2-D Left-Side Quaternion Discrete Fourier Transform: Fast Algorithm

We describe a fast algorithm for the 2-D left-side QDFT which is based on the concept of the tensor representation when the color or four-componnrnt quaternion image is described by a set of 1-D quaternion signals and the 1-D left-side QDFTs over these signals determine values of the 2-D left-side QDFT at corresponding subset of frequency-points. The efficiency of the tensor algorithm for calculating the fast left-side 2-D QDFT is described and compared with the existent methods.  The proposed algorithm of the 2r×2r-point 2-D QDFT uses 18N2 less multiplications than the well-known methods: • column-row method • method of symplectic decomposition.  The proposed algorithm is simple to apply and design, which makes it very practical in color image processing in the frequency domain.  The method of quaternion image tensor representation is uique in a sense that it can be used for both left-sida and right-side 2-D QDFTs. 3 Inroduction – Quanterions in Imaging  The quaternion can be considered 4-dimensional generation of a complex number with one real part and three imaginary parts. Any quaternion may be represented in a hyper-complex form Q = a + bi + cj + dk = a + (bi + cj + dk), where a, b, c, and d are real numbers and i, j, and k are three imaginary units with the following multiplication laws: ij = −ji = k, jk = −kj = i, ki = −ik = −j, i2 = j2 = k2 = ijk = −1.  The commutativity does not hold in quaternion algebra, i.e., Q1Q2≠Q2Q1.  A unit pure quaternion is μ=iμi+jμj+kμk such that |μ| = 1, μ 2 = −1 For instance, the number μ=(i+j+k)/√3, μ=(i+j)/√2, and μ=(i-k)/√2  The exponential number is defined as exp(μx) = cos(x) + μ sin(x) = cos(x) + iμi sin(x) +jμj sin(x) +kμk sin(x) 4 RGB Model for Color Images 5  A discrete color image fn,m in the RGB color space can be transformed into imaginary part of quaternion numbers form by encoding the red, green, and blue components of the RGB value as a pure quaternion (with zero real part): fn,m = 0 + (rn,mi + gn,mj + bn,mk) Figure 1: RBG color cube in quaternion space.  The advantage of using quaternion based operations to manipulate color information in an image is that we do not have to process each color channel independently, but rather, treat each color triple as a whole unit. Calculation of the left-side 1-D QDFT  Let fn =(an,bn,cn,dn)=an +ibn +jcn +kdn be the quaternion signal of length N. The left-side 1-D quaternion DFT ( LS QDFT) is defined as 6 If we denote the N-point LS 1-D DFTs of the parts an, bn, cn, and dn of the quaternion signal fn by Ap, Bp, Cp, and Dp, respectively, we can calculate of the LS 1-D QDFT as If the real part is zero, an =0, and fn =(0,bn,cn,dn)=an +ibn +jcn +kdn , the number of operations of multiplication and addition can be estimated as Multiplications and Additions for the left-side 1-D QDFT  In the general case of the quaternion signal fn, the number of operations of multiplication and addition for LS 1-D QDFT can be estimated as 7 The number of operations for the left-side 1-D QDFT can be estimated as Here, we consider that for the fast N-point discrete paired transform-based FFT, the estimation for multiplications and additions are and two 1-D DFTs with real inputs can be calculated by one DFT with complex input, (1) Number of multiplications: Special case 8 The number of operations of multiplication and addition equal or 8N operations of real multiplication less than in (1). The direct and inverse left-side 2-D QDFTs  Given color-in-quaternion image fn,m =an,m +ibn,m +jcn,m +kdn,m , we consider the concept of the left-side 2-D QDFT in the following form: 9 1. Column-row algorithm: The calculation of the separable 2-D N×N-point QDFT by formula 2. The calculation the LS 2-D QDFT by the symplectic decomposition of the color image requires 2N N-point LS 1-D QDFTs. Each of the 1-D QDFT requires two N-point LS 1-D DFTs. Therefore, the column-row method uses 4N N-point LS 1-D DFTs and multiplications or The inverse left-side 2-D QDFT is: Example: N×N-point left-side 2-D QDFT 10 Figure 2. (a) The color image of size 1223×1223 and (b) the 2-D left-side quaternion discrete Fourier transform of the color-inqiuaternion image (in absolute scale and cyclically shifted to the middle). Tensor Representation of the regular 2-D DFT Let fn,m be the gray-scale image of size N×N.  The tensor representation of the image fn,m is the 2D-frequency-and-1D-time representation when the image is described by a set of 1-D splitting-signals each of length N 11 The components of the signals are the ray-sums of the image along the parallel lines Each splitting-signals defines 2-D DFT at N frequency-points of the set on the cartesian lattice Example: Tensor Representation of the 2-D DFT 1-D splitting-signal of the tensor represntation of the image 512×512 12 Figure 3. (a) The Miki-Anoush-Mini image, (b) splitting-signal for the frequency-point (4,1), (c) magnitude of the shifted to the middle 1-D DFT of the signal, and (d) the 2D DFT of the image with the frequency-points of the set T4,1. Tensor Representation of the left-side 2-D QDFT Let fn,m =an,m +ibn,m +jcn,m +kdn,m be the quaternion image of size N×N, (an,m =0). In the tensor representation, the quaternion image is represented by a set of 1-D quaternion splitting-signals each of length N and generated by a set of frequencies (p,s), 13 The components of the signals are defined as Here, the subsets Property of the TT: Example: Tensor Representation of the 2-D LS QDFT The splitting-signal of the tensor represntation of the color image 1223×1223: 14 Figure 5. The 123-point left-side DFT of the (1,4) quaternion splitting-signal; (a) the real part and (b) the i-component of the signal. Figure 4. Color image and (a,b,c) components of the splitting-signal generated by (1,4). Example: Tensor Representation of the 2-D LS QDFT 15 Figure 7. (a) The real part and (b) the imaginary part of the left-side 2-D QDFT of the 2-D color-in-quaternion `girl Anoush" image. Figure 6. (a) The 1-D left-side QDFT the quaternion splitting-signal f1,4,t (in absolute scale), and (b) the location of 1223 frequency-points of the set T1,4 on the Cartesian grid, wherein this 1-D LS QDFT equals the 2-D LS QDFT of the quaternion image. μ=(i+2j+k)/√6 Tensor Transform: Direction Quaternion Image Components Color image can be reconstructed by its 1-D quaternion splitting-signals or direction color image components defined by 16 Statement 1: The discrete quaternion image of size N×N, where N is prime, can be composed from its (N+1) quaternion direction images or splitting-signals as Color-or-Quaternion Image is The Sum of Direction Image Components 17 Figure 8: (a) The color image and direction images generated by (p,s) equal (b) (1,1), (c) (1,2), and (d) (1,4). The Paired Image Representation: Splitting-Signals and Direction Quaternion Image Components The tensor transform, or representation is redndant for the case N×N, where N is a power of 2. Therefore the tensor transform is modified and new1-D quaternion splitting-signals or direction color image components are calculated by 18 Statement 2: The discrete quaternion image of size N×N, where N=2r, r>1, can be composed from its (3N−2) quaternion direction images as Here JʹN,N is a set of generators (p,s). Such representation of the quaternion image is called the paired transform; it is unitary and therefore not redundant.