Rotation Invariant Feature Matching - Based on …aht/articles/0027-0021.pdf · Rotation Invariant...

Rotation Invariant Feature Matching- Based on Gaussian Filtered Log Polar Transform

and Phase Correlation

Anders HastUppsala University

Department of Information TechnologyDivision of Visual Information and Interaction

Uppsala, SwedenEmail: [email protected]

Andrea MarchettiConsiglio Nazionale delle RicercheIstituto di Informatica e Telematica

Pisa, ItalyEmail: [email protected]

Abstract—Rotation invariance is an important property forany feature matching method and it has been implemented indifferent ways for different methods. The Log Polar Transformhas primarily been used for image registration where it is appliedafter phase correlation, which in its turn is applied on the wholeimages or in the case of template matching, applied on majorparts of them followed by an exhaustive search. We investigatehow this transform can be used on local neighborhoods of featuresand how phase correlation as well as normalized cross correlationcan be applied on the result. Thus, the order is reversed andwe argue why it is important to do so. We demonstrate acommon problem with the log polar transform and that manyimplementations of it are not suitable for local feature detectors.We propose an implementation of it based on Gaussian filtering.We also show that phase correlation generally will perform betterthan normalized cross correlation. Both handles illuminationdifferences well, but changes in scale is handled better by thephase correlation approach.

I. INTRODUCTION

The aim of this paper is to present a rotation invariantfeature detector based on the Log Polar Transform (LPT) andto discuss implementation issues. It turns out that the LPToften is implemented in a way not suitable for the algorithmwe present. Moreover, image registration techniques usingLPT and Phase Correlation (PC) are usually applied on wholeimages or larger parts of them, while we apply it on a smalllocal neighborhood. Furthermore, we reverse the order so thatthe LPT us applied first on a circular neighborhood assuringthat the same area in both images for corresponding pointsare sampled, even if the images are rotated . We compare themethod using Normalized Cross Correlation (NCC) after LPTand we show that PC generally performs better.

The contributions of this paper are twofold: 1) an imple-mentation of the LPT based on Gaussian Filtering (LPTGF )instead of higher order interpolation; 2) a discussion andcomparison of rotation invariant feature matching techniquesbased on LPTGF together with PC or NCC.

II. IMAGE REGISTRATION

PC have been extensively used for image registration [1],[2], [3] where some achieve orientation invariance by applying

a LPT after the Fast Fourier Transform (FFT) [4], [5], [6], [7].It should be noted that they perform the operations on thewhole images or larger parts of them and they are thereforenot doing feature matching in the sense we aim to do it.Moreover, we do not scan the second image in order to findthe best match, like in template matching. Instead we compareall feature points in one image with all feature points in theother image.

A. Rotation Invariance

Schmid and Mohr [8] deal with the problem of differentorientations using a ’local jet’ [9], i.e. the a complete setof differential invariants, which locally describes the image.Lowe [10] propose to use local image gradient directions forthe same purpose in the implementation of SIFT [10], [11].Bay et al. [12] propose to use Gaussian weighted waveletresponses to construct a vector indicating the orientation forSURF. Other detectors like FREAK [13] and BRISK [14]estimate the rotation by summing the estimated local gradientsover selected pairs.

B. Matching

The feature points themselves can be Harris corners [15],difference of Gaussians (like in SIFT) or the determinant ofthe Hessian matrix (used for SURF) and many others (see anoverview of Interest point detectors [16], [17], [18]). Carneiroand Jepson[19], [20] propose phase based local features basedon the response from steerable filters [21], where samplingaround the interest point is done in equally spaced orientationsaround the feature point. This approach is further elaboratedby Ulusoy and Hanckok [22]. Zokai and Wolberg [23] use theLPT on 10% − 25% of the images by cropping the centralregion of one image and then they perform template matchingby applying an exhaustive search though every pixel in theother image, by cropping out a region using the LPT in orderto find the matches that gives the highest response using crosscorrelation. In any case, none of these methods are like theapproach we propose in this paper.

Finally, when the tentative corresponding pairs have beenestablished, the transformation (or homography [24]) between

8th International Symposium on Image and Signal Processing and Analysis (ISPA 2013) September 4-6, 2013, Trieste, Italy

Image ProcessingPattern Recognition 100

söndag 17 mars 13

Fig. 1. The area around a feature point (blue dot) in two images to bematched is shown as two quads that overlap. A 90◦ rotation causes the regionto cover different areas in their corners. The circular area, however, alwayscovers the same part, regardless of the rotation.

images is computed using RANSAC [25] or some of its manyvariants [26], [27], [28], [29], [30].

III. LPTGF PC AND LPTGF NCC

We present the prosed feature matching technique and alsoemphasize the differences between existing methods. Usuallythe FFT is done on whole images or larger parts and then theLPT is used to handle the rotations [7]. Finally the Inverse FastFourier Transform (IFFT) is performed. We propose to do theLPT first and after that the FFT is performed, and finally theIFFT, on each local neighborhood as discussed in the followingsubsections.

A. The Log-Polar Transform

The LPT is usually constructed so that a quadratic inputregion is sampled and transformed and the output region is alsoquadratic. However, due to its polar nature not all pixels in theoutput region will be set, leaving some regions unfilled. Thismakes perfectly sense for the image registration methods basedon the FFT as the output of the FFT is quadratic and the LPTis performed on that region and finally the cross correlationincluding the IFFT is performed. Nevertheless, we proposeto work on rather small circular neighborhoods around eachfeature point and therefore we can do the LPT first. This isone of the main advantages we would like to stress because itwill handle rotations better. If there exists a rotation betweenthe images then the square regions to be matched will coverdifferent areas in their corners as shown figure 1. Nonetheless,a circular area will always cover the same part regardless ofthe orientation. This will make sure that even if one of thetwo regions to be matched is rotated, the region that the FFTworks on (the output from LPT) is always the same aroundthe feature points. The input area to the LPT can have anysize but the output should have side length that is a multipleof two, for an example 8, 16, 32 or 64. In this paper we choseto work on 32 for reasons discussed later.

Another problem with quadratic input areas to the LPT isthat there will be some unfilled parts that will be rather large,which will have an negative impact on the matching. Thereforeit is better two use a circular input area for the LPT and hencethe FFT must be done afterwards.

Figure 2 shows the already discussed details. a) is the inputimage, which is transformed at its centre where there is a singlewhite pixel. b) shows the output of the LPT implemented inOpenCV. Note how the white central pixel is shown as a blobin the right. Moreover, the corners of a) is visible in the right

a

b c

Fig. 2. a) Input image. b) Result of the LPT in OpenCV. c) The proposedLPTGF method. Note how the center pixel in a) appears like a blob in b)but a broad line in c) just as it should. Also note how the right part of b) isnot completely filled as the input region is quadratic, while in c) it is circularand hence the whole quadratic region is filled.

of b), leaving some parts unfilled (in black). c) shows theresult of the proposed Gaussian filtered LPT (LPTGF ), whichis sampled in a circular fashion so that there are no unfilledblack corners. Furthermore, the corners of quadratic area arenot present. We chose a different filling order compared toOpenCV, which is no problem as long as the same order isapplied in the matching. In any case, our result is rotated 90◦

depending on the scan order. The white central pixel in a)is visible as a band in the top in c) just as it should as theLPTGF samples in all directions around that centre point.The band is uneven as the pixel is square, but the filteringwill smooth it, which is desirable for the subsequent matching.Hence, OpenCV interpolates badly and we tried many otherMatlab R© implementations found on the web and some of themexhibited exactly the same behavior as in b), while others didnot show the white pixel at all.

If the output is distorted it will have an impact on thesubsequent PC, especially of there exists an rotation betweenthe images. Since there is a log function involved the pixelsclose to the center are larger in the output than pixels furtheraway and hence it is more important that they are correctlyrepresented. It will be even more important for small neigh-borhoods as in our case, compared to when matching wholeimages as the center then will occupy only a fraction of theresult.



To be sure that every pixel in the output is set, the image istraversed pixel by pixel in a scanline fashion. The position x, yis transformed into coordinates in the input image that is beingsampled. Some pixels will be oversampled and some undersampled and therefore it is important to perform some kindof interpolation. It has been proposed to use Gaussian filtersto change scale in images: [31] and therefore this approachwas used in our implementation of the LPT. In figure 2, b)has been computed using bicubic interpolation. The quality ofthe result, except for the error previously discussed, is quitesimilar to c), where a 5 × 5 mask was used with σ = 0.6.By changing σ it is possible to obtain a more blurred output,which often is desirable in order to cope with small differencesin transformations between the regions, and in the tests σ = 1.0was used.

Algorithm 1 shows an implementation of LPTGF usingGaussian filtering. The sampling is performed using algorithm2, which samples using a 2D Gaussian kernel. The unweightedGaussian function G at point x0, y0, sampling at xi, yj isdefined as

G(i, j) = e−((xi−x0)2+(yj−y0)2)/2σ2

(1)

Hence, this function is multiplied with each sample in the im-age at relative integer points xi, yi and the result is normalizedby the sum of all G(i, j).

input : I : Image; x0, y0 : point; r : radius; s : Size oflog polar image; m : Mask size; σ : Standarddeviation;

output: lp : Log polar Image;κ1 ← log(r + 1)/(s− 1);κ2 ← s/(2 ∗ π);for i← 1 to s do

for j ← 1 to s doλ← e((i−1)∗κ1) − 1;x← λ ∗ cos(j/κ2) + x0;y ← λ ∗ sin(j/κ2) + y0;lp(i, j)← Sample2D(I, x, y,m, σ);

endend

Algorithm 1: LPTGF Log Polar Transform of a circulararea in a image using Gaussian Filtering.

B. Phase Correlation

PC aims at finding the relative offset between two similarimages. For an example two aerial images taken with a shortdistance in time will have an overlapping area containing thesame surface on the ground. The displacement is computedusing the inverse Fourier transform of the normalized cross-power spectrum of the two images, which is a two-dimensionalDirac delta function ρ(x, y) = δ(x −∆x, y −∆y). The peaklocation corresponds to the displacement (∆x,∆y) betweenthe two images and the magnitude of the pulse depends onhow similar the images are. Nevertheless, this approach onlyworks for image translation and not for rotation or differencesin scale. Rotation and scaling can be handled by the LPT,which performs nonlinear and nonuniform sampling of thespatial domain. The rotation is handled by the nonlinear polarmapping and scaling by nonuniform logarithmic scaling. Thepeak location is now displaced depending on rotation and scale,

input : I : Image; x0, y0 : point; m : Mask size; σ :Standard deviation;

output: s : Sample;x← bx0c;y ← by0c;m← bm/2c;gs ← 0;s← 0;for i← −m to m do

for j ← −m to m dog ← e(−(((x+i)−x0)

2+((y+j)−y0)2)/2σ2);gs ← gs + g;s← s+ g · I(x+ i, y + j);

endends← s/gs;

Algorithm 2: Sample2D. Sample an image using GaussianFiltering

correspondingly, because scaling and rotation in the Cartesiandomain corresponds to translation in the log-polar domain.

C. Matching

The FFT part of the PC is performed on the result ofthe LPT for each feature point. This result is now stored asa feature vector before the actual matching. The size of thevector is m×m× n, where n is the number of feature pointsin one image and m is the size of the square region that theFFT computes. Now each point in the first image A must bematched to each point the second image B. The equation forthe PC requires that the conjugate is computed for one of theFFTs and this can be pre computed in one of the feature vectorsas well.

In the matching (see algorithm 3) each m × m sizedfeature is multiplied element vise (Hadamard multiplication◦) and then the IFFT is computed and the magnitude rof the maximum peak is subsequently compared for eachfeature. After comparing one feature in image A with allfeatures in image B the highest magnitude tells which oneis the best match. These will be regarded as tentative matches.Nonetheless, this do not prevent that one feature in image Bcan have several tentative matches in image A. However, thisis also the case for many other feature matching schemes andwill be handled by RANSAC that finds the final true matches.

We will refer to this algorithm as LPTGF PC (see algo-rithm 3). We also did tests using Normalized Cross Correlation(NCC) [32] on the result from LPTGF directly. Hence, no FFTwas done and in the Results section we refer to this algorithmas LPTGF NCC (see algorithm 4). Note that this is basicallythe same approach as proposed by Wolberg and Zokai [6],[23] except that they do template matching trying to find thelocation of the template, which covers a major part of the firstimage, traversing every pixel in the other image in order tofind the best match.

IV. RESULTS

A number of experiments were set up to examine how wellthe proposed approaches work. The images were all cropped



input : Ia : Image a; Ib : Image b;output: T : Tentative matches;Ha ← Harris(Ia);Hb ← Harris(Ib);for i← 1 to |Ha| doFa(i)← FFT (LPT (Ha(i)));

endfor j ← 1 to |Hb| doFb(i)← Conj(FFT (LPT (Hb(i))));

endρ′ ← 0for i← 1 to |Ha| do

for j ← 1 to |Hb| doR ← Fa(i) ◦ Fb(j)/ |Fa(i) ◦ Fb(j) + ε|;ρ← max(IFFT (R));if ρ > ρ′ then

ρ′ ← ρ;T (i)← j;

endend

endAlgorithm 3: LPTGF PC. Feature matching based onPhase Correlation and the Log Polar Transform.

input : Ia : Image a; Ib : Image b;output: T : Tentative matches;for i← 1 to |Ha| doLPa(i)← LPT (Ha(i));

endfor j ← 1 to |Hb| doLPb(i)← LPT (Hb(i));

endρ′ ← 0for i← 1 to |Ha| do

for j ← 1 to |Hb| doρ← NCC(LPa(i),LPb(j));if ρ > ρ′ then

ρ′ ← ρ;T (i)← j;

endend

endAlgorithm 4: LPTGF NCC. Feature matching based onNormalized Cross Correlation and the Log Polar Transform.

so that matching was performed on the inner circular part. Thisassures that the area to be matched is equal for each rotation0◦ − 360◦, which was done with a step of 6◦. Every imagewas thus matched to a rotated variant of itself.

The experiments were conducted on a number of imagesshown in table I and the results from rotation is in columna. However, only one of the rotated images (60◦) are shownin the table. It was also tested how change in illuminationand size affected the results. The results are in columns band c, respectively. The illumination was changed accordingto Inew = 0.75Iold + 0.25, so that the range was diminishedto 75% while making the whole image 25% brighter. The sizewas downscaled 25% and run both in an experiment of its ownas well as together with the change in illumination. The latteris in column d.

We chose to use the Harris corner detector in the tests,but nothing prevents the use any other feature detector ofyour choice. The top 600 strongest responses of the detectorwere chosen and are shown as green crosses in the imagesin the table. The number of inliers was computed for eachstep using a version of RANSAC (Optimal RANSAC [33] thatfinds all the inliers in each run so that the experiments willbe repeatable. Moreover, Harris is known to not be invariantto scale. However, since the transformation is know for eachrotation it was determined how many feature points η in thereference image actually were present at the correspondinglocation in the rotated image. The number of inliers aftermatching η′ was divided by η so that the percentage of truematches was obtained, which are shown in the table. Hence,the result is not dependent on the corner detector itself, but onhow well the matching works.

V. DISCUSSION

The results in table I show that LPTGF PC generallyperforms better than LPTGF NCC. The result from a. andb. show that both are not sensitive to light changes as PCperforms correlation on phase rather than amplitude and NCCdiminish the difference in illumination by normalization. Whenthere is a difference in scale (c. and d.), both methods performsless well, just as expected. The size of the feature vectoris to small to handle large scale changes. However, one canbuild a multi resolution scale space pyramid of different scaledversions to be matched as many others do [10], [23], [12]. Thiswill of course increase the amount of time needed to do thematching, but will make sure that the algorithm will be ableto handle large scale differences.

The two bottom rows in the table show photos taken on theground, one in Florence and the other in Venice, both in Italy.Obviously, the approach handles scale differences sufficientlygood for matching but less efficiently compared to the fourtop rows showing aerial photos. The reason is probably dueto the fact that these photos exhibit many details that are verysimilar and the response from the matcher is therefore alsoquite similar.

In the experiments 32 × 32 feature vectors were usedand the size might make the approach to slow for real-timeapplication of video sequences etc. However, the purpose wasto find a feature matcher that does a good job for stitchingof aerial photos where speed is not crucial. More important isthat the matcher is invariant to rotations, while differences inillumination can be removed by illumination correction [34]and the image scale is often quite similar for such images.

We conducted many tests changing the size of the radius rand the conclusion was that around 32 was optimal. Hencea rather large area is transformed using the LPTGF . Thetests also showed that this radius could be halved withoutmuch change in the result for images of almost equal scale.Moreover, the size s of the input to the PC could be decreasedto 16× 16 for such images with little change in the result. Infact even 8×8 often worked well for some cases. Nevertheless,the aim was to find a size that works also for changes in scaleto be on the safe side. Increasing to 64 × 64 will yield evenbetter results.



Images a b c d

Reference image Rotated Rotated with changed Rotated and scaled Rotated and scaled withto be matched illumination changed illumination

PC NCCa 100.00% 88.61%b 99.99% 88.61%c 96.95% 48.70%d 96.95% 48.70%

408× 408

PC NCCa 99.88% 90.57%b 99.88% 90.57%c 86.82% 45.25%d 86.81% 45.24%

408× 408

PC NCCa 99.80% 90.22%b 99.80% 90.22%c 78.38% 48.76%d 78.38% 48.76%

408× 408

PC NCCa 99.14% 83.81%b 99.10% 83.78%c 84.16% 28.38%d 83.78% 28.44%

570× 570

PC NCCa 99.67% 84.94%b 99.68% 84.95%c 92.48% 41.04%d 92.47% 41.03%

650× 650

PC NCCa 99.58% 93.38%b 99.57% 93.38%c 58.62% 51.33%d 58.62% 51.33%

1224× 1224

PC NCCa 96.07% 90.94%b 96.07% 90.94%c 34.88% 22.51%d 34.34% 22.79%

816× 816 a b c d

TABLE I. c©MIBAC-ICCD, AEROFOTOTECA NAZIONALE, FONDO RAF. c©GOOGLE EARTH c©ANDERS HAST. THE REFERENCE IMAGE TO THE LEFTIS MATCHED WITH IMAGES A,B,C AND D, USING BOTH THE LPT-PC AND LPT-NCC. THE RESULTS SHOW THAT LPT-PC GENERALLY WORKS BETTER.



VI. CONCLUSION

The method proposed in this paper is different from previ-ously proposed techniques in the following ways:

• The log polar transform is implemented using Gaus-sian filtering, which will make it produce better resultsthan many implementations available.

• The LPT is performed on a small circular neighbor-hood, so that the region will not change when rotated.

• The LPT is performed before the FFT, which willmake it possible to have a circular input region as theLPT will produce the required quadratic input regionfor the FFT.

• No exhaustive search in the image is necessary likein template matching. Instead the feature points canbe found by any feature detector, such as Harriscorners, Difference of Gaussians, or determinant ofthe Hessian. Hence, each feature point in one imageis compared to each feature point in the other imagein order to determine the best match.

The LPTGF PC generally performed better than theLPTGF NCC and both were able to handle rotations anddifferent illumination well. Both algorithm also cope with scaledifferences, but larger differences should be handled using amulti resolution scale space pyramid of different scales, justas many other algorithms do. Nevertheless, the conclusion isthat the LPTGF is very well suited for handling rotations andthat PC can be used for matching of the result from LPTGF .

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