New boundary effect free algorithm for fast and accurate image arbitrary scaling and rotation

Transcription

1 ew boundary effect free algorithm for fast and accurate image arbitrary scaling and rotation Leonid Bilevich a and Leonid Yaroslavsky a a Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, 69978, Tel Aviv, Israel ABSTRACT A new fast DCT-based algorithm for accurate image arbitrary scaling and rotation is described The algorithm is free from boundary effects characteristic for FFT-based algorithm and ensures perfect interpolation with no interpolation errors The algorithm is compared with other available algorithms in terms of the interpolation accuracy, computational complexity and suitability for real time applications Keywords: scaling, rotation, DCT, convolution, fast algorithm ITRODUCTIO Scaling and rotation are basic image processing tasks with wide range of applications The rotation is usually implemented through either three-pass method (with D translations) or non-separable DFT-domain method Presently, for image scaling& rotation available are spline methods 3,4 implemented in spatial domain and discrete sinc-interpolation methods implemented as a convolution in DFT domain 5,6 The spline methods work in image domain and tend to introduce image blurring The discrete sinc-interpolation methods are potentially perfectly accurate but heavily suffer from boundary effects We propose a novel DCT 7 -based scaling& rotation algorithm that implements the ideal discrete sinc-interpolation and at the same time is very substantially less vulnerable to boundary effects In video processing, a crucial issue is its computational complexity Generally, there is a trade-off between the accuracy and computational complexity In this paper, we, along with introducing a new improved DCT-based method of image arbitrary scaling& rotation, compare its computational complexity with the complexity of spatial domain interpolation methods and DFT-domain methods The definitions of different D DCT-related transforms used in this paper are provided in Appendix A DCT-DOMAI SCALIG AD ROTATIO ALGORITHM Inverse Scaled Rotated Discrete Cosine Transform (IScRotDCT) Suppose the input real image a m,n of size has to be scaled by arbitrary scaling factor σ and rotated by arbitrary angle θ (in radians, positive θ corresponds to counter-clockwise rotation) We need to choose a method to perform this scaling& rotation The Inverse Scaled Rotated Discrete Fourier Transform (IScRotDFT) 5,6 suffers from heavy boundary effects due to periodicity of Further author information: (Send correspondence to Leonid Bilevich) bilevich@engtauacil; yaro@engtauacil;

2 DFT In order to eliminate these effects, we mirror-reflect the input image a m,n to double length (in both dimensions) and apply IScRotDFT to the mirror-reflected image Constraining the output image ã k,l to be real-valued, we obtain the Inverse Scaled Rotated Discrete Cosine Transform (IScRotDCT) defined as follows: ã k,l = σ (0) (0) α C(/) r,s () ] cos σ (C(k +/)+S(l+/) )r ] cos σ (C(l+/) S(k +/) )s, where ã k,l is the scaled and rotated signal, (0) is the unified length given by: is the rounding operator: C and S are the rotation factors :, are the centering factors : α C(/) r,s = (C +S) σ is the half-spectrum given by: α C(/) r,s = α C r,s (0) = min(, σ ), () σ σ = CEIL(σ) σ < σ = FLOOR(σ), (3) S = sinθ, C = cosθ, (4) σ, = (C S) σ and α C r,s is the DCT spectrum of the input image a m,n : α C r,s = DCT(a m,n ) = 0 (0) (0) 0 /4 / / /4 / / (0) / / (0) /4 / / /4 m=0 n=0 otes about the derivation of Eq (): σ, (5), (6) a m,n cos π (m+/)r ] cos π (n+/)s ] (7) The centering factors and are used in order to map the center of the input image to the center of the output image The last row and the last column of the spectral coefficients α C r,s were halved in order to improve the speed of convergence (to zero) of the scaling& rotation point-spread function (PSF) ote also that for integer σ and θ = 0 the Eq () represents the DCT-domain zero-padding scaling algorithm 5,8

3 Implementation through DCT-domain convolution Conversion of IScRotDCT to convolution Let s define the modified centering factors : = (C +S) σ σ, = (C S) σ σ (8) Also, let s define a spectrum αr,s C(/),ZP spectrum αr,s: C that is obtained by zero-padding or truncation of DCT α C(/),ZP r,s = for σ and 0 4 αc r,s αc r,s 0 σ αc r,s αc r,s 4 αc r,s α r,s αr,s C 0 αc r,s αc r,s 4 αc r,s αc r,s αc r,s 4 αc r,s 0 0 σ (9) α C(/),ZP r,s = 0 4 αc r,s αc r,s 0 σ σ αc r,s αc r,s αr,s C σ αc r,s σ 4 αc r,s αc r,s αc r,s 4 αc r,s αc r,s αc r,s 4 αc r,s (0) for σ < Then the Eq () can be converted to the form of convolution: ã k,l = σ cos ( C l k ) Skl ]] () σ σ { αr,s C(/),ZP cos C ( s r ) Srs+( r s ( cos C (k r) (l s) ) ]] +S(k r)(l s) σ σ { αr,s C(/),ZP sin C ( s r ) Srs+( r s ( sin C (k r) (l s) ) ]] +S(k r)(l s) σ sin ( C l k ) Skl ]]

4 + σ σ σ { αr,s C(/),ZP sin cos σ C ( s r ) Srs+( r s ( C (k r) (l s) ) +S(k r)(l s) { αr,s C(/),ZP cos C ( s r ) Srs+( r s ( sin C (k r) (l s) ) ]] +S(k r)(l s) + σ cos ( S k l ) Ckl ]] σ σ { αr,s C(/),ZP cos cos σ σ { αr,s C(/),ZP sin ]] S ( s r ) Crs+( r + s ( S (l r) (k s) ) +C(l r)(k s) S ( s r ) Crs+( r + s ( sin S (l r) (k s) ) ]] +C(l r)(k s) σ sin ( S k l ) Ckl ]] σ σ { αr,s C(/),ZP sin cos σ σ { + αr,s C(/),ZP cos ]] S ( s r ) Crs+( r + s ( S (l r) (k s) ) +C(l r)(k s) S ( s r ) Crs+( r + s ( sin S (l r) (k s) ) ]] +C(l r)(k s) ]] Regular and flipped convolutions From Eq () we see that computation of the scaled and rotated image ã k,l boils down to computation of four regular convolutions of the form where α () r,s is given by either α C(/),ZP r,s sin σ b () k,l = σ α () r,sh () k r,l s, () C ( s r ) Srs+( r )]] s

5 or α C(/),ZP r,s cos C ( s r ) Srs+( r )]] s, h () r,s is given by either ( sin C r s ) +Srs ]] π ( or cos C r s ) +Srs ]], and to computation of four flipped convolutions of the form σ b () l,k = σ that are converted to the transpose of the regular convolution: b () l,k = ( where α () r,s is given by either or h () α C(/),ZP r,s α C(/),ZP r,s sin b () k,l ) T = σ α () r,sh () l r,k s, (3) σ α () r,sh () k r,l s S ( s r ) Crs+( r + )]] s cos S ( s r ) Crs+( r + )]] s, r,s is given by either ( sin S r s ) +Crs ]] π ( or cos S r s ) +Crs ]] For simplicity of notation for the following derivation, let s denote b () k,l α r,s () by α r,s, h () r,s and h () r,s by h r,s T, (4) and b() k,l by b k,l, α () r,s and Even and odd regular convolutions Let s convert all four versions of h r,s to the sum of two summands: ( h r,s = sin C r s ) +Srs ]] (5) π = sin C( r s )] ] cos σ Srs +cos C( r s )] ] sin σ Srs, h r,s = cos = cos ( C r s ) +Srs ]] (6) ] σ Srs sin C( r s )] ] sin σ Srs, C( r s )] cos ( h r,s = sin S r s ) +Crs ]] (7)

6 = sin S( r s )] ] cos σ Crs +cos S( r s )] ] sin σ Crs, h r,s = cos = cos ( S r s ) +Crs ]] (8) ] σ Crs sin S( r s )] ] sin σ Crs S( r s )] cos We note from Eqs (5)-(8) that in all four cases the first summand is an even function of r and s and the second summand is an odd function of r and s, so we can represent the function h r,s in the following form: h r,s = h e r,s +h o r,s, (9) where h e r,s is the even function of r, s (ie, h e r,s = h e r,s, h e r,s = h e r, s, h e r,s = h e r, s) and h o r,s is the odd function of r, s (ie, h o r,s = h o r,s, h o r,s = h o r, s, h o r,s = h o r, s) (ote also that the multiplier generating the product h o r,s is either sinπsrs/(σ)] or sinπcrs/(σ)], so h o r,0 = 0 for all r and h o 0,s = 0 for all s) So the regular convolution b k,l = σ σ can be represented as a sum of even and odd regular convolutions: where b e k,l is the even regular convolution given by b e k,l = α r,s h k r,l s (0) b k,l = b e k,l +b o k,l, () σ σ and b o k,l is the odd regular convolution given by b o k,l = σ σ α r,s h e k r,l s () α r,s h o k r,l s (3) DCT-based convolution The even and odd regular convolutions can be computed using DCT-based algorithms: and ( IDCT ξ CC p,q ηp,q CI ) ( +IDcS/CT ξ SC ) ( +IDcST ξ SS b e k,l = σ +IDC/cST ( ξ CS b o k,l = σ IDcS/CT ( ξ CS p,qηp,q CI p,qηp,q CI p,qη CI p,q ( IDcST ξ CC p,q ηp,q SI ) ( IDC/cST ξ SC p,qηp,q SI ) )] p,qη SI p,q ) +IDCT ( ξ SS p,qη SI p,q)], ) (4) (5)

7 where ξ CC p,q = DCT(α r,s ), ξ SS p,q = DcST(α r,s ), (6) ξ SC p,q = DcS/CT(α r,s ), ξ CS p,q = DC/cST(α r,s ), (7) ηp,q CI = DCT I ( h e ) r,s, η SI p,q = DcST I ( h o r,s), (8) and different DCT-related transforms involved in these formulae are defined in Appendix A 3 COMPUTATIOAL COMPLEXITY We assume that both the input image a m,n and the output image ã k,l are D real matrices In order to compute D scaled image, rotated image and scaled& rotated image by DCT or DFT based methods, one needs to generate the cos/sin or exp factors, perform real or complex additions/ multiplications and compute DCT or DFT transforms The computational complexity of multiplication of complex numbers is 4 real multiplications and real additions, in total 6 real floating-point operations (flops) The D DFT is implemented by the Fast Fourier Transform (FFT) algorithm that needs log flops9 The D separable DFT algorithm needs log flops The numerically stable D DCT algorithm needs 3 log 9 floating point operations (flops) and the numerically stable D DCT type-i algorithm needs 3 log 8 9 flops0 The D non-separable DCT algorithm needs log flops Also, the following implementation details were taken into account: The D DFT-based scaling algorithm was implemented separably by D scaling of rows followed by D scaling of columns For the three-pass rotation with translation using splines we noticed that the spline coefficients have to be computed only at the first pass and only for the top half of rows At the second and third passes these coefficients are reused Computational complexity of discussed image scaling/rotation methods is listed in Table From Table it is clear that the computational complexity of the DCT-based scaling& rotation method is higher than that of the low-order splines From the other hand, the DCT-based scaling method is equivalent to very high-order spline and provides very accurate scaling results that outperform those of low-order splines In order to provide very high quality scaling using spline interpolation, one has to use a high-order spline, eg, B-spline of order p with computational complexity (4p+)+/( σ )] flops per one output sample In this case the complexity increases with the order of spline and becomes comparable and even higher that the complexity of the DCT-based scaling So the DCT-based scaling& rotation method is a natural choice for very accurate image scaling& rotation with reasonable computational complexity

8 Table : Comparison of computational complexities of D scaling, rotation and scaling& rotation algorithms Type of the method D scaling implemented with central B-spline of degree p (p odd) umber of flops per one output sample ( ) (4p+) + σ D DFT-based scaling 3 log σ with DFT-based cyclic convolution 5,6 + σ 6 ( (0)) log 9 5 D DFT-based scaling 90 3 log σ with DFT-based linear convolution 3 + σ 6 ( (0)) log 9 5 Rotation with three-pass algorithm with translation implemented 7p + with central B-spline of degree p (p odd) Rotation with three-pass algorithm with translation implemented in DFT domain,4 3 log Rotation with three-pass algorithm with translation implemented in DCT domain 5,5 log DFT-based rotation (with DFT-based cyclic convolution) 30 9 log 0 7 DFT-based scaling& rotation 3 log σ (with DFT-based cyclic convolution) 5,6 + σ 6 ( (0)) log 9 5 DCT-based 330log σ 55 scaling and rotation + σ 4 ( (0)) ] log 7 ] 7 ] 7 ] 4 EXPERIMETAL VERIFICATIO AD PERFORMACE COMPARISO In this Section we compare the DCT-based scaling&rotation algorithm with conventional scaling& rotation algorithms Example of rotated image The image rotated by angle θ = 45 by the DCT-based scaling& rotation algorithm is demonstrated in Figure We observe that the rotated image is free of boundary effects and the corners of the rotated image contain information that was mirrorreflected from the central part of the image Also, we can verify that the rotated image is centered correctly Comparison of different scaling&rotation methods As test images for checking D scaling, an image text and a pseudo-random image with uniform spectrum of size were used In order to evaluate the accuracy of image resampling by the suggested DCT-domain scaling & rotation algorithm in comparison with conventional bilinear/ bicubic 6 -based scaling & rotation algorithms, we employed iterative Scaling/ Rotation & DeRotation/ DeScaling sequence of operations (zooming-in of the original image by factor σ (σ ) and rotation by angle θ followed by rotation by the negative angle θ and zooming-out by the reciprocal factor /σ) For the ideal resampling procedure the resulting zoom&rotate-back image has to be identical to the original image The results of 0-step iterative zoom&rotate-back of text image (σ =, θ = ) by bilinear/ bicubic/ DCT-based scaling & rotation algorithms are shown in Figure

9 Figure : Demonstration of image rotated by the DCT-based scaling&rotation algorithm Rotated image (σ =, θ = 45 ) The results of 0-step iterative zoom&rotate-back of random image (σ =, θ = ) by bilinear/ bicubic/ DCT-based scaling & rotation algorithms are shown in Figure 3 Evolution of image spectra in such an iterative Scaling/ Rotation & DeRotation/ DeScaling test one can evaluate from comparison of spectra of zoom&rotate-back -iterated images The magnitudes of DFT-spectra after 0-step iterative zoom&rotate-back of random image (σ =, θ = ) by bilinear/ bicubic/ DCT-based scaling & rotation algorithms are shown in Figure 4 Comparing the presented results one can see that the bilinear/ bicubic scaling & rotation methods tend to blur scaled & rotated images, also, DFT-based scaling & rotation methods suffer from boundary artifacts The suggested DCT-based scaling & rotation method produces perfectly sharp images and is virtually free of boundary effects, demonstrating virtually perfect accuracy of resampling 5 COCLUSIO A novel DCT-domain algorithm for image scaling& rotation by arbitrary scaling factors and angles is presented It was demonstrated that the proposed algorithm is virtually free of boundary effects and ensures virtually perfect reconstruction accuracy Thanks to the availability of fast FFT-type algorithms for computing DCT and DCT-related transforms, the algorithm has a reasonable computational complexity and represents a valuable alternative to known scaling& rotation algorithms in image and video processing applications that need a very high resampling accuracy

10 (a) Test image (b) Iterative Scaling/Rotation & DeRotation/DeScaling (bilinear algorithm) (c) Iterative Scaling/Rotation & DeRotation/DeScaling (bicubic algorithm) (d) Iterative Scaling/Rotation & DeRotation/DeScaling (DCT algorithm) Figure : Iterative Scaling/Rotation & DeRotation/DeScaling of the test text image (a) by the bilinear algorithm (b), by the bicubic algorithm (c), and by the DCT-domain scaling&rotation algorithm (d) (after 0 iterations)

11 (a) Test random image (b) Iterative Scaling/Rotation & DeRotation/DeScaling (bilinear algorithm) (c) Iterative Scaling/Rotation & DeRotation/DeScaling (bicubic algorithm) (d) Iterative Scaling/Rotation & DeRotation/DeScaling (DCT algorithm) Figure 3: Iterative Scaling/Rotation & DeRotation/DeScaling of the test random image (a) by the bilinear algorithm (b), by the bicubic algorithm (c), and by the DCT-domain scaling& rotation algorithm (d) (after 0 iterations)

12 (a) Magnitude of DFT spectrum of the test image (b) Magnitude of DFT spectrum after iterative scaling & rotation (bilinear algorithm) (c) Magnitude of DFT spectrum after iterative scaling & rotation (bicubic algorithm) (d) Magnitude of DFT spectrum after iterative scaling & rotation (DCT algorithm) Figure 4: Magnitude of DFT spectrum of iterative Scaling/Rotation & DeRotation/DeScaling of the test random image (a) by the bilinear algorithm (b), by the bicubic algorithm (c), and by the DCT-domain scaling&rotation algorithm (d) (after 0 iterations)

13 APPEDIX A DEFIITIOS OF DCT-RELATED TRASFORMS DCT (k = 0,,, l = 0,,, r = 0,,, s = 0,, ) α r,s = DCT(a k,l ) (9) ] (k +/)r = a k,l cos π cos π (l+/)s ] k=0 l=0 DC/cST (k = 0,,, l = 0,,, r = 0,,, s =,, ) α r,s = DC/cST(a k,l ) (30) ] (k +/)r = a k,l cos π sin π (l+/)s ] k=0 l=0 DcS/CT (k = 0,,, l = 0,,, r =,,, s = 0,, ) α r,s = DcS/CT(a k,l ) (3) ] (k +/)r = a k,l sin π cos π (l+/)s ] k=0 l=0 DcST (k = 0,,, l = 0,,, r =,,, s =,, ) α r,s = DcST(a k,l ) (3) ] (k +/)r = a k,l sin π sin π (l+/)s ] k=0 l=0 DCT I (k = 0,,, l = 0,,, r = 0,,, s = 0,, ) α r,s = DCT I (a k,l ) (33) = a 0,0 +( ) r a,0 +( ) s a 0, +( ) r+s a, ( + a k,0 cos π kr ) ( + a 0,l cos π ls ) k= +( ) s +4 k= k= l= ( a k, cos π kr a k,l cos l= )+( ) r ( π kr ) ( cos π ls ) ] l= ( a,lcos π ls )

14 DcST I (k =,,, l =,,, r =,,, s =,, ) α r,s = DcST I (a k,l ) (34) ( = a k,l sin π kr ) ( sin π ls ) k= l= IDCT (k = 0,,, l = 0,,, r = 0,,, s = 0,, ) a k,l = IDCT(α r,s ) (35) { = ] (k +/)r α 0,0 + α r,0 cos π r= + α 0,s cos π (l+/)s ] s= ] (k +/)r +4 α r,s cos π cos π (l+/)s ] } r= s= IDC/cST (k = 0,,, l = 0,,, r = 0,,, s =,, ) a k,l = IDC/cST(α r,s ) (36) { = ] ( ) l α 0, +( ) l (k +/)r α r, cos π r= + α 0,s sin π (l+/)s ] +4 s= r= s= α r,s cos π ] (k +/)r sin π (l+/)s ] } IDcS/CT (k = 0,,, l = 0,,, r =,,, s = 0,, ) a k,l = IDcS/CT(α r,s ) (37) { = ] ( ) k (k +/)r α,0 + α r,0 sin π r= +( ) k α,scos π (l+/)s ] +4 r= s= s= α r,s sin ] (k +/)r π cos π (l+/)s ] } IDcST (k = 0,,, l = 0,,, r =,,, s =,, )

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