Log-polar transform-based wavelet-modified maximumaverage correlation height filter for distortion invariance

in a hybrid digital–optical correlator

Amit Aran,1 Naveen K. Nishchal,2 Vinod K. Beri,1 and Arun K. Gupta1,*1Photonics Division, Instruments Research and Development Establishment, Dehradun-248 008, India

2Department of Physics, Indian Institute of Technology Guwahati, Guwahati, India

*Corresponding author: akgupta@irde.res.in

Received 25 June 2007; revised 2 October 2007; accepted 3 October 2007;posted 4 October 2007 (Doc. ID 84437); published 12 November 2007

We discuss and implement a log-polar transform-based distortion-invariant filter for automatic targetrecognition applications. The log-polar transform is a known space-invariant image representation usedin several image vision systems to eliminate the effects of scale and rotation in an image. For in-planerotation invariance and scale invariance, a log-polar transform-based filter was synthesized. In cases ofin-plane rotation invariance, peaks shift horizontally, and in cases of scale invariance, peaks shiftvertically. To achieve out-of-plane rotation invariance, log-polar images were used to train the wavelet-modified maximum average correlation height (WaveMACH) filter. The designed filters were imple-mented in the hybrid digital–optical correlation scheme. It was observed that, for a certain range ofrotation and scale differences, the correlation signals merge with the strong dc. To solve this problem ashift was introduced in the log-polar image of the target. The use of a chirp function for dc removalhas also been discussed. Correlation peak height and peak-to-sidelobe ratio have been calculated asmetrics of goodness of the log-polar transform-based WaveMACH filter. Experimental results arepresented. © 2007 Optical Society of America

OCIS codes: 070.0070, 070.4550, 070.5010, 100.3008.

1. Introduction

Recognition of distorted targets is a challengingtask in pattern recognition applications [1–4]. Thematched filter is the earliest and simplest of all cor-relation filters. Several filter design techniques havebeen proposed to extend this fundamental concept[1–10]. Two approaches have been reported in theliterature for rotation-invariant filter synthesis [1–7].In one approach, geometric invariance properties areemployed for filter formulation, and in the other ap-proach a number of rotationally distorted images aretrained, which results in a filter. The circular har-monic function (CHF) filter approach employs thegeometric invariance properties while the syntheticdiscriminant function (SDF) approach uses a number

of rotationally distorted images for training. Severalvariations of SDF have been reported [6–10].

Mahalanobis et al. [6] introduced the maximumaverage correlation height (MACH) filter in 1994,which has proved to be a powerful correlation filter.The MACH filter can be designed to maximize thecorrelation peak height, peak sharpness, and noisesuppression while also being tolerant to distortions inthe target that fall between the distortions in thetraining set. Nevel and Mahalanobis [8] reported twoextensions of the MACH filter as extended MACHand generalized MACH. Bhuiyan et al. [10] reporteda matched spatial filter-based algorithm to detect,classify, and track single and multiple targets fromgray-scale image sequences. They used a combinationof a MACH filter and a polynomial distance classifi-cation correlation filter.

Young and his co-workers in 1993 outlined anoptical–digital hybrid correlator system that allowsthe potential for a multikilohertz reference template

0003-6935/07/337970-08$15.00/0© 2007 Optical Society of America

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search on data acquired at video rates [11,12]. In thishybrid approach, the input data are digitally Fouriertransformed and multiplied with the digitally presyn-thesized distortion-invariant filters. This is called aproduct function, which is then displayed over anelectrically addressed spatial light modulator (SLM).The SLM is illuminated with a coherent beam of laserlight, and its optical Fourier transform is obtainedthrough a lens, which results in correlation outputs.This approach exploits the asymmetry that existsbetween the speed requirements for performing theFourier transform of input signal and that requiredfor template search [13].

The application of wavelets in optical pattern rec-ognition has been reported [14–20]. Recently, Goyalet al. [18,19] reported a wavelet-modified MACH fil-ter, called WaveMACH filter, for recognition of in-plane and out-of-plane rotated images. Use of wavelettransform improved the performance of a MACH fil-ter by reducing the number of filters required foridentifying a rotated target at any angle between 0°and 360° and enhancing the autocorrelation peak in-tensity. Recently, Bone et al. [21] proposed a methodof detecting targets in still images despite any kind ofgeometric distortion. They employed a log r � � map-ping to give invariance to in-plane rotation and scaleby transforming rotation and scale variations of thetarget object into vertical and horizontal shifts. TheMACH filter was trained on the log r � � map ofthe target for a range of orientations.

In this paper, we discuss and implement a log-polartransform-based distortion-invariant filter in a hy-brid digital–optical correlator. For in-plane rotationinvariance (0°–360°) and scale invariance (50%–200%) a log-polar transform-based filter was synthe-sized. In the case of in-plane rotation-invariance,peaks shift horizontally, while in the case of scaleinvariance, peaks shift vertically. To achieve out-of-plane rotation invariance (0°–360°), log-polar imageswere used to train the WaveMACH filter. The prob-lem of merging of correlation signals with the strongdc and its solution for optical implementation is dis-cussed. The use of a chirp function for dc removal isalso explained. Correlation peak height and peak-to-sidelobe ratio are calculated as metrics of goodness ofthe log-polar transform-based WaveMACH filter. Ex-perimental results are presented.

2. Log-Polar Transform

The log-polar transform is a space-invariant imagerepresentation used in many image vision systemsto eliminate the effects of scale and rotation in animage. In this representation, pixel separation in-creases linearly with distance from a central point.After applying the log-polar transform operation, a ro-tated and scaled image is converted into a correspond-ing log-polar image, which is rotation invariant andnearly scale invariant. The logmap uses a variation onthe basic x–y grid sensor used in conventional imageprocessing [22]. However, any orientation changecauses row shifting in the log-polar image.

The image mapped from the Cartesian space intothe polar space is called log-polar mapping. Consid-ering z as the Cartesian plane with coordinates xand y and w as the polar plane with coordinates uand v, the complex logarithmic mapping can bewritten as [21]

w � log z, (1)

z � x � iy, (2)

where

r � �x2 � y2�1�2, � � arctan�y�x�.

After substitution, w can be written as

w � log r � i� � u � iv, (3)

where u � log r and v � �.Log-polar mapping offers in-plane rotation and

scale invariances in addition to some other features,such as wide field of view, reduced pixel count, and ahighly focused central area [21–23]. Changes due toscale and rotation in a Cartesian coordinate systemmanifest themselves as shifts in a log-polar coordi-nate system. The rotation and scale changes resultin horizontal and vertical shifts. However, log-polarmapping is not shift invariant, therefore rotation andscale invariance properties hold only if they are withrespect to the origin of the Cartesian image space.

Log-polar images were used to train the Wave-MACH filter and were used in the hybrid digital–optical correlator architecture. In such a correlationscheme, input target is log-polar transformed, andthen its Fourier transform is obtained. Thus, ob-tained frequency spectrum is multiplied with theWaveMACH filter, and the resulted product functionis encoded for optical implementation. The opticalFourier transformation of the product function is cap-tured, which produces correlation signals along witha strong dc. It was observed that the correlation sig-nals merge with the strong dc. This problem occurredonly for a certain range of rotated (�15° from center)and scaled ��10%� images. To solve this problem sothat correlation signals do not merge with dc, thelog-polar image was shifted.

Figure 1(a) shows the original image of Barbara ofsize 256 � 256 pixels, and Fig. 1(b) shows the log-polar image in the r � � coordinate. After convertingan image from Cartesian to log-polar coordinates, allthe information is found contained up to a certainlength, �2r, where r is the radius of the innermostcircle, as depicted in Fig. 1(a). The rest of the log-polar transformed image, as shown in Fig. 1(b), isfound to be redundant. Therefore, we appropriatelyshifted the log-polar image. Figure 1(c) shows theshifted log-polar image. The shift in the log-polarimage introduces the separation between correlationsignals and the strong dc without losing any relevantinformation. Thus placement of the intensity-sensing

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device captures the correlation outputs. In this paper,we implemented this technique for capturing the cor-relation outputs.

3. WaveMACH Filter

A distortion-invariant filter is basically a linear com-bination of preprocessed training images, where thepreprocessing emphasizes high frequencies in at-tempting to produce a delta function correlationpeak. The MACH filter, a rotation-invariant filter, is

a variant of the SDF filter [6]. The filter is designedconsidering the expected distortions in the target ob-ject. The MACH filter maximizes the relative heightof the average correlation peak with respect to theexpected distortions. The filter is created by maximiz-ing a performance metric called the average similar-ity measure (ASM). The smaller the value of theASM, the more invariant the response of the filter.The MACH filter f is given by [6]

f � S�1m, (4)

where S is a diagonal matrix, called ASM, and m isthe average of the training images that have under-gone a two-dimensional Fourier transformation. TheASM is defined as

S �12�

i�1

N

�Xi � M��Xi � M�*, (5)

where Xi are the individual training images in theFourier domain. The symbol * is used to indicate theconjugate transpose and M is the mean of the train-ing images.

Wavelet transform has been attracting increasedattention in the optical pattern recognition commu-nity because of its attractive multiresolution, denois-ing, and feature extraction capabilities [14–16]. Dueto the edge-enhancement property, the wavelet-matched filter produces a sharper correlation. Theconcept of a wavelet-matched filter has been moti-vated by its improved discrimination properties com-pared to the classical matched filter and its improvedsignal-to-noise ratio compared to the phase-only filter[15]. Goyal et al. [18] combined the advantages ofwavelet and MACH filter and named the Wave-MACH filter.

In this study, we used a two-dimensional Mexican-hat wavelet function with the MACH filter. TheMexican-hat wavelet function is a second derivate ofthe Gaussian function and is represented as [14]

h�x, y� � �1 � �x2 � y2��exp��x2 � y2

2 . (6)

Fig. 2. Experimental setup used. SF, spatial filter; CL, collimat-ing lens; FT lens, Fourier transform lens; PBS, polarizing beamsplitter; SLM, spatial light modulator.

Fig. 1. (a) Image of Barbara, (b) log-polar image, (c) shifted log-polar image.

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In the frequency domain it is given as

H�u, v� � 4�2�u2 � v2�exp<�2��u2 � v2�=. (7)

The MACH filter defined in Eq. (4) is multiplied withthe frequency spectrum of the Mexican-hat waveletfunction, as given in Eq. (7). Thus the WaveMACHfilter is expressed as

WaveMACH � S�1m�H�u, v��2. (8)

The waveMACH filter has been reported to be in-plane, out-of-plane rotation invariant, and scaleinvariant [18–20]. The use of wavelet transformimproves the performance of the MACH filter. Itreduces the required number of filters for identify-ing a rotated and scaled target and enhances thediscrimination capability.

4. Experimental Results

The log-polar transform-based correlation filters syn-thesized were implemented in a hybrid digital–opticalcorrelator. The basic experimental setup used isshown in Fig. 2. A beam from a laser diode source� � 670 nm� is passed through a spatial filter and

then through a lens CL �f � 75 mm�, which expandsand collimates the beam. An input target was con-verted into log-polar coordinates and then digitallyFourier transformed and multiplied with the presyn-thesized WaveMACH filters, trained with log-polar im-ages. The multiplied product results in a complexfunction, which needs to be encoded onto an electri-cally addressed spatial light modulator (SLM) [with a832 � 624 resolution and a pixel size of 30 � 30 m,from Holoeye, Germany] working in the transmissivemode. For complex data encoding we implemented thescheme proposed by Davis et al. [24,25]. The encodedcomplex data were then displayed on the SLM to ob-tain the optical Fourier transformation through the FTlens �f � 105 mm�. The transformed intensity is thencaptured using a CCD camera (having a 752 � 582resolution and a pixel size of 6.5 � 6.25 m fromSony, Japan) connected to a personal computerthrough a frame-grabber card.

The product functions obtained after the use ofgenerated filters in the simulations were displayed onthe SLM one by one. Figure 3 shows the images offour tanks. Tank 1 was used as the true class imagefor synthesizing the filter, and Tanks 2–4 are thefalse class images. All the images are 45 � 25 pixels.The images have been zero padded to make them128 � 128 pixels. Figures 4(a)–4(d) show the true

Fig. 3. Images of tanks. Tank 1 corresponds to the true class image and Tanks 2–4 correspond to false class images.

Fig. 4. (a)–(d) True class images at angles of in-plane rotation 1°, 90°, 240°, and 330°, respectively, (e) false class image, and (f)–(j)corresponding correlation outputs.

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class images at angles of in-plane rotation 1°, 90°,240°, and 330°, respectively, Fig. 5(e) false class im-age, and Figs. 5(f)–5(j) corresponding correlation out-puts. Figures 5(a)–5(d) show the true class images atthe various scale differences of 80%, 130%, 150%, and180%, respectively, Fig. 5(e) false class image, andFigs. 5(f)–5(j) corresponding correlation outputs. Forfull out-of-plane rotation invariance (0°–360°) log-polar transformed images have been used to train theWaveMACH filter. Figures 6(a)–6(d) show the trueclass images at angles of out-of-plane rotation of 60°,150°, 270°, and 330° respectively. Fig. 6(e) false classimage, and Figs. 6(f)–6(j) corresponding correlationoutputs. The rotation angles and scaled versions werechosen arbitrarily. In all the experimental results,the central bright region is the zero order term, thedc. Two bright sharp spots �1 and �1 diffractionorders, corresponding to two autocorrelation peaks,are obtained when the filters are correlated with

trained images belonging to the true class. For falseclass images no peaks are obtained.

To discard one of the autocorrelation peaks and thestrong dc, usually chirp encoding is implemented.Due to chirp encoding the correlation signals are fo-cused in three different planes. Thus placing the CCDcamera at a particular plane only the desired auto-correlation peak is recorded. We used a chirp functionas defined in [18], with the designed filter. Figures7(a)–7(c) show the true class images at angles of in-plane rotation 0°, 60°, and 120°, Fig. 7(d) false classimage, and Figs. 7(e)–7(h) corresponding correlationoutputs obtained after introducing chirp function. Itwas observed that using a chirp function did helpcapture a single autocorrelation peak but could notalleviate the blurred dc. This may be attributed to theshift introduced in the log-polar image. In this case, ashift of the log-polar mapped image is not possibleover a distance of more than the available space

Fig. 5. (a)–(d) True class images at scale differences 80%, 130%, 150%, and 180%, respectively, (e) false class image, and (f)–(j)corresponding correlation outputs.

Fig. 6. (a)–(d) True class images at angles of out-of-plane rotation 60°, 150°, 270°, and 330°, respectively, (e) false class image, and (f)–(j)corresponding correlation outputs.

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shown in Fig. 1(b). It also limits the separation be-tween autocorrelation peak and defocused�blurreddc. Thus use of a chirp function for dc removal is notappropriate for the log-polar transform-basedWaveMACH filter.

Goyal et al. reported the WaveMACH filter forrecognition of in-plane and out-of-plane rotated im-ages [18,19]. Using WaveMACH, seven and ninefilters are required to achieve full in-plane and out-of-plane rotation invariance, and six filters for scaleinvariance from 55% to 220%, respectively. Log-polar transform offers in-plane rotation and scaleinvariance around its center, since rotation or scalechanges simply produce vertical or horizontal shiftsin the polar space. In this study, we used a log-polartransform-based WaveMACH filter to achieve fullin-plane and out-of-plane rotation invariance andscale invariance from 50% to 200%. In this case toachieve full in-plane rotation and scale invariancefrom 55% to 220% only one filter is required and

for complete out-of-plane rotation invariance, onlysix filters are required. The correlation results arecomparable with the WaveMACH filter [18,19].Therefore, the designed filter would reduce thememory requirement for filter storage in a practicalsystem.

5. Performance Measure

To characterize the goodness of the log-transformed-based WaveMACH filter in identifying the distortedtargets, we calculated the values for correlationpeak height and peak-to-sidelobe ratio. Figure 8shows the plot for peak height versus the in-planerotation angle for Tank 1 as the true class imageand Tanks 3 and 4 as the false class images. Figure9 shows the plot for peak height versus scale differ-ences for Tank 1 as the true class image and Tanks2 and 3 as the false class images. Figure 10 showsthe plot for peak height versus the out-of-plane ro-

Fig. 8. (Color online) Plot of correlation peak height versus angleof in-plane rotations with Tank 1 as true class and Tanks 3 and 4as false class targets.

Fig. 7. (a)–(c) True class images at angles of in-plane rotation 0°, 60°, and 120°, respectively, (d) false class image, and (e)–(h)corresponding correlation outputs obtained after introducing chirp function.

Fig. 9. (Color online) Plot of correlation peak height versus per-centage scale variations with Tank 1 as true class and Tanks 2 and3 as false class targets.

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tation angle for Tank 1 as the true class image andTanks 2 and 4 as the false class images.

It is said that, for valid targets, the MACH filtermaximizes the peak-to-sidelobe ratio (PSR), definedas [4, Chap. 6]

PSR �peak �

�, (9)

where � and � are the mean and standard deviationof the correlation values in some neighborhood of thepeak. We calculated the values of PSR after employ-ing the log-transformed-based WaveMACH filter. Forcalculating the sidelobes we used an area of 25 �25 pixels and for measuring peaks we used an area of5 � 5 pixels. Figure 11 shows the plot for PSR versusthe angle of out-of-plane rotation.

6. Conclusion

A log-polar transform-based WaveMACH filter wasdesigned and implemented in a hybrid digital–optical correlator architecture for achieving the in-plane and out-of-plane rotation and scale invariance.In cases of rotation invariance (0°–360°), peaks shifthorizontally, while in cases of scale invariance (50%to 200%), peaks shift vertically. For out-of-plane ro-tation invariance log-polar images were used to trainthe WaveMACH filter. The problem of merging ofcorrelation signals with the strong dc, and its solu-tion for optical implementation have been discussed.The use of a chirp function for dc removal has alsobeen explained. Correlation peak height and thepeak-to-sidelobe ratio have been calculated as met-rics of goodness of the proposed log-polar transform-based WaveMACH filter. The designed filter wouldreduce the memory requirement for filter storage in apractical system. With the use of log-polar cameras,real-time application of this filter would become areality [26]. Experimental results have been pre-sented in support of the proposed idea.

The authors are grateful to Shri S. S. Sundaram,Director, Instruments Research and DevelopmentEstablishment Dehradun for his encouragement andgiving permission to present and publish this paper.

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