2D Wedge Filter Design and Applications in Oriented Line Extraction

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1 D Wedge Filter Design and Applications in Oriented Line Extraction RADU MATEI Faculty of Electronics Telecommunications and Information Technology Gheorghe Asachi Technical University Bd Carol I nr11 Iasi ROMAIA rmatei@ettituiasiro Abstract:- An analytical design method is proposed for two-dimensional recursive zero-phase wedge filters It relies on a frequency mapping applied to a 1D IIR low-pass maximally-flat prototype filter of a specified shape The design is presented in two versions one entirely analytical based on the bilinear transform and another involving a numerical approximation step This paper focuses on the design method and provides simulation results of oriented line extraction from a real-life image to prove the usefulness of these filters in image processing Key-Words:-D IIR filter design directional filters frequency transformations approximation 1 Introduction The field of two-dimensional filters has known a steady development due to their applications in image processing and various design methods have been proposed [1] A commonly used design technique is to start with a 1D prototype filter and transform it in order to obtain a D filter with desired frequency response The existing design methods for D IIR filters rely on 1D analog prototypes using spectral transformations from s to z plane via bilinear or Euler method followed by z to ( z1 z) mappings []-[4] There are several types of filters with orientationselective frequency response useful in various tasks such as edge detection motion analysis etc In [5] a filter bank for directional image decomposition was proposed A class of linear operators with directional response was introduced in [6] Wedge filters named so due to their wedge-like shape in the frequency plane find useful applications in feature extraction eg in texture classification [7] Steerable wedge filters were described in [8] Different design and implementation methods for FIR and IIR fan and wedge filters were introduced in papers like [9]-[14] In this work two design methods in the frequency domain are proposed for a class of D zero-phase wedge filters The idea of this design approach was first introduced in [15] Two ideal wedge filters in the frequency plane are displayed in Fig1 The filter in Fig1(a) has its frequency response along axis The angleaob will be called aperture angle A more general wedge filter is shown in Fig1(b) with an aperture angle BOD oriented along an axis CC ' which forms an angleaoc with the frequency axis O A general wedge filter is considered with specified values for the aperture angle and orientation angle of its longitudinal axis as shown in Fig1(b) A maximally-flat prototype filter will be used for design We approach here only zero-phase filters particularly useful in image filtering since they are free of phase distortions Wedge Filter Design Method 1 Prototype Filters and Frequency Transformations Let us consider the transfer function H ( z) of an IIR filter of ordergiven by: M P( z) i j H ( z) piz qjz (1) Q( z) i1 j1 We can derive a zero-phase prototype filter (with real-valued transfer function) which preserves only the magnitude characteristic of H ( z ) while its phase is zero throughout the frequency domain [ ] Let us consider the magnitude given by the absolute value of H ( z) H ( e ) taken from (1): j M H ( e ) p exp( jn) q exp( jm) n j () n m Using a symbolic calculation software we derive a j rational approximation Ha ( ) of H ( e ) in powers m ISB:

2 f ( s s ) a s s tg s tg s (8) S Two design procedures will be next presented one entirely analytical and another using a numerical approximation step Fig 1 Ideal wedge filters specified in the frequency plane: (a) along axis ; (b) oriented at an angle j of since H ( e ) has even parity An efficient rational approximation is Chebyshev-Padé method The function Ha ( ) assuming the numerator and denominator of the same degree 4k (k ) can be factored into functions of the form: Hp ( ) b b b 1 a a (3) Usually the value 4 is satisfactory for accuracy and we take H ( ) H ( ) The function H ( ) a p is a rational approximation of the frequency response magnitude of prototype (1) Let us consider the zero-phase maximally-flat low-pass IIR prototype function in the complex variable s derived from the general expression (3) where s : s 1895s Hp ( s) (4) s 53357s This simple prototype will be next used to design a zero-phase wedge filter with specified parameters A wedge filter having O as longitudinal axis results using the 1D to D frequency transformation (for ): f ( 1 ) a 1 (5) The coefficient a1 tg( ) is a measure of the aperture angle More generally if the wedge filter axis has an orientation specified by an angle about axis the filter results by rotating the axes of the plane ( 1 ) by an angle as defined by the following linear transformation: 1 cos sin 1 sin cos (6) where 1 are the original and 1 the rotated variables The 1D to D frequency mapping can be written: f ( ) a tg tg (7) Since s 1 j 1 and s j the mapping (7) in the complex plane ( s1 s) takes the following form: a Design Method Using the Bilinear Transform In the general case of an oriented wedge filter replacing in (3) by the expression (8) we derive the D filter frequency response in s 1 s namely HS ( s1 s ) a ratio of two-variable polynomials of 4- th degree The next step is mapping HS ( s1 s) onto the complex plane ( z 1 z ) using bilinear transform For our purposes the sample interval is T 1 so the double bilinear transform for variables s 1 s in the complex plane ( s1 s) has the form: s1 ( z1 1) ( z1 1) s ( z1) ( z 1) (9) substituting s 1 s from (9) in the function HS ( s1 s) found in the previous step we find a rational function HZ ( z1 z) in z 1 z The designed D filter will have noticeable distortions towards the frequency plane limits compared to the ideal response This is due to frequency warping effect of the bilinear transform expressed by the frequency mapping: ( T ) arctg T (1) where is the frequency of the discrete-time filter and a is the frequency of the continuous-time filter In order to correct this error we apply a prewarping using the inverse of mapping (1) Taking the sampling period T 1 in (1) we substitute the mappings: tg tg (11) 1 1 a A more suitable rational approximation of mappings (11) can be obtained using again the Chebyshev- Padé method: tg (1 1 ) g( ) (1) which is very accurate in the frequency range [ ] and has the low-order advantage Using (7) (11) we get the transformation including frequency pre-warping for 1 : a tg( ) tg( ) tg 1 f P ( 1 ) tg( 1 ) tg tg( ) (13) Substituting the rational approximation (1) written for both frequency variables 1 and into (13) we get a rational expression of frequency transformation f P ( 1 ) in 1 and ISB:

Then taking into account that s1 j1 and s j in the complex plane ( s1 s) we obtain the frequency transformation f P ( s1 s ) : as1 (1 1 s ) tg s(1 1 s1 ) fp ( s1 s ) tg s1 (1 1 s ) s(1 1 s1 ) (14)

3 Then taking into account that s1 j1 and s j in the complex plane ( s1 s) we obtain the frequency transformation f P ( s1 s ) : as1 (1 1 s ) tg s(1 1 s1 ) fp ( s1 s ) tg s1 (1 1 s ) s(1 1 s1 ) (14) This 1D to D mapping includes the frequency prewarping along the two axes Applying the double bilinear transform (9) after simplification we obtain a mapping F : : af ( z1 z) tg g( z1 z) B ( z1 z) F( z1 z ) tg f ( z z ) g( z z ) A ( z z ) (15) where f ( z1 z) and g( z1 z) are symmetric in z 1 z : f ( z z ) 14z z 1z z 14z 14z 1z g( z z ) 14z z 1zz 14z 14z 1z 14 (16) ext the term template will be used meaning the coefficient matrices of the numerator and denominator of a transfer function H ( z1 z ) Denoting by B and A the 33 templates for the numerator B ( z1 z) and denominator A ( z1 z) we get: 14tg14 1tg 14tg14 A 1 1 (17) 14tg14 1tg 14tg14 The matrix B is A clockwise rotated with 9 : then 9 B A If matrixmcorresponds to g( z1 z ) 9 M corresponds to f ( z1 z) B M 9 tg M A 9 tg and M M Finally from (15) the frequency mapping can be expressed as: T T F ( z z ) z Bz z Az (18) where z 1 and z are the vectors: 1 z [ z z 1 z z ] z z z z z [ 1 ] (19) and the 55 matrices BB Bp and AA A resulted by convolution preserve the property 9 BA We apply the mapping (18) directly to the 1D prototype (3) obtaining the D wedge filter transfer function in z 1 and z : T T W W H ( z z ) Z B Z Z A Z () W where the vectors Z1 and Z have the form: 1 1 Z [ z z z 1] 1 Z [ z z z 1] with 9 ; the matrices AW and aa (1) BW result as: B b ( AA) ba ( AB) ba ( BB) W A AAaa ( AB) ( BB) W () Even if the templates result relatively large this is the price paid for ensuring a good linearity of the wedge filter shape in the frequency plane The frequency pre-warping has increased the filter order but the filter templates result as a convolution of smaller matrices ( ) being partially separable At least the numerator of prototype (3) may have real roots and can be factored involving convolution of smaller matrices For instance the numerator in (4) is factored as: ( ) (1 195 ) (1 194 ) (3) and is realized as a convolution of the two 55 templates taking into account (18): A 195a B A 194a B (4) Using the prototype (4) we designed a wedge filter with an aperture angle and orientation angle 5 For these specifications we get the values a tg( )=349 tg = 765 The frequency response and contour plot of the designed filter are shown in Fig 3 Design Method Using umerical Approximations The second design method for zero-phase wedge filters starts again from a zero-phase 1D prototype of the general form (3) We use again the 1D-D frequency mapping (7) Since (3) is a rational function of the design method requires finding the discrete approximation of the function F ( ) f ( ) a tg tg 1 (5) Fig Frequency response and contour plot of an oriented flat-top wedge filter with aperture and orientation ISB:

4 The approximation results using the change of variable : 1 arccos x1 arccos x (6) and F ( 1 ) is mapped into a functiong ( x1 x) The next step is to find a bivariate Taylor series expansion of G ( x1 x) Using again the symbolic calculation we easily determine this series expansion in x 1 x Then we return to the former variables by substituting back x1 cos1 and x cos into G ( x1 x) We obtain an approximation of F ( 1 ) in powers of cos1 and cos Applying trigonometric identities we find F ( 1 ) as: F ( 1 ) amn cos( m1 n ) (7) mn where is chosen to ensure a desired precision Usually we can take the value The coefficients amn depend on angle and have polynomial expressions in variable tg Let us design a wedge filter with specifications given in Section II ie prototype (4) with parameters: a tg( )=349 tg = 765 Our method gives the approximation: F ( 1 ) a [ cos( ) 1134cos( ) (cos( ) cos( )) (cos( ) cos( )) (cos( ) cos( )) cos( ) 14584cos( ) (cos( ) cos( ))] 1 1 which corresponds to the 55 template: Wa (8) (9) found by identifying coefficients of the D Z transform from (8) Once obtained the 1D to D frequency mapping of the form F ( 1 ) the next step is to substitute it in Hp ( ) from (3) Thus we get the D function: H ( ) B( ) A( ) (3) The templates B and A result according to Hp ( ) in (3) as: Bb Eb1 Wb bww (31) AEa1Wb aww The symbol stands for matrix convolution ande is a 9 9 matrix with zero elements and central element equal to 1 The 9 9 matrix Wb is obtained by bordering the 5 5 matrixwwith zeros in order to be summed with matriceseand WW An advantage of the second design method over the first one is that it avoids the use of bilinear transform which is known to introduce large distortions unless a frequency pre-warping is applied The second design approach is somewhat simpler but requires the use of bivariate Taylor series expansion for a given orientation angle 4 Fan Filter Design Example The proposed method can be used as well in designing fan filters We consider two types of fan filters specified as in Fig3 (a) (b) The filter with the shape shown in Fig3 (a) is ideally described in the frequency plane as: H F ( ) (3) otherwise This fan filter is a particular wedge filter with so a 1 tg ; the mapping (7) reduces to: f ( ) (33) 1 1 In this particular case the templatewresults as: W (34) The fan filter based on the prototype (4) shown in Fig3(c) visibly preserves the maximally-flat response For the second fan filter type shown in Fig3(b) and 4 so we obtain a1 and tg 1 ; the frequency transformation (7) thus simplifies to: f ( ) ( ) ( ) (35) In this particular case we obtain a particular matrix W and the final filter templates result again using relations (31) The stability of the D filters designed using the proposed methods will be studied in further work For D filters it is generally difficult to take stability constraints into account during approximation stage ISB:

(a) (b) (a) (b) (c) Fig 3 (a) (b) Two versions of ideal fan filter bandpass regions; (c) fan filter frequency response Various methods were developed to order to separate stability from approximation

a specified orientation from a given image The spectrum of a straight line is oriented in the plane ( 1 ) at an angle of with respect to the line direction Fig4 shows a real gray-scale image

lines with spectrum oriented along filter characteristic remain unchanged while all the rest are more or less blurred due to directional filtering The filter aperture angle was 5 and three different

applications as well The advantage of these analytical design methods over other approaches is that the frequency response of the resulted D wedge filter depends explicitly on the specified

(c) (d) Fig 4 (a) Grayscale straw texture image; (b) (c) (d) filtering result using aperture 5 and three orientations: 6 3 3 4 Conclusion A design method was proposed for D IIR zerophase wedge

5 (a) (b) (a) (b) (c) Fig 3 (a) (b) Two versions of ideal fan filter bandpass regions; (c) fan filter frequency response Various methods were developed to order to separate stability from approximation problem If the D filter becomes unstable some stabilization procedures are needed 3 Applications and Simulation Results A wedge filter can be used in directional image filtering to select lines with a specified orientation from a given image The spectrum of a straight line is oriented in the plane ( 1 ) at an angle of with respect to the line direction Fig4 shows a real gray-scale image representing a straw texture The straws have random directions and choosing different filter orientations we can select the straws with roughly the same orientation and filter out the rest Only the lines with spectrum oriented along filter characteristic remain unchanged while all the rest are more or less blurred due to directional filtering The filter aperture angle was 5 and three different orientations were used This simple example was given just to illustrate the potential use of wedge filters in pattern recognition applications Further research will approach other real-life applications as well The advantage of these analytical design methods over other approaches is that the frequency response of the resulted D wedge filter depends explicitly on the specified parameters (orientation aperture prototype flatness and steepness) and therefore the filter is adjustable The design need not be remade every time again from the start for different specifications (c) (d) Fig 4 (a) Grayscale straw texture image; (b) (c) (d) filtering result using aperture 5 and three orientations: Conclusion A design method was proposed for D IIR zerophase wedge filters oriented along a specified direction They are based on a 1D LP prototype with given frequency response A 1D to D frequency mapping is applied to the prototype One method uses the bilinear transform and frequency prewarping to compensate for distortions The other avoids the use of bilinear transform and includes instead a numerical approximation Both methods are fairly accurate and can be used to design wedge filters with various specifications References: [1] WS Lu A Antoniou Two-Dimensional DigitalFilters CRC Press 199 [] A Pendergrass SK Mitra EI Jury Spectral Transformations for Two-Dimensional Digital Filters IEEE Transactions on Circuits andsystems CAS-3 pp6-35 [3] K Hirano JK Aggarwal Design of Two- Dimensional Recursive Digital Filters IEEE Transactions on Circuits and Systems CAS-5 Dec 1978 pp [4] L Harn BA Shenoi Design of Stable Two- Dimensional IIR Filters Using Digital Spectral Transformations IEEE Transactions on Circuits and Systems volcas-33 pp May 1986 ISB:

6 [5] RH Bamberger MJT Smith A Filter Bank for the Directional Decomposition of Images: Theory and Design IEEE Transactions on SignalProcessing Vol 4 pp [6] PE Danielsson Rotation-Invariant Linear Operators With Directional Response 5 th Int Conference on Pattern Recognition Miami December 198 [7] JM Coggins AK Jain A Spatial Filtering Approach to Texture Analysis Pattern RecognitionLetters Vol3 (3) 1985 [8] E Simoncelli H Farid Steerable Wedge Filters for Local Orientation Analysis IEEE Transactions on Image Processing Vol 5(9) Sep 1996 pp [9] R Ansari Efficient IIR and FIR Fan Filters IEEE Transactions on Circuits and Systems Vol34 (8) Aug 1987 pp [1] Z Weiping S akamura An Efficient Approach for the Synthesis of -D Recursive Fan Filters Using 1-D Prototypes IEEE Trans Signal Processing Vol44(4) pp Apr 1996 [11] G Qunshan MS Swamy On the Design of a Broad Class of D Recursive Digital Filters with Fan Diamond and Elliptically-Symmetric Responses IEEE Transactions on Circuits and SystemsII Vol41(9) Sep1994 pp [1] GS Mollova Analytical Least Squares Design of -D Fan Type FIR Filter International Conference on Digital Signal Processing 1997 Vol pp [13] TQ Hung HD Tuan TQ guyen Design of Half-Band Diamond and Fan Filters by SDP IEEE International Conferemce on Acoustics Speech and Signal Processing ICASSP 7 Honolulu 15- April 7 Vol III pp [14] AH Kayran RA King Design of Recursive and onrecursive Fan Filters With Complex Transformations IEEE Trans on Circuits and Systems CAS-3 (1) 1983 pp [15] R Matei Design Method for Wedge-Shaped Filters Proceedings of the International Conference SIGMAP 9 Milano Italy pp 19-3 ISB:

20th European Signal Processing Conference (EUSIPCO 2012) Bucharest, Romania, August 27-31, 2012

20th European Signal Processing Conference (EUSIPCO 2012) Bucharest, Romania, August 27-31, 2012 th European Signal Processing Conference (EUSIPCO ) Bucharest, Romania, August 7-3, RECURSIVE FAN-YPE FILER DESIGN FROM D ANALOG RANSFER FUNCIONS Radu Matei Gh.Asachi echnical University of Iasi, Romania