tures. Motivated by the coding scheme in JPEG, the entropy distribution is derived from
the power spectrum of local patches around each pixel position in a statistically sound
manner. As the total number of bits for coding the image and for representing it with local
features may be different, we measure incompleteness by the Hellinger distance between
pH(x) and pc(x). We will derive a procedure for measuring incompleteness of possibly
mixed sets of local features and show results on standard datasets using some of the most
popular region and keypoint detectors, including Lowe, MSER and the recently pub-
lished SFOP detectors. Furthermore, we will draw some interesting conclusions about
the complementarity of detectors.
1 Introduction
Local image features play a crucial role in many computer vision tasks, most importantly as
an input for camera calibration and object recognition. The basic idea of using local features
is to represent the image content by small, possibly overlapping, independent parts which
are robust to a number of image distortions up to varying degrees. By identifying such parts
in different images of the same scene or object, it becomes computationally feasible to make
reliable statements about both image geometry and scene content.
Many algorithms for detecting salient and stable features are available, providing a range
of alternative methods with different definitions of salience and stability. Among the differ-
ent detectors are methods for extracting basic geometric structures like junctions, circles and
edges, and methods for extracting dark and bright blobs. Whilst the particular requirements
c
2009. The copyright of this document resides with its authors.
It may be distributed unchanged freely in print or electronic forms.

2 FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES
Lowe blobs Line segments SFOP junctions Entropy density pH(x)
pc(x) for Lowe blobs pc(x) for line segments pc(x) for SFOP junctions pc(x) for all three
Lowe blobs Line segments SFOP junctions Entropy density pH(x)
pc(x) for Lowe blobs pc(x) for line segments pc(x) for SFOP junctions pc(x) for all three
Figure 1: Sets of extracted features on example images Bricks and Beach, capturing different
types and amount of image information, and corresponding entropy densities pH(x) and
feature coding densities pc(x). We illustrate Lowe blobs [17], straight edge segments [6],
SFOP junction features [5, using a = 0 in eq. (7)].
for saliency and stability vary strongly depending on the application, we argue that com-
pleteness of a feature detector is mostly application independent: The image content should
be coded well by the detected features.
Let us illustrate the idea on the two simple images depicted in Figure 1. In the top row,
the detector by Lowe [17] seems to cover much of the areas representing the visible objects.
However, the characteristic contours and properties of the bricks are better coded by the
SFOP junctions [5] and especially the edge segments. For coding all relevant parts of the
image, one would probably decide to use all three detectors. Indeed, the three exemplary
detectors chosen here are highly complementary by design. In the third row of Figure 1, it
obviously becomes more difficult to code the whole image content with local features. We
see that the Lowe detector covers the piece of rock quite well and also captures the roughness
of the surge, but fails in coding the borders between water and dry and wet sand, respectively.
The SFOP junctions are more focused on the discriminant parts of such contours, but also
clearly lack the completeness of contour coverage provided by the edges.
The major part of this contribution consists in deriving a useful measure d for the in-
completeness of a particular set of local features in terms of image coding. We require d

FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES 3
to take small values if a feature set covers relevant image content in a similar manner as a
good compression algorithm would. Conversely d should take increasing values for features
covering unimportant content as well as for relevant content which is not covered. Therefore
we propose to compute two densities over the image domain, both of which are also illus-
trated in Figure 1: An entropy density pH(x) based on local image statistics, and a feature
coding density pc(x) which is directly computed from each particular set of local features.
The incompleteness is determined as the distance d between pH(x) and pc(x), namely the
Hellinger metric.
After a short overview on the related work in section 2 we will give a derivation of both
densities in section 3 together with an explanation of the proposed scheme. In order to
show the feasibility and practical relevance of the scheme, we use various feature detectors
and well-known image datasets, as explained in section 4. We finally conclude with a brief
summary in section 6.
2 Related Work
Local feature detectors A broad range of local feature detectors with quite different prop-
erties is available today. We will shortly introduce some of them here and refer to the detailed
review given in [27]. The scale invariant blob detector proposed by Lowe [17] is by far the
most prominent one. As it is based on finding local extrema of the response of the Lapla-
cian, which has the well known Mexican hat form, it conceptually aims at extracting dark
and bright blobs on characteristic scales of an image. The Hessian affine detector introduced
by Mikolajczyk and Schmid [20] based on the theory of Lindeberg [15] also relies on the
second derivatives of the image function over scale space. Some detectors directly determine
image regions that are characterized by their boundary, such as the intensity- and edge-based
region detectors by Tuytelaars and Van Gool [28] and the Maximally Stable Extremal Re-
gions (MSER) by Matas et al. [19].
Another family of detectors is based on the second moment matrix, or structure tensor,
computed from the squared gradients of the image functions. It has initially been used by
the classical detectors of Harris and Stephens [10] and Förstner and Gülch [7]. Two schemes
for exploiting the structure tensor over scale space have been proposed: In [20], scale is still
determined by investigating the second derivatives, but the location is determined using the
structure tensor. Lindeberg [15] uses the junction model of [7], determines the differentiation
scale by analyzing the local sharpness of the image and chooses the integration scale by
optimizing the precision of junction localization. The structure tensor may also be utilized
for extracting edge segments, as for example in [6]. We do not detail different methods for
detecting line segments here and refer to Heath et al. [11], for example.
The image content covered by such detectors, and hence the proportion of information
that is thereby coded, varies strongly depending on the type of detector, as depicted in Fig-
ure 1. This variation is natural, as pointed out by Triggs [26] who stated that there is no such
thing as a generic keypoint detector. The concepts of the detectors are different, sometimes
complementary, in spite of aiming at the same tasks, mainly matching and recognition.
There has been high effort in comparing the performance of local feature detectors. A
generally accepted criterion is the repeatability of extracted patches under viewpoint and
illumination changes in mostly planar scenes [21] and on 3D structured objects [22], but
also localization accuracy [9, 29] and the general impact on automatic bundle adjustment [3]
has been studied.

4 FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES
Benchmarks usually evaluate each method separately, not addressing the positive effect
when using a combination of multiple detectors, which may be very useful in many applica-
tions [3]. For example, Bay et al. [1] propose to use edge segments together with regions for
calibrating images of man-made environments with poor texture. We experienced a surpris-
ingly strong benefit when using mostly “complementary” combinations of detectors in the
framework of [1]. A tool for measuring this complementarity however is still lacking.
The intent of this contribution is to develop an evaluation scheme for measuring in how
far the detectors cover the image content completely and whether they are complementary in
this sense.
Image statistics For evaluating the completeness of detectors, the question arises which
kind of complete but yet sparse coding of an image we take as a reference. The classi-
cal, biologically inspired view of Marr [18] is to extract the so-called primal sketch, which
mainly refers to the blobs, edges, ends and zero-crossings detectable in an image, and is
hence achieved to a certain extend by using a combined set of different feature detectors.
Marr’s approach has been supported by the more recent work of Olshausen and Field [24].
An excellent but yet compact insight into such aspects of image statistics can be found in
Mumford [23].
Another approach is motivated from information theory: If we split the image into
patches of equal size, a complete but preferably sparse coding would focus on patches with
high information content, measured in terms of entropy in the sense of Shannon [25], where
entropy is defined as the average (or expected) amount of information transmitted when
transferring a signal from sender to receiver.
The entropy of a signal depends on the probability distribution of the events, which is
usually not known and hence has to be estimated from data. For example, for pixels in lo-
cal image patches, the probability distribution may well be modeled by Gaussian densities
representing each pixel intensity in the patch. The correlations between pixels, however,
are not known a priori and difficult to estimate, so the joint probability distribution is un-
known. J.-F. Bercher [12] proposed a general method for estimating the entropy of a signal
by modeling the unknown distribution as an autoregression process, focusing on finding a
good approximation with tractable numerical integration properties.
The widely accepted JPEG image coding scheme is also based on entropy encoding. To
account for the unknown pixel correlations, the data compression is carried out after a Dis-
crete Cosine Transformation (DCT) of the image, computed on local 8 8 patches or on a
wavelet transform of the complete image. The coefficients are then coded in blocks. A com-
parison of image coding schemes is given in Davis and Nosratinia [2], for example. Grazzini
et al. [8] found that maxima of local entropy play a major part for extracting singular man-
ifolds, that is statistically important parts, in infrared satellite images. They also addressed
the problem of scale, and proposed to use larger patch sizes and weight pixels according to
their distance to the center pixel before computing the spectrum.
In contrast to using image statistics as a prior for image analysis, we aim at taking a
certain aspect only, namely the local entropy as a reference for evaluating the ability of
feature detectors to completely cover the image.

FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES 5
3 Theory
Local feature detection can be considered as coding of image content. Measuring com-
pleteness of local feature extraction may consequently be considered as a comparison with
classical coding schemes. This requires to develop
1. an entropy density pH(x) using local image statistics. If the image can be coded with
H [bits] we can derive the number of bits per image region R by H =
R
x2R pH(x)dx.
2. a feature coding density pc(x). It is derived from a particular set of detected local
features, assuming each feature is representative for a certain image area.
The two densities can then be compared. In case pc is close to pH , the image is efficiently
covered with features, and the completeness is high. We hereby require busy parts of images
to be densely covered with features, and smooth parts not to be covered with features.
Furthermore, we have to take into account that many detectors are able to find features
of different scales at the same image position. This is easy to realize regarding pc, as each
feature represents a certain image area, thus the bits for coding the image feature are spread
over that area. For the entropy density pH , we assume that the coding is hierarchical, like in
wavelet transform coding where information from different scales is integrated.
3.1 Entropy density
We determine the entropy density pH based on small image patches of different size, or scale,
respectively. Given a square image patch g(i; j); i; j = 1; :::;N, we only have one sample of
the distribution of image content in that patch. We therefore assume the image patch to be
representative for a large image. Specifically, we assume that it is a subsection of an infinite
doubly periodic image with period N in both directions, allowing us to represent it in the
frequency domain. Furthermore we assume the image to be the noisy version of a Gaussian
process.
We can derive the entropy from the covariance matrix Sgg of the N2 intensity values,
which itself can be derived from the autocorrelation function and noise variance s2n :
H(g) =
1
2
log2 2pe
jSggj
s2n
[bits/patch] (1)
Due to Parseval’s identity, the determinant jSggj can easily be derived from the power spec-
trum P(u) = jDCT(g(x))j2 via jSggj=Õun0P(u;v). Note that we omit the DC term P(0), as
the covariance matrix captures only deviations from the mean. We use the Discrete Cosine
Transform (DCT) instead of the Fast Fourier Transform in order to reduce the effects of patch
borders. We assume the powerspectrum to be additively composed of the powerspectra of
the signal and the noise. Thus it theoretically is limited from below by the noise variance s2n
of the graylevels. We therefore use the regularized estimate for the power spectrum of the
signal
bP(u) = max

P(u)s2n ;0


: (2)
We then determine the entropy of a single pixel in the image patch from
H(g) =
1
2N2 åun0
max

log2 2pe
max

P(u)s2n ;0


s2n
;0
!
[bits/pixel] : (3)

6 FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES
excluding frequencies where the entropy would be negative. We assume the noise to be larger
than the rounding error introduced by graylevel quantization, thus sn  e=
p
12 0:29 [gr],
where e is the unit of the graylevels. This setup guarantees the entropy to be H(g) 0.
We now integrate the expected number of bits per pixel over scales. Let the entropy of
a pixel x based on patch size N be H(x;N). As a first choice, we determine the expected
number of bits per pixel over several scales by the equally weighted sum
H(x) =
S
å
s=1
H(x;1+2s) (4)
We still have to investigate whether this is an optimal choice. In our experiments we use
S = 7, thus the patch size is limited to 3 N  129. Observe that omitting the DC term in a
lower scale is compensated by the coding in the higher scales.
Finally we obtain the entropy density by normalizing H(x):
pH(x) =
H(x)
åyH(y)
(5)
The expected number of bits in a certain region R therefore is H åx2R pH(x), where H is the
total number of bits for the complete image. However, for comparing the entropy distribution
over the image with a particular set of local features, we can not use absolute values (i.e. bits
per pixel), but only relative values. This is why we take pH(x) as reference. In the right
column of Figure 1, the density pH(x) is depicted for two example images.
3.2 Feature coding density
Feature detectors usually deliver sets of features which are representative for a certain region.
For each feature type we need a certain amount of bits coding the respective region. Let a
feature f require c( f ) [bits], being responsible for a region R ( f ). Assuming a uniform
density of the bits per pixel within each region, we obtain a feature coding map from all F
features
c(x) =
F
å
f=1
1R ( f )(x)
jR ( f )j
c( f ) (6)
where 1R ( f )(x) is an indicator function, being 1 within the region R ( f ), and 0 outside.
In our case we analyse keypoint features f which are characterized by their position m f
and their scale s f , or a scale matrix S f in case of affine invariant features. In case we analyse
straight edge segments (xS;xE) f we use m f = (xS; f +xE; f )=2 and a covariance matrix with
major semi-axis jxE; f xS; f j=2 in the direction of the edge and minor semi-axis sa = 1 [pel].
Therefore it is reasonable to replace the uniform distribution by a Gaussian and derive the
feature coding map by
c(x) =
F
å
f=1
c( f )G(x;m f ;S f ) (7)
The actual density to be compared with the entropy density pH(x) is then
pc(x) =
c(x)
åy c(y)
: (8)

FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES 7
Aerial image Brodatz Cartoon Forest Mountain Tall building Kitchen
Figure 2: Example images from the datasets used for the experiments.
3.3 Evaluating completeness of feature detection
In case the empirical feature coding density pc(x) would be identical to the entropy density
pH(x), coding of the image with features would cover the image in the same manner as
using image compression. Of course image features may use less or more bits for coding the
complete image, depending on the coding of the individual feature, so we do not compare the
absolute number of bits per pixel, but their densities. We use Hellinger’s metric d(p(x);q(x))
of two densities p(x) and q(x) for measuring the difference between pH(x) and pc(x):
d(pH(x); pc(x)) =
s
1
2åx
p
pH(x)
p
pc(x)
2
(9)
4 Experiments
Applied feature detectors. For providing a good spectrum of the available methods, we
selected most of the prominent feature detectors presented in Mikolajczyk et al. [21], in-
cluding both the non-affine and affine versions of the scale-invariant Harris and Hessian
detectors from [20]. The implementations are taken from the website maintained by the au-
thors of [21]. For the Lowe detector, we used the original sourcecode kindly provided by the
author, but starting with the original instead of the double image resolution for comparabil-
ity. Additionally, we used scale invariant SFOP junction features [5], and a classical edge
detector based on the theory in [6], which uses a minimum edge length of ten pixels. The
SFOP junctions are obtained as a subset by restricting to a = 0 [5, eq. (7)].
Image data. Our objective is to compare the completeness of these feature detectors on a
wide variety of images. For that purpose, we used the fifteen natural scene category dataset
[13, 14] and added the well-known Brodatz texture dataset, a collection of cartoon images
and a set of subimages of an aerial image as further categories. We present here results for
a subset of these datasets only, as depicted by some example images in Figure 2. Our main
conclusions however hold for all datasets.
Experimental setup. We compute the incompleteness measure d for each of the images
and each of the detectors mentioned above. As we are interested in the effect of combin-
ing different features, especially for finding evidence about their complementarity, we also
consider combinations of features. However, as evaluating all possible combinations is in-
tractable within the scope of this paper, we focus on the potential of feature detectors to
complement the Lowe detector for now.
Clearly, completeness raises when the significance level of a detector is lowered, which
conversely reduces repeatability and performance in practice. We used the default settings

8 FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES
Figure 3: Experimental results for separate (top row) and combined (bottom row) feature
detectors. Abbreviations used in the bottom legend are introduced in the top legend.
as applied in the existing benchmarks for the standard detectors, and kept the settings for
the SFOP detector so that the number of features is comparable to that of Lowe. It is im-
portant to note that too low significance levels would distort the results, as noisy features on
homogeneous areas would increase the proposed metric. In order to minimize such effects,
the actual image noise should be taken into account, either by noise removal in the image or
consideration of the estimated noise in the algorithm [4, 16].
5 Results
Figure 3 shows the average difference d between the entropy density and the feature coding
density for each operator and seven image categories. In the plots, the mean value over
all images of an image category is given, together with the 1-s -confidence region, denoted
by the vertical black bars. Note that this is the standard deviation of the samples, not the
the standard deviation of the mean values, which would clearly be smaller. The distance d
is related to the angle between the vectors
p
pH and
p
pc. We have plotted a boundary at
d0 =
p
1=
p
2 which corresponds to an angle of 90, indicating a threshold beyond which
we treat densities as sufficiently different. The value d0 is indicated by a level line in both
diagrams.
5.1 Results for separate detectors
The average completeness per image category is quite similar for the Lowe, Harris affine
and Hessian affine detectors: It ranges around 0.37 to 0.39, with a significantly better result
on the Brodatz dataset ( 0:25) and worst results for Mountain and Tall Buildings. The
standard deviation however is significantly lower for the Lowe detector than for the other
two. Higher completeness w.r.t. the Brodatz images can be observed for all other detectors
as well, which can be understood from the fact that these pictures possess a strong uniform

FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES 9
structuredness.
The SFOP junctions show significantly higher completeness in all categories than the
previous three. The distance measure on average is more than 25% smaller. The standard
deviation is comparable to that of Lowe.
The MSER detector achieves best overall completeness as a single detector for Brodatz
and Forest, although on average the results are comparable with that of the SFOP junctions.
Compared to its overall results, MSER has a quite bad completeness for Aerial, and - due
to the similarity of the texture - also for Mountain. Edge features alone show a rather bad
completeness in most cases, which may be explained from the fact that on most images, only
the boundaries of informative image regions are extracted. On datasets dominated by visible
contours like Kitchen and Building however, coverage by edges is at least average.
In the upper plot of Figure 3 we also show results for the Harris and Hessian affine
detectors compared to the so called Harris- and Hessian-Laplace detectors [20]. For the
latter ones, the final estimation of affine parameters for the local pixel neighbourhood is left
out, so that the resulting patches are circular instead of elliptical. We expected better results
for the affine versions, which has not been confirmed by the experiments.
5.2 Results for combinations of features
Our expectation on the results for pairwise combinations was the following: Combining the
Lowe with the Hessian affine detector should not yield significant improvement, as both use
the Laplacian to identify the optimal scale. For MSER we expect stronger improvements,
as it looks for homogeneous regions. The combination with the Harris and SFOP junction
detector should give the best completeness, as they rely on the structure tensor and hence
search for points with many gradients instead of bright or dark blobs.
The results for pairwise combinations are shown left in the bottom row of Figure 3. The
improvement achieved by combining Lowe with Hessian affine features is only slight, i.e.
below 10% except for Brodatz and Cartoon. Other than expected, the Harris affine detector
complements Lowe no better than Hessian affine, which is very surprising. A possible reason
for this observation is that Harris affine is no “pure” junction detector: Characteristic scales
for detected junction points are searched in the Laplacian scale space. Moreover, the Harris
affine detector is known to find multiple keypoints with overlapping regions at junctions.
The SFOP junction detector seems to be the best pairwise complement to the Lowe
detector. Here the improvement on the Lowe detector alone is at least 10%, often over
20%. This becomes most obvious for Mountain. The MSER detector complements Lowe
better than Harris and Hessian, but not as significant as the SFOP junctions.
Using more different detectors at once, we get slight improvements with MSER upon the
pairwise combination of Lowe and SFOP junctions, as can be seen from the yellow bars in
the bottom row of Figure 3. The result of these three seems to be optimal in our experiments:
it is not noticeably changed when adding more detectors.
6 Conclusion and outlook
We have proposed an intuitive scheme for measuring completeness of local features in the
sense of image coding. To achieve this, we derived suitable estimates for the distribution of
relevant image information and the coverage by a set of local features over the image domain,

10 FÖRSTNER ET AL.: COMPLETENESS OF CODING WITH FEATURES
which we compared by the Hellinger’s distance. This enables us to analyze the completeness
of different sets of local features over arbitrary image sets.
We made a number of interesting observations: Affine detectors do not seem to system-
atically improve completeness compared to scale and rotation invariant detectors.
The Harris affine detector shows significantly lower complementarity w.r.t. the Lowe de-
tector than the SFOP junctions. It actually behaved very similar to the Hessian affine detector
in our experiments. Probably the fact that the Harris affine detector uses the Laplacian for
scale selection is one reason for this. The SFOP junctions complement the Lowe detector
most significantly, while the effect for MSER is only slightly worse. A very good and stable
complement is achieved when combining Lowe blobs, MSER regions and SFOP junctions.
The proposed entropy density pH(x) gives rise to a new scale invariant keypoint detector
which locally maximizes the entropy over position and scale. We will analyse its complete-
ness w.r.t. image coding, similar as proposed in this paper, and its applicability as a keypoint
detector for matching and object detection.
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