2-D Signals and Systems 2-D Fourier Transform Digital Image Processing Lectures 3 & 4 M.R. Azimi, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017 M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Definitions and Extensions: 2-D Signals: A continuous image is represented by a function of two variables e.g. x(u; v) where (u; v) are called spatial coordinates and x is the intensity. A sampled image is represented by x(m; n). If pixel intensity is also quantized (digital images) then each pixel is represented by B bits (typically B = 8 bits/pixel). 2-D Delta Functions: They are separable 2-D functions i.e. 1 (u; v) = (0; 0) Dirac: δ(u; v) = δ(u) δ(v) = 0 Otherwise 1 (m; n) = (0; 0) Kronecker: δ(m; n) = δ(m) δ(n) = 0 Otherwise M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Properties: For 2-D Dirac delta: 1 1- R R x(u0; v0)δ(u − u0; v − v0)du0dv0 = x(u; v) −∞ 2- R R δ(u; v)du dv = 1 8 > 0 − For 2-D Kronecker delta: 1 1 1- x(m; n) = P P x(m0; n0)δ(m − m0; n − n0) m0=−∞ n0=−∞ 1 2- P P δ(m; n) = 1 m;n=−∞ Periodic Signals: Consider an image x(m; n) which satisfies x(m; n + N) = x(m; n) x(m + M; n) = x(m; n) This signal is said to be doubly periodic with horizontal and vertical periods M and N, respectively. Only MN samples that are within the fundamental period are distinct. M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform In general case x(m + M1; n + N1) = x(m; n) x(m + M2; n + N2) = x(m; n) ∆ where D = M1N2 − M2N1 6= 0 is number of linearly independent t t (distinct) samples. Then M1 = [M1 N1] and M2 = [M2 N2] represent the displacement from any sample to the corresponding samples of two other periods. Note that D = det[M1 M2]. Examples of doubly periodic 2-D signals with horizontal and vertical periods M = 3 and N = 3, and general periodic 2-D signals with period t t vectors M1 = [7; 2] and M2 = [−2; 4] are shown in below figures. n2 n2 N2 N1 n1 n1 M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform 2-D Linear System and Shift Invariance: 1-Linearity: The system defined by input-output mapping y(m; n) = T [x(m; n)] is said to be linear iff the operator T satisfies the following conditions. That is, if y1(m; n) = T [x1(m; n)] and y2(m; n) = T [x2(m; n)] then T [ax1(m; n) + bx2(m; n)] = ay1(m; n) + by2(m; n) for any arbitrary a and b, i.e. any linear combination of two or more inputs gives the same combination of their corresponding outputs. 2-Spatial Invariance: The system defined by y(m; n) = T [x(m; n)] is said to be space invariant iff the operator T satisfies y(m; n) = T [x(m; n)] then T [x(m − k; n − l)] = y(m − k; n − l) 8k; l M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform 2-D Impulse Response and 2-D Convolution: The response of a 2-D system to a 2-D Kronecker delta input is the 2-D impulse response i.e. h(m; n; k; l) =∆ T [δ(m − k; n − l)] For an imaging system it represents the image of an ideal point source. Thus, it is also called point spread function (PSF). PSF is real and non-negative since input-output of an imaging system represent positive quantity e.g., intensity of light. For a linear system, the output can be obtained from its PSF by applying superposition rule, y(m; n) = T [x(m; n)] X X = T [ x(k; l)δ(m − k; n − l)] k l X X = x(k; l)T [δ(m − k; n − l)] k l X X = x(k; l)h(m; n; k; l) k l If the system is further spatiallyM.R. Azimi invariantDigital then Image Processing h(m; n; k; l) = T [δ(m − k; n − l)] = h(m − k; n − l; 0; 0) = h(m − k; n − l) 2-D Signals and Systems 2-D Fourier Transform If the system is also spatially invariant then h(m; n; k; l) = T [δ(m − k; n − l)] = h(m − k; n − l; 0; 0) = h(m − k; n − l) Thus, for a 2-D LSI system convolution sum becomes X X y(m; n) = x(k; l)h(m − k; n − l) k l X X = x(m − k; n − l)h(k; l) k l = x(m; n) ∗ ∗h(m; n) Imaging systems with separable and circularly symmetric impulse response or PSF are often encountered. Graphical interpretation of 2-D convolution operation is illustrated in the figure. Using 1st equation above, PSF is first rotated about the origin by 180◦, then shifted by m,n and overlayed on image x(k; l). The sum of the products of the arrays fx(k; l)g and fh(m − k; n − l)g which occupy a common region gives the result at (m; n). M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Figure: Output at (m; n) is the sum of products of quantities in the shaded area. Example: Graphical convolution of two 2 × 2 and 3 × 2 arrays h(m; n) and x(m; n). M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Remarks 1 The result of the convolution of two 2-D arrays of sizes (M1 × N1) and (M2 × N2) is another 2-D array of size (M1 + M2 − 1) × (N1 + N2 − 1). 2 The result of convolution can be verified by checking the identity P P y(m; n) = (P P h(m; n))(P P x(m; n)) m n m n m n This property of convolution is known as area conservation. In this example, P P y(m; n) = 32; P P h(m; n) = 2, m n m n P P x(m; n) = 16 which verifies the above relation. m n M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Stability of 2-D LSI Systems Definition: A 2-D system is said to be bounded-input bounded-output (BIBO) stable iff a bounded input results in a bounded output, i.e. BIBO jx(m; n)j < M < 1 =) jy(m; n)j < N < 1; 8m; n Theorem: A necessary and sufficient condition for BIBO stability of a 2-D LSI system is that the PSF must be absolutely sumable i.e. 1 1 P P jh(m; n)j = P < 1 m=−∞ n=−∞ Remark: Note that most of 2-D signals (or images) are spatially limited unlike the temporal 1-D signals that have no limits in time (streaming data). Thus, the properties of the output image beyond the spatial bounds of the system are irrelevant to the usefulness of the system. From practical viewpoint and not a mathematical one it is therefore possible to assess the stability of a system by evaluating the sum over the bounds of the output image and requiring this to be finite for BIBO stability. M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Causality of 2-D LSI Systems Causality is a requirement for all realizable 1-D systems and output y(m) is dependent only on those values of x(n) for which n ≤ m (i.e past and present). This is always the case when the independent variable is time. When the variables are not temporal rather are spatial, this restriction does no longer hold and the output signal can have left as well as right dependencies on the input signal. However, when processing 2-D images arriving in real-time (e.g., raster scan or video) causality comes to play. Definition: A 1-D sequence fh(n)g is causal if h(n) = 0; 8n < 0, anticausal if h(n) = 0; 8n > 0 and noncausal if it is neither causal nor anticausal. A 2-D sequence fh(m; n)g is causal if its 1-D map h~(k) is causal. Since there can be many mappings from 2-D to 1-D, the definition of causality depends on the choice of mapping. M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Definition 1: A 2-D sequence fh(m; n)g is said to be strongly causal if h(m; n) = 0 fm < 0; 8ng [ fn < 0; 8mg In this case the region of support (ROS) is Quarter Plane. Definition 2: A 2-D sequence fh(m,n)g is said to be weakly causal if h(m; n) = 0 fn < 0; 8mg [ fn = 0; m < 0g i.e. Non-Symmetric Half Plane (NSHP) ROS. M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Definition 3: A 2-D sequence fh(m; n)g is said to be semicausal if it is causal in one dimension and noncausal in the other i.e. h(m; n) = 0 n < 0; 8m i.e.