Radial Fourier Analysis (RFA) Descriptor with Fourier-based Keypoint Orientation

S. C. F. Lin, C. Y. Wong, G. Jiang, M. A. Rahman & N. M. Kwok
International Journal of Image Processing (IJIP), Volume (8) : Issue (6) : 2014 397
Radial Fourier Analysis (RFA) Descriptor with Fourier-based
Keypoint Orientation
S. C. F. Lin stephen.lin@unsw.edu.au
School of Mechanical and Manufacturing Engineering
The University of New South Wales
UNSW Sydney, NSW, 2052, Australia
C. Y. Wong chin.wong@unsw.edu.au
G. Jiang guannan.jiang@unsw.edu.au
M. A. Rahman md.arifur.rahman056@unsw.edu.au
N. M. Kwok nmkwok@unsw.edu.au
Abstract
Local keypoint detection and description have been widely employed in a large number of
computer vision applications, such as image registration, object recognition and robot localisation.
Since currently available local keypoint descriptors are based on the uses of statistical analysis in
spatial domain, a local keypoint descriptor, namely Radial Fourier Analysis (RFA) keypoint
descriptor, is developed with the use of spectral analysis in frequency domain. This descriptor
converts image gradients around SIFT keypoints to frequency domain in order to extract the
principle components of the gradients and derive distinctive descriptions for representing the
keypoints. Additionally, a keypoint orientation estimate is also introduced to improve the rotational
invariance of the descriptor rather than simply adopting SIFT keypoint orientations. The
introduced orientation estimate employs the starting point normalisation of Fourier coefficients,
which are frequency responses, to deduce rotating angles that ensure keypoint correspondences
are aligned at the same orientation. Through experiments and comparisons, RFA descriptor
demonstrates its outstanding and robust performances against various image distortions.
Particularly, the descriptor has extremely reliable performances in dealing with the images, which
are degraded by blurring, JPG compression and illumination changes. All these indicate that
spectral analysis has strong potential for local keypoint description.
Keywords: Local Keypoint Descriptor, Keypoint Orientation, Fourier Transform, Keypoint
Matching, SIFT Descriptor.

1. INTRODUCTION
Local keypoint detection and description have been frequently utilised as an integral system to
accomplish diverse computer vision tasks due to its invariance capabilities against image
distortions caused by many types of geometric and photometric transformations. For example,
they have been widely applied to basic computer vision tasks including image retrieval [1], image
registration [2] and object recognition [3]. As an extension from these basic tasks, they can be
specifically designed to carry out object categorisation [4], texture classification [5] and face
recognition [6]. In engineering, their applications could be related to medical diagnosis [7], robot
localisation [8], land cover surveillance [9] and machine operation monitoring [10].
From the perspective of computer vision, local keypoint detection and description should be
treated as independent processes as they produce different outputs and serve different purposes.
Many keypoint detectors [11] have been proposed to search stable and repeatable keypoints in
images, and keypoints generally contain the information of position and other supplementary
measures to localise and define the properties of keypoints. One of the supplementary measures
could be keypoint orientation that is commonly assigned to keypoints for rotational invariance. For
example, SIFT detector [3] estimates keypoint orientation by creating a histogram of the image
gradient orientations around a keypoint. Furthermore, ORB detector [12] defines keypoint
orientation as the vector between a keypoint position and the intensity centroid of local image
patch of the keypoint. Taylor and Drummond [13] compute keypoint orientation by simply
choosing the longest vector among the vectors that are formed by intensity differences around a
keypoint. All these keypoint orientation estimates are established on gradient and statistical
approaches in spatial domain. It is advantageous if a different keypoint orientation estimate, such
as using spectral analysis in frequency domain, can be developed to provide accurate keypoint
orientations.
For keypoint description, it extracts the features concealed in the local image patches around
keypoints and analyses the features to represent that keypoints. A variety of keypoint descriptors
[14] have been devised to be invariant to geometric and photometric transformations such as
image distortions caused by blur, rotation, scale, illumination and JPG compression changes.
Many keypoint descriptors, such as SIFT [3], PCA-SIFT [15], and GLOH [14] descriptors, utilise
histograms to determine the distributions of local image patches around keypoints in order to
extract and analyse features, while achieving results that are invariant to image distortions.
Summation of image gradients is also a robust approach in keypoint description, which is
demonstrated by SURF descriptor [16]. Furthermore, BRISK descriptor [17] uses the
comparisons of local image intensities around keypoints as features to represent the keypoints.
Similar to the issue raised in keypoint orientation estimates, most keypoint descriptors are based
on voting or statistical approaches in spatial domain. A local keypoint descriptor that extracts and
analyses features in a different domain is essential as it could expand the diversity of local
keypoint descriptions to increase its distinctiveness.
To address the overdevelopment of statistical analysis in spatial domain for keypoint orientation
estimations and keypoint descriptions, this work relies instead on spectral analysis in frequency
domain. For the keypoint orientation estimate, it utilises Fourier transform to convert the image
intensities of local image patches to frequency domain from spatial domain. Afterward, starting
point normalisation is performed on the intensity frequency to derive keypoint orientations in order
to enhance the rotational invariance of keypoints. The proposed keypoint descriptor, which is
named as Radial Fourier Analysis (RFA) descriptor, transforms the image gradients of local
image patch to frequency domain, then the gradient frequency is decomposed to provide a
unique keypoint description. Both of the transformed frequencies are not inverted back to spatial
domain as the keypoint orientation and description are directly carried out in frequency domain.
The rest of the paper is organised as follows. In Section 2, the principle of SIFT keypoint detector
is briefly revised, and a keypoint orientation estimate is developed to replace SIFT keypoint
orientation. Section 3 presents the procedures to generate RFA descriptor with the reasons
declared for each procedure. This is followed by Section 4, where an experiment setup to

evaluate the performances of the new descriptor is detailed. The performance results and
comparisons are illustrated in Section 5. Significant observations from the results are compared
and highlighted for discussions. Finally, in Section 6, all these works are concluded by suggesting
possible research directions in accordance with the performances of the proposed keypoint
descriptor and orientation estimate.
2. IMAGE KEYPOINT DETECTION
In this section, SIFT keypoint detector is briefly explained as SIFT keypoints are utilised to
generate RFA descriptors. In order to improve the rotational invariance of SIFT keypoints, a novel
estimate of keypoint orientation using Fourier normalisation is proposed to replace SIFT keypoint
orientation.
2.1 SIFT Keypoint Detector
SIFT descriptor has been concluded by [18] [19] as the state-of-the-art in the research and
development of local keypoint descriptor. As such, SIFT keypoints are used to localise salient
points because of its trustworthy characteristics as well as providing a fair comparison base of
describing power between our descriptor and SIFT descriptor. Fundamentally, SIFT keypoints are
detected by utilising Difference of Gaussian (DoG) operator that was initially introduced by
Crowley and Parker [20]. However, Lowe’s extension of the operator [3] to detect scale invariant
keypoint in a given image is more widely used and is adopted in this work.
SIFT keypoint detector involves the following three steps: (1) scale-space extrema detection; (2)
keypoint localisation; and (3) orientation assignment. The first step is DoG operation, where input
image is convoluted with Gaussian kernels at different scales to produce a series of Gaussian
images. Adjacent images in the series are subtracted from each other, forming DoG images. A
window is then employed to search the local maximums and minimums in the DoG
images, and these extremes are labelled as potential keypoints. The second step filters potential
keypoints by measuring their contrast and edge response, within which the keypoints with low
contrast and high edge response are discarded. The remaining keypoints are subsequently re-
localised to sub-pixel accuracy and identified as SIFT keypoints. In the final step, the dominant
gradient orientation of each SIFT keypoint is estimated through binning the gradient orientations
around the keypoint into a histogram and the peak of the formed histogram is taken as the
gradient orientation of keypoint. The dominant gradient orientations are estimated and assigned
to SIFT keypoints to enhance its rotational invariance ability. The last two steps are expanded
from DoG operator for stabilising the keypoints in matching and recognition tasks.
2.2 Keypoint Orientation Estimate by Fourier Analysis
Other than histogram-based estimate of keypoint orientation, such as SIFT keypoint orientation,
RFA descriptor adopts a simple and effective approach to compute keypoint orientation with
Fourier analysis. This Fourier-based keypoint orientation estimate consists of three processing
steps: (1) intensity sample collection; (2) Fourier transform; and (3) orientation estimate.
In the first processing step, the neighbouring image intensities around a keypoint are circularly
sampled and arranged into a vector. The procedure of this intensity sample collection is
presented in Algorithm 1. The intensity sample collection is preformed at the image scale where
the keypoint is detected. For each sampling angle, image intensities are sampled along the
radial direction to represent the intensity signatures at that angle. This sampling process is
performed at counter-clockwise direction till a full circular sampling is completed. The intensity
vector, , has the image intensity samples arranged in an ascending order with respect to
angular direction, and each sampling angle is represented by image intensities. The intensity
vector, , can be described as,

where, , and is the image intensity at a given
sampling position . An illustration of this intensity sample collection is shown in Figure 1
with a radial sampling range of 5 pixel radius (i.e., ) and an angular sampling interval of
(i.e., ) for demonstration purpose. Each black line connecting the red circles is a
set of samples along the radial direction for that sampling angle. These sets of samples are then
arranged into the intensity vector, , with the order from to as mentioned
previously.
ALGORITHM 1: Intensity Sample Collection for Orientation Estimate by Fourier
Input gray image,
Input keypoint position,
Set radial sampling range,
Set angular sampling interval,
Set sampling angle index,
Initialise to a zero vector of the length
For to do
Determine sampling angle
For to do
Determine the sampling position
Sample image intensity at
Distribute the image intensity into
End for
End for
Output
FIGURE 1: Intensity sample collection for keypoint orientation estimate. The red circles indicate the
positions of samples and the blue cross sign represents the location of keypoint. Each black line connecting
the red circles is a set of samples for that sampling angle.

For the second processing step, a discrete Fourier transform is performed on the obtained
intensity vector, , that is,
where, . Here, denotes the Fourier coefficient (i.e., frequency
response) based on the information given by that is obtained by Algorithm 1. Note that, the
Fourier coefficients, , are shifted to the centre.
The final processing step estimates the keypoint orientation by using the phase of first positive
Fourier coefficient (i.e., ). The phase is calculated by,
where, and denote the real and imaginary parts of the Fourier coefficient
respectively. Since this calculated phase is in frequency domain, it must be converted back to
spatial domain to properly express the phase as keypoint orientation with regard to the
arrangement of image intensity samples in the intensity vector, . The conversion from
phase, , to keypoint orientation, , is defined as,
The multiplication of implies that the phase is converted to spatial domain from
frequency domain. After the conversion, the term aligns the converted phase to
coincide with the angular sampling interval and radial sampling range in intensity sample
collection.
The principle of this orientation estimate is based on the property of starting point of Fourier
coefficients. For example, a complex signal is defined as and its Fourier coefficients are
defined as . Considering the signal is shifted by an amount of in its sequence that merely
changes the starting point of the signal to from . The shifted signal is expressed
as , whose Fourier coefficients are
for . Thus, shifting the starting point of the signal causes a linear phase shift
in all Fourier coefficients that depends on the coefficient index (i.e., ) and the amount of shift
(i.e., ).

It can be seen from Algorithm 1 that the keypoint orientation estimate is transformed to starting
point normalisation as the neighbouring image pixels around a keypoint are sampled and
arranged in the order corresponding to angular direction. Then, Eq. 3 determines the amount of
linear phase shift that is imposed on the Fourier coefficients, , obtained in Eq. 2. Lastly, Eq.
4 converts the linear phase shift to the change of starting point in the intensity vector, . It
also reduces the computation cost by avoiding the use of inverse Fourier transform to determine
the change of starting point in spatial domain.
Another issue to be considered while applying Fourier transform is the relationship between input
and output of the transform. As the input of Fourier transform is a complex function, the output
coefficients are difficult to predict and do not obey any symmetries since they are the sum of all
the real and imaginary parts of the input. However, if the input of Fourier transform is a real
function, the real and imaginary parts of the output coefficients are even and odd functions
respectively. These two symmetries make the magnitude and phase of the output coefficients to
become even and odd functions respectively.
For our keypoint orientation estimate, the phase calculated in Eq. 3 could either account for the
first positive or negative Fourier coefficients (i.e., or ) as they are the same in
magnitude but opposite in direction. Nevertheless, the intensity sample collection in Algorithm 1
samples image intensities in counter-clockwise direction and analyses a non-zero area. The first
positive Fourier coefficient, , is therefore chosen to estimate keypoint orientation as well as
to define its direction. The use of this coefficient is also suggested by Folkers and Samet [21] to
normalise the Fourier coefficients that undergo the change in starting point.
3. IMAGE KEYPOINT DESCRIPTION: RFA DESCRIPTOR
In principle, keypoint descriptor aims to represent a keypoint by extracting main information within
a minimum sampling window that is centred at the keypoint. The extracted information is further
analysed to outline its features while maintaining the discrimination among other keypoint
descriptors. The information can be extracted either in spatial domain or frequency domain with
the uses of statistical or spectral analysis. For example, SIFT descriptor computes the image
gradient orientations within the local image patch of a keypoint in spatial domain. Then, a
statistical analysis is applied as the gradient orientations are distributed into orientation
histograms that yield a 128 dimensional vector for representing the keypoint. In this work, RFA
descriptor obtains the Fourier coefficients of image gradients to determine the main components
of local image patches through spectral analysis in frequency domain.
The RFA descriptor is generated with the following steps: (1) gradient sample collection; (2)
Fourier transform; and (3) principle frequency extraction. As previously mentioned, RFA
descriptors are generated by using SIFT keypoints that contain the details of position, scale and
dominant gradient orientation. In particular, the keypoint orientation estimated by Fourier analysis
replaces the dominant gradient orientation in SIFT keypoints as it is more reliable and accurate.
Firstly, the local image gradients around a keypoint are circularly sampled in a manner that is
similar to the intensity sample collection for Fourier keypoint orientation estimate. Here, image
gradients specifically mean both horizontal and vertical gradients. The gradient sample collection
process is detailed in Algorithm 2. Through the algorithm, it can be seen that the keypoint
orientation is taken into account to determine the position of gradient samples to ensure that
image gradients are correctly sampled with the same starting point as its corresponding keypoint.
Figure 2 displays the sampling pattern of this gradient sample collection with a radial sampling
range of 5 pixel radius (i.e., ) and an angular sampling interval of (i.e., )
for demonstration purpose. A slightly different sampling pattern is applied in gradient sample
collection that the image gradients along angular direction are treated as the signatures at that
sampling radius. It is shown by each black line that connects the red circles in Figure 2. As for the
output of this step, 16 gradient vectors are obtained and they are express as,

where, sampled horizontal and vertical gradients are substituted into the real and imaginary parts
of these complex vectors respectively. For Eq. 6, it has the variables
and , where and . The radial sampling range, ,
indicates the number of gradient vectors and is set to this value due to the suggestion given by
the SIFT descriptor performance evaluation for local keypoint matching [22].
ALGORITHM 2: Gradient Sample Collection for RFA Descriptor
Input gray image,
Input keypoint details,
Set radial sampling range,
Set angular sampling interval,
Set sampling angle index,
For to do
Initialise to a zero complex vector of the length
For to do
Determine the sampling angle,
Determine the sampling position
Determine and sample image gradients at
Distribute the image gradients into
End for
End for
Output
FIGURE 2: Gradient sample collection for RFA descriptor. The red circles indicate the positions of samples,
the blue cross sign represents the location of keypoint and the black arrow directs keypoint orientation.
Secondly, prior to Fourier transform, unit vector normalisations are separately employed on the
real and imaginary parts of each of the obtained gradient vectors, . The normalisations are
used to increase gradient contrast while preserving the characteristics of the gradients. Then, a
discrete Fourier transform is independently performed on each of the obtained gradient vectors to

observe the Fourier coefficients (i.e., frequency responses) at different radial distances from the
keypoint. The transformations are defined as,
for . The variable denotes the Fourier coefficient of the
gradient vector that is based on the information given by the in Eq. 6. Again, these Fourier
coefficients have been shifted to centre and are the output of the second step.
The last step manipulates the Fourier coefficients, , in order to extract the principle
frequencies in these coefficients. It translates the coefficients into more meaningful analysis and
distinctive representation of the neighbourhood of keypoints. RFA descriptor is generated by
extracting four coefficients with the indices of from each set of the obtained
Fourier coefficients, . The descriptor, , is mathematically expressed as,
for and . Consequently, it results in a 128 dimensional vector
since the real and imaginary parts of the extracted coefficients are treated as individual numbers.
The vector is then normalised to unit length to suppress the influences induced by illumination
changes. This normalised vector becomes the RFA descriptor.
It is essential to avoid or minimise the information loss associated with the generation of keypoint
descriptor. For example, SIFT descriptor uses the gradient orientations weighted by their gradient
magnitudes to analyse the local image gradients around keypoints. While the magnitude and
orientation information of the gradients are preserved, this methodology also changes the
characteristics of the gradients through the weighting operation. To prevent this situation, both
horizontal and vertical gradients are taken into account for the gradient sample collection in the
generation of RFA descriptor. Therefore, the information of gradients can be entirely preserved
and used for Fourier analysis without altering its characteristics from the original.
During the manipulation of Fourier coefficients in the last step of RFA generation, all the centre
coefficients (i.e., ) have been ignored as these coefficients straddle in the line between
positive and negative frequencies. Phrased differently, the coefficients contain the responses
from both positive and negative frequencies that make them unstable and sensitive to geometric
translation changes [23]. In addition, the same property is applied to the last Fourier coefficients
(i.e., ).
Basically, the low frequency band of a spectrum represents the approximation of the spectrum
itself, and the high frequency band stands for the detail of the spectrum. According to this
principle, RFA descriptor analyses the low frequency band of local image gradients around
keypoints by extracting and utilising the four coefficients as explained in the last step of RFA
generation. Specifically, the extraction of the four coefficients acts like a low-pass filter
convoluting with image gradients in spatial domain, but the extraction directly doing so in
frequency domain. High frequency band is discarded because the details of local image gradients
would be extremely dissimilar in the cases that input images are not exactly identical and the fact
that images are smoothed by different Gaussian scales during the generation of RFA descriptor.
Whereby, the descriptor extracts low frequency coefficients to describe the principle component
in local image gradients.

Since the input of the Fourier transform in Eq. 7 is a complex function, the resulting Fourier
coefficients do not obey any symmetrical properties as mentioned previously. To properly analyse
low frequency band, for RFA descriptor, it is necessary to include the coefficients at both of the
positive and negative frequencies. They represent for the same frequency response and contain
the information at that frequency, but only in opposite direction or sign.
4. EXPERIMENT SETUP
This section describes the image database, matching techniques and evaluation metrics used to
manifest the performances of RFA descriptor.
4.1 Image Database
The image database from a performance study of keypoint descriptors [14] was adopted to
evaluate the performances of RFA descriptor with the keypoint orientation estimated by Fourier
analysis. The image database includes a few sets of images that are individually affected by blur,
illumination, JPG compression, rotation, and scale distortions. Thus, RFA descriptor’s
performances can be completely examined under these image distortions. Example images of
this database are shown in Figure 3.
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j)
FIGURE 3: Example images of the database used for the evaluations of RFA descriptor. Blur changes: (a)
Bikes and (b) Trees. JPG compressions: (c) UBC. Illumination changes: (d) Leuven. Viewpoint changes: (e)
Graffiti and (f) Wall. Rotation and scale changes: (g) Bark and (h) Boat. Rotation changes: (i) Building and (j)
Text.
4.2 Matching Technique
In order to obtain descriptor matches, it requires a similarity measure and a matching strategy to
calculate the likeliness between descriptors and determine most likely descriptor matches among
a pool of descriptors respectively. There are numerous similarity measures [24] [25] and matching
strategies [26] [27] available for the matching of keypoint descriptors.
In this paper, however, the most familiar similarity measure and matching strategy are utilised to
generate descriptor matches. They are Euclidean distance and threshold-based nearest
neighbour (TNN) matching strategy respectively. As soon as the Euclidean distances between a
descriptor and a pool of descriptors are calculated, TNN matching strategy would assess the
descriptor pair that has the shortest distance between them. If the distance of that pair is below a
pre-defined threshold, the pair is then recognised as a match, otherwise, no match is generated.

4.3 Evaluation Metrics
The RFA descriptor matches generated by Euclidean distance with TNN matching strategy are
used to measure the performances of RFA descriptor. The matches are evaluated by recall-
precision, which was initially employed in [28] to verify the performances of keypoint descriptors.
It is the most popular performance metrics in the evaluation of keypoint descriptors because the
major examinations on descriptor matches are their sensitivity (i.e., matching rate) and accuracy
(i.e., matching accuracy). Recall represents the sensitivity of the generated descriptor matches in
a given image pair that contains the same scene. It is defined as the ratio between the number of
correct matches and the number of correspondences, such that,
where, the correct matches and correspondences are verified by projecting keypoints from its
image to the corresponding image. If the projected keypoints are equal as or close to their
corresponding keypoints, then they are recognised as correct matches or correspondences.
Another metrics in recall-precision is 1-precision that indicates the accuracy of generated
matches and is the ratio between the number of false matches and the number of total matches.
It is expressed as,
5. EXPERIMENTAL RESULTS
This section presents the performance evaluations of RFA descriptor under various image
distortions. Apart from the evaluations of RFA descriptor, the performances of other benchmark
descriptors, such as SIFT, VLFeat SIFT and PCA-SIFT, are also provided for comparison
purposes. Since PCA-SIFT requires a training process, it was trained by 26,000 image patches
that are not related to the image database before it is used in the following evaluations. Twenty
and 128 dimensional PCA-SIFT descriptors are illustrated in the evaluations for the completeness
of the comparisons. The former is recommended by its author and the latter is to fairly compare
with the other descriptors as they have dimensions of 128. Significant observations and
discussions about the evaluations are also given in this section.
The descriptors were assessed while the images of Figure 3(a) and Figure 3(b) were blurred at
different degrees. The performance results obtained by these blurred images demonstrate the
abilities of the descriptors against blur changes, and they are presented in Figure 4. From the
recall-precision graphs, it can be seen that RFA descriptor particularly outperforms the other
keypoint descriptors in high precision region, between 0 and 0.5 of 1-precision. Since the
smoothing effect caused by blurring occurs at high frequency band, the RFA descriptor is
benefited from using Fourier transform and low-pass filter that remove the high frequency band of
local image gradients around keypoints. Thus, the performances of RFA descriptor is hardly
degraded by the image distortions induced by blur changes.
The quality of Figure 3(c) is degraded by JPG compression, which is based on discrete cosine
transform to trade off image quality for a smaller file size. Figure 5 displays the performance
results of the descriptors under this image quality degradation. As expected, RFA descriptor
demonstrates a better performance than the other descriptors in this case since discrete cosine
transform is a variant of Fourier transform. Both JPG compression and RFA descriptor process
their input image in frequency domain and manipulate information with frequency decomposition.
Whereby, the remaining information after JPG compression can be easily extracted and used
with RFA descriptor.

The performance results from altering the illuminations of Figure 3(d) are depicted in Figure 6.
Once more, it is observed that RFA descriptor surpasses the other descriptors in high precision
region. The high recall at high precision region indicates the representations of image gradients
by RFA descriptor are distinct and accurate under this type of image distortion. It is advantaged
by the unit vector normalisation prior to Eq. 7 as the normalisation minimises the effect of
illumination changes by locally standardising the magnitudes of horizontal and vertical gradients.
Another contributor for this high performance is the linearity of Fourier transform that allows the
scaling of an input signal to be factorised, then it is normalised by the unit vector normalisation in
the last step of RFA descriptor generation.
Figure 7 shows the performances of the descriptors against the image distortions caused by
viewpoint changes. As Figure 3(e) and Figure 3(f) were captured at different viewpoints,
sophisticated geometrical transforms are imposed onto these images, increasing the possibility of
generating incorrect matches. In Figure 7(a), the performance of RFA descriptor is extensively
higher than the other descriptors in high precision regions. Although the other descriptors have
higher recall in low precision region, it should be noted that their recall approximately starts to rise
at the 1-precision of 0.7 and it can be said that the most of their matches are incorrect. With
Figure 7(b), however, RFA descriptor only demonstrates an acceptable performance, which is in-
between SIFT and VLFeat SIFT descriptors, as serious image deformations have been imposed
in this dataset.
(a) (b)
FIGURE 4: Recall-Precision evaluations for blur changes using: (a) Bikes and (b) Trees.
FIGURE 5: Recall-Precision evaluations for JPG compression using UBC.

FIGURE 6: Recall-Precision evaluation for illumination changes using Leuven.
(a) (b)
FIGURE 7: Recall-Precision evaluations for viewpoint changes using: (a) Graffiti and (b) Wall.
Figure 8(a) and Figure 8(b) present the performance results of the descriptors when the images
of Figure 3(g) and Figure 3(h) simultaneously experienced the image distortions caused by
rotation and scale changes. From Figure 8(a), RFA descriptor has achieved a performance that is
much lower than SIFT and VLFeat SIFT descriptors. In Figure 8(b), RFA descriptor has a higher
recall in high precision region when compared with the other descriptors. It is an acceptable
performance, especially if accuracy is more important than sensitivity in keypoint matches.
(a) (b)
FIGURE 8: Recall-Precision evaluations for rotation and scale changes using: (a) Bark and (b) Boat.

Apart from the performance results given in Figure 7 and Figure 8, Figure 9 shows the
performances of the descriptors when they were subjected under the influences of purely rotation
changes. From both the recall-precision graphs, RFA descriptor demonstrates equivalent
performances as VLFeat SIFT descriptor, but it outperforms SIFT and PCA-SIFT descriptors by
significant margins. It is a solid evidence that our keypoint orientation estimate provides a reliable
and robust reference orientation and allows the local image patches of descriptor matches to be
rotated to a similar orientation. With the assistance of this keypoint orientation estimate, RFA
descriptor has been able to easily deal with the image distortion induced by rotation changes.
(a) (b)
FIGURE 9: Recall-Precision evaluations for rotation changes using: (a) Building and (b) Text.
6. CONCLUSION
In this paper, a keypoint orientation estimate and a keypoint descriptor have been proposed to
analyse the local image patch around keypoints in frequency domain. The keypoint orientation is
estimated from the information of image intensities with the starting point normalisation of Fourier
coefficients. For the keypoint descriptor, it extracts low order Fourier coefficients of image
gradients to approximate overall appearance of the gradients. Meanwhile, image detail that
resides in high order Fourier coefficients are ignored as excessive detail would make the
descriptor superfluously distinctive.
When evaluating the performances of RFA descriptor, it was found that the descriptor
outperforms benchmark descriptors, such as SIFT, PCA-SIFT and VLFeat SIFT descriptors, in
images suffering from a variety of image distortions. RFA descriptor is particularly stable and
reliable in dealing with the image distortions caused by blur, rotation, illumination and JPG
compression changes. This is evidenced from the experimental results presented in this paper. In
addition, RFA descriptor encodes image gradients into a distinctive and accurate representation
as its matches always yield a recall that rapidly rises up in high precision region, indicating high
sensitivity and accuracy during matching.
Through this paper, it has been proven that spectral analysis, such as Fourier transform, Fourier
normalisation and low-pass filtering, can be employed to estimate keypoint orientations and to
describe local keypoints. This research shows that spectral analysis serves well as an alternative
or supplementary tool for statistical analysis of keypoint description and orientation estimation.
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