2D cross-Ψ B -energy operator for images analysis J.C. CEXUS, and A.O. BOUDRAA, and A. BAUSSARD, and F.H. ARDEYEH, and E.H.S. DIOP Abstract—In this paper, the Cross-Teager-Kaiser Energy Operator (CTKEO) used for interaction measure between two signals is generalized for complex-valued patterns and images. This new operator called 2D-CTKEO and noted Ψ 2B [., .] is useful for some image processing problems. Some properties of 2D-CTKEO showing the interest of such operator for image analysis are provided. Based on Ψ 2B [., .] operator, two similarity functions, called Simil 1 2B and Simil 2 2B , are constructed. The effectiveness of Ψ 2B [., .] operator is demonstrated on noise-free and noisy images. These results show that Ψ 2B [., .] is useful for detection of interest points such as corners and edges and for template matching. I. I NTRODUCTION Despite its simple definition, Teager-Kaiser Energy Oper- ator (TKEO) [1], [2] has found many applications in various fields of signal processing. An interesting application of TKEO is the demodulation of AM-FM signals [3], [6]. As the definition of TKEO is simple, a variety of generalizations have been made. Quite similar to TKEO, an energy-like operator called Cross-TKEO (CTKEO), that takes two argu- ments, has been introduced [2]. CTKEO is used as detector of transients in the noise [7] and for time delay estimation [8]. TKEO has also been extended to two dimensions, termed 2D-TKEO, to make it suitable as image filter [9], [14]. In this paper, 2D-TKEO and CTKEO are combined to define a new energy-like operator that measures the interaction or the similarity between two images. This operator is useful for detection of interest points such as corners or edges. Also, this operator can be used for image matching or registration. II. 1D-TKEO TKEO is defined as a local energy measure for oscillating (simple harmonic) signals [2]. This operator, Ψ 1R [.], com- putes the energy of a real-valued signal x(t) as follows: Ψ 1R [x(t)] = [ ˙ x(t)] 2 - x(t)¨ x(t) , (1) where ˙ x(t) and ¨ x(t) are the first and the second time derivatives of x(t) respectively. A discrete-time version of TKEO is given by [1], [3]: Ψ 1R [x(n)] = x 2 (n) - x(n + 1)x(n - 1) . (2) TKEO is nearly instantaneous because only three samples are required to compute the energy at each instant n. This J.C. CEXUS, and A. BAUSSARD, and F.H. ARDEYEH : ENSI- ETA / E 3 I 2 EA 3876, 2, rue Franc ¸ois Verny, 29806 Brest, France {cexusje,baussaal}@ensieta.fr A.O. BOUDRAA, and E.H.S. DIOP : Ecole Navale / IRENav EA 3634, BCRM Brest, CC 600, Brest 29240, France {boudra, el-hadji.diop}@ecole-navale.fr excellent time resolution permits to capture the energy fluc- tuations. TKEO is used to separate x(t) into amplitude envelope and instantaneous frequency signal to accomplish monocompo- nent AM-FM signal demodulation [3], [4]. TKEO is less computationally complex and has better time resolution than other classical demodulation approaches such as the Hilbert transform [15]. TKEO has been successfully used in various speech processing applications [5] or time-frequency analysis of nonstationary signals [6]. III. 1D-CTKEO To measure the interaction between two real signals CTKEO has been introduced [2]. Let x(t) and y(t) be two real signals, CTKEO, Ψ 1R [., .], is defined as follows [2]: Ψ 1R [x(t),y(t)] = ˙ x(t)˙ y(t) - x(t)¨ y(t) , (3) Ψ 1R [y(t),x(t)] = ˙ y(t)˙ x(t) - y(t)¨ x(t) . (4) If x(t) and y(t) represent displacements in some generalized motions, Ψ 1R [., .] has dimensions of energy and thus it can be viewed as a cross-energy between x(t) and y(t) [2]. In general this operator is not commutative: Ψ 1R [x(t),y(t)] = Ψ 1R [y(t),x(t)] [2], [16]. CTKEO, Ψ 1B [., .], has been ex- tended to complex-valued signals in [16]. More precisely, let x(t)≡x and y(t)≡y be two complex signals, cross-Ψ 1B - energy operator is given by [16]: Ψ 1B [x, y]= 1 2 (Ψ 1C [x, y]+Ψ 1C [y,x]) , (5) where Ψ 1C [x, y] is defined by [16]: Ψ 1C [x, y]= 1 2 [˙ x ∗ ˙ y +˙ x ˙ y ∗ ] - 1 2 [x ¨ y ∗ + x ∗ ¨ y] , (6) where ‘ ∗ ’ denotes the complex conjugate. Ψ 1B [x, y] is a symmetric bilinear form and Ψ 1B [x, x] is the associated quadratic form [16]. Ψ 1B [., .] is a real quantity, as expected for an energy operator [16]. A discrete version of equation (5) is given by [17], [18]: Ψ 1B [x(n),y(n)] =  k∈{i,r} x k (n)y k (n) - 0.5[x k (n +1)y k (n -1) + y k (n +1)x k (n -1)] , (7) where x(n)= x r (n)+jx i (n) and y(n)= y r (n)+jy i (n) with (x r ,x i ,y r ,y i )∈ℜ and j denotes the imaginary unit. As for TKEO, Ψ 1B [., .] operator is also nearly instantaneous which allows to capture the interaction energy fluctuations between two signals. Ψ 1B [., .] has been used in signal detection [8] and for dynamic analysis of a sequence of scintigraphic cardiac images [17], [18]. Proceedings of the 4th International Symposium on Communications, Control and Signal Processing, ISCCSP 2010, Limassol, Cyprus, 3-5 March 2010 978-1-4244-6287-2/10/$26.00 ©2010 IEEE