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Radon transform of a straight line segment (a) to a sinogram in Radon space (d) with parameters …n; b; k†=(256, 0, 1).This is obtained by resampling the 2D Fourier spectrum (b) of the image, in n angular slices, into its 1D Fourier spectrum (c), andthen taking n 1D DFTs across (c). With a spatial enlargement factor of b ˆ 0, the sinogram exhibits some aliasing, which can bereduced by zero supplementation of the data.Figure 8. Radon transform of an image (a) with parameters …n; b; k† = …256; 0; k†.

b is set to zero resulting in aliasing andinterperiod artefacts. Increasing k decreases the radial sampling increment by 2k from to =2k . This reduces the aliasinge€ects but increases the size of the reordered grid by 2k .A FAST HOUGH TRANSFORM FOR THE PARAMETRISATION OF STRAIGHT LINES USING FOURIER METHODSalgorithm as a 1D ®ltering operation on the 1D Fourierspectrum. In the next sections we will discuss the®ltering processes used to enhance the peak structuresin the sinogram.1D FilteringPeaks in the Radon transform of f …r† are correlationpeaks indicating rays, parametrised by and , alongwhich straight line segments of f …r† lie.

The image is121usually edge enhanced with a 2D convolutional ®lterprior to application of the Radon transform. This can beequivalently applied as a ®ltering operation in the 2DFourier space. Further, by invoking the ®lter theorem[17], it can be reduced to a 1D ®ltering operation, bytaking projections of the 2D ®lter function. If the ®lter iscircular symmetric in the frequency domain then a crosssection can be taken and applied to each angular slice inthe 1D Fourier spectrum independent of the angle. Anyother post transform processes can be combined into the®lter to produce a net ®lter function.Figure 9.

Radon transform of Figure 8(a) with parameters …n; b; k† ˆ …256; b; 1†. k is set at 1. Increasing b increases the size of theFourier ®eld by 4b , thus decreasing the sampling increments in the 2D Fourier space Px ; Py by 1=2b . This reduces the aliasinge€ects and interperiod artefacts present in the sinogram. Higher values of b result in a greater improvement in the sinogram, butalso result in a huge increase in the computing time required.122C.W. HO ETAL.Figure 10. 2D di€erence of Gaussian (DOG) functions and corresponding 1D DOG functions. (a) …de ; di †=(80, 50).

(b) …de ; di † =(120, 75). (c) …de ; di †=(160, 100). (d) …de ; di †=(200, 125).Leavers and Boyce [5] propose a convolutional ®lter,which enhances the peak structure of the butter¯ydispersions in the sinogram, based on the angle betweenthe boundaries of the butter¯y distribution. Thisconvolution ®lter has a high positive response todistributions which have their largest value in the centerpixel, which falls o€ to approximately 50% to either sideand vanishes rapidly above and below the center pixel.However, the butter¯y distribution is determined by thelength of the line in the image and the number ofangular slices taken, and so each ®lter has to be tailoredto the parameters of the mapping.

A low number ofsamples results in a very broad butter¯y distribution,whereas a very large number of samples results in amuch thinner distribution. A more ecient approach isto apply a 1D ®lter to the resampled 1D Fourier space.This can be incorporated into our algorithm to ®lter theimage whilst in the Fourier space.Filter theoremThe ®lter theorem [17] states that the projection of the2D convolution of two functions is the 1D convolutionof the projections of each of the functions.

Sinceconvolution in the spatial domain is equivalent tomultiplication in the Fourier domain, the ®lteringprocess can be incorporated into our algorithm as a1D ®lter function, whilst in the Fourier domain.If we wish to convolve the 2D continuous functionf …r† with a ®lter function h …r†:g…r† ˆ f …r† h…r†…16†where the number of asterisks denotes the dimensionallyof the convolution.Therefore,F2 ‰g…r†Š ˆ F2 ‰ f …r†Š F2 ‰h…r†Š…17†This can be rewritten using the operator notation for thecentral slice theorem.

From Eqn (11):F1 R2 ‰g…r†Š ˆ F1 R2 ‰ f…r†Š F1 R2 ‰h…r†Š…18†Applying the Radon transform operator using Eqn (7):F1 ‰lg …; †Š ˆ F1 ‰lf …; †Š F1 ‰lh …; †Š…19†Therefore,lg …; † ˆ lf …; † lh …; †…20†Thus, the projection of a 2D convolution of twofunctions is the 1D convolution of the projections ofeach of the functions.

The same relationship holds for2D correlation. Therefore the projection of a 2D®ltering or correlation operation can be derived bysimply performing 1D ®ltering or correlation of theprojections of the original functions.It is computationally faster to perform this convolution as a ®ltering process whilst in the frequencydomain. Therefore, from Eqn (20) we haveg …; † ˆ f …; † h …; †…21†1D di€erence of Gaussian ®lterThe peak structure of the sinogram can be enhanced bybandpass ®ltering the 1D Fourier spectrum to enhanceA FAST HOUGH TRANSFORM FOR THE PARAMETRISATION OF STRAIGHT LINES USING FOURIER METHODS123slices.

Thus only one slice through the 2D frequencydomain ®lter is required to form the 1D DOG ®lter. Thisis applied to each angular slice in the 1D Fourierspectrum prior to it being conjugate mirrored.The DOG ®lter is a function which approximates ther2 Gaussian operator and acts as a second-di€erentialoperator on the image intensities to produce an edgemap of the reduced resolution image [24]. In thefrequency domain this is equivalent to a bandpass®ltering operation. The DOG ®lter, h…x; y†, can bewritten in the space domain as:h…x; y† ˆde2 exp…ÿde2 …x2 ‡ y2 ††ÿdi2 exp…ÿdi2 …x2 ‡ y2 ††Figure 11.

1D Di€erence of Gaussian (DOG) function h …; †at angle with parameters …de ; di †=(120, 75) for m=256. Thisis equivalent to an angular slice from the center to the edge ofthe corresponding 2D DOG function. The ®lter is applied toeach slice of the resampled 1D Fourier spectrum prior to itbeing conjugate mirrored.the high frequency components of the butter¯y distribution.

This was implemented using a 1D di€erence ofGaussian (DOG) ®lter, which can be formed by takingslices through the 2D frequency domain DOG ®lter.However, since the 2D DOG ®lter is circular symmetricin the 2D frequency space, the set of projectionsobtained about its center will be identical for all angular…22†where1de;i ˆ p2e;i…23†where e and i represent the excitatory and inhibitorystandard deviations of the two Gaussian functions.The DOG ®lter can be expressed in the frequencydomain as:F2 ‰h…x; y†Š ˆH…p†ÿP2x ‡ P2yˆ expde2!ÿ expÿP2x ‡ P2ydi2!…24†Figure 12. Radon transforms and 1D Fourier spectrums of the image in Figure 8(a) with Gaussian white noise of zero mean andvariance 0.05 added: (b),(e) un®ltered (c),(f) ideal (d),(g) ®ltered with parameters …de ; di † ˆ …120; 75†: …n; b; k† ˆ …256; 1; 1†. Thisattenuates the low and high frequency components in the Fourier spectrum (d) to produce a peak enhanced sinogram (g).

This canbe thresholded to produce the maxima in Radon space (f).124C.W. HO ETAL.Figure 13. Radon transform of a real 8-bit grayscale image (a) with parameters …n; b; k†=(256, 1, 1). Not all the maxima points inthe un®ltered sinogram (e) are visible compared with the sinogram (f) produced by transforming the edge enhanced image (b). Theedge enhancement operation on the image is equivalent to a bandpass ®ltering operation in the 1D Fourier space (d) prior to a 1DDFT being taken.de and di produce less pronounced positive and negativeswings either side of the zero crossing of the ®lter andlead to a greater reduction in the resolution of theimage.Taking angular slices by substituting for p ˆ ^n:h …; † ˆH…p†jpˆ^nÿ… cos †2 ‡… sin †2ˆ expd 2e22ÿ… cos † ‡… sin †di2 2 2ÿÿÿexpˆ exp2dedi2ÿ exp!Results of 1D ®ltering!…25†This is independent of , since the DOG ®lter is acircular symmetric 2D ®lter.

Therefore by specifying thevalues of de and di we can de®ne our desired ®lter. Marrand Hildreth [24] show that the best approximation tothe r2 of a Gaussian operator occurs if the ratiobetween the standard deviations of the inhibitory andexcitatory Gaussians, i and e is 1.6. Figure 10 shows2D DOG functions and their corresponding 1D DOGfunctions for various values of (de ; di ). Larger values ofFigure 11 shows the 1D DOG ®lter function h …; † atangle with parameters (de ; di )=(120,75) for m=256.This is independent of and so it is applied to eachangular slice along m.

The resampled Fourier spectrumis conjugate mirrored after being sampled and so it isonly necessary to ®lter the sampled half of the Fourierspectrum. The ®lter of size m n is dependent on theparameters (n; b; k) of the mapping and the size of theinput function. Thus the ®lter function will change withthe length of the sampled half Fourier spectrum, m andwill have to be scaled with di€erent values of m toproduce the desired 1D DOG function.This ®lter enhances the peak structures in thesinogram by attenuating the low frequency componentsA FAST HOUGH TRANSFORM FOR THE PARAMETRISATION OF STRAIGHT LINES USING FOURIER METHODS125Figure 14.

E€ect of applying di€erently scaled DOG ®lters to the 1D Fourier spectrum of the image in Figure 13(a) usingparameters …n; b; k†=(256, 1, 1). The ®lter with …de ; di † = (200, 125) gives a much sharper sinogram (f) than the sinogram (e)corresponding to …de ; di †=(120, 75). Note that the circular washer has not been detected but the edges of the image have, resultingin spurious maxima points.of the 1D Fourier spectrum. Figure 12 shows the Radontransform and un®ltered 1D Fourier spectrum of theimage in Figure 8(a) with Gaussian white noise of zeromean and variance 0.05. Parameters are (n; b; k)=(256, 1, 1), so m=256 and (de ; di )=(120, 75). The Fourier spectrum is bandpass ®ltered to attenuate the boththe low and high frequency components in the Fourierspectrum.

This produces a peak enhanced sinogramwhich can be thresholded to locate the maxima points inthe sinogram.When applying the Radon transform to real-worldimages, the image is usually ®rst edge enhanced prior totransformation (Figure 13). However, when evaluatingthe Radon transform using Fourier methods the edgesmay be equivalently detected with a 2D bandpass ®lterin the 2D frequency space. This can be combined intoour ®lter to produce a net ®lter function. As with the 2DDOG ®lter the values of (de ; di ) need to be tailored tothe image being transformed. Increasing the values of deand di highpasses the 1D Fourier spectrum to preservehigh frequencies and attenuate low frequencies. Figure14 shows the e€ects of the ®ltering the 8-bit grayscaleimage of Figure 13(a), with parameters (n; b; k)=(256, 1, 1).

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