- 10.1 INTRODUCTION. - 10.2 THE FRAME GRABBER. - Figure 10.1 Block diagram of frame grabber. - The pixel clock in the frame grabber defines the sampling interval of the A/D converter and generates an internal Hsync signal. - 10.3 DIGITAL IMAGE REPRESENTATION. - 10.4 CAMERA CALIBRATION. - Figure 10.2 Digital image representation. - 10.4.1 Lens Distortion. - x i = mx o (10.1a). - y i = my o (10.1b). - r i = mr o + d 1 r o 3 (10.2a). - r i = x i 2 + y i 2 , r o = x o 2 + y o 2 (10.2b) Higher odd order terms of r o may be added, but normally they will be negligible. - By multiplying Equation (10.2a) by cos φ and sin φ, we get (since x = r cos φ, y = r sin φ). - x i = mx o + d 1 x o (x o 2 + y o 2 ) (10.3a) y i = my o + d 1 y o (x o 2 + y o 2 ) (10.3b) This results in the well-known barrel (positive d 1 ) and pin-cushion (negative d 1 ) distortion.. - Figure 10.3 (a) Object and (b) image coordinates. - x u = x d + dx d (x d 2 + ε 2 y d 2 ) (10.4a) y u = y d + dy d (x d 2 + ε 2 y d 2 ) (10.4b) where ε is the aspect ratio between the horizontal and vertical dimensions of the pixels.. - The origin of the xy-coordinates is at the optical axis. - When transforming to the frame-store coordinate system XY (see Section 10.3) by the transformation. - x = X − X s (10.5a). - y = Y − Y s (10.5b). - Equation (10.4) becomes. - The magnitude of d is of the order of pixels per mm 3. - A mispositioning of the sensor chip in the camera could add further to these values. - The problem is then to find the coordinates of the image center. - Figure 10.4 Perspective transformation. - 10.4.2 Perspective Transformations. - Figure 10.4 shows a lens with the conjugate object- and image planes and with object and image distances a and b respectively. - b (10.7a). - b (10.7b). - Equation (10.7) is the perspective transformation and must be taken into account when e.g. - 10.5 IMAGE PROCESSING. - 10.5.1 Contrast Stretching. - Figure 10.5 Grey-level histogram (a) before and (b) after contrast stretching. - This is called contrast stretching and the resulting histogram when applied to the image of Figure 10.5(a) is given in Figure 10.5(b).. - 10.5.2 Neighbourhood Operations. - f (ξ, η)h(x − ξ, y − η) dξ dη (10.9) When the variables x, y are not continuous but merely discrete values m, n as the pixel numbers in the x- and y-direction as of a digitized image, the double integral has to be replaced by a double sum:. - f (m, n)h(i − m, j − n) (10.10). - Figure 10.6 (a) convolution and (b) 3 × 3 convolution filter h. - 10.5.3 Noise Suppression. - Figure 10.7 (a) 3 × 3 average mask and (b) image smoothing using local average mask. - Figure 10.7(a) shows this local neighbourhood average mask and Figure 10.7(b) illustrates the application of the mask to part of an image. - Referring to Figure 10.7(b) we find the value of the output pixel which replace the input pixel corresponding to the centre position of the mask to be. - This remove some of the noise without smoothing out too much of the detail in the original image.. - Of the other noise-suppression methods we mention median filtering. - 10.5.4 Edge Detection. - x (10.11). - The best approximation of the (partial) derivative in the x-direction we can give is therefore simply. - f (x, y) (10.12a) and in the y-direction. - f (x, y) (10.12b) The gradient S is a vector. - S = S x e x + S y e y (10.13). - Figure 10.8 (a) intensity distribution I across two edges. - S = S x 2 + S y 2 (10.14a). - (10.14b) The gradient magnitude and direction can be found from the two components along any two orthogonal directions.. - S = S x 2 + S y 2 (10.17). - S y | (10.18) From Figure 10.9 we see that Equation (10.16a) is obtained by applying the convolution mask illustrated in Figure 10.9(b) (left) and similarly Equation (10.16b) is obtained by applying the mask in Figure 10.9(b) (right). - Then the location of the maximum. - Figure 10.9 Convolution masks for estimation of the partial derivatives with (a) Roberts;. - of the quadratic curve is given by. - (10.19) To avoid detecting false edges, the magnitude of f (i) can be compared to a pre- determined threshold value and edges excluded with a maximum derivative below this threshold. - ∂y 2 (10.20). - (10.21) From Figure 10.8(c) we see that the second derivative crosses zero at the position of the edge. - Figure 10.10 shows the result of an experiment for detecting a straight edge using an interlaced frame transfer CCD video camera. - Figure 10.10(a, b) shows the result from detecting a vertical and a horizontal edge respectively. - In the left part of Figure 10.10(a) we see the effect of line jitter. - When the edge is horizontal as in Figure 10.10(b) a better result is obtained although the resolution in the vertical direction is poorer.. - 10.6 THE DISCRETE FOURIER TRANSFORM (DFT) AND THE FFT. - Figure 10.10 Detection of edge using the Sobel operator with sub-pixel accuracy. - g s [n]δ(x − np) (10.22). - i2π npu } (10.23). - G s (u) exp { i2π npu } du (10.24) In practice we can have only a finite number N of samples and therefore Equation (10.23) becomes. - i2π npu } (10.25). - This inserted into Equation (10.25) gives the sampled frequency domain representation:. - i2π nk/N } (10.26). - To develop the inverse relation, we start with Equation (10.24). - G[k] exp { i2π nk/N } (10.27). - Clearly, the DFT is only an approximation to the actual spectrum of the underlying analogue signal. - u − n p (10.28). - Equ- ation (10.28) shows that the spectrum of the sampled function g s is periodic with period 1/p. - p ≥ 2W (10.29). - Equation (10.29) can also be stated as. - f p ≥ f n (10.30). - The Nyquist theorem states that to recover a band-limited signal, the sampling frequency must at least be twice the bandwidth and is given by Equation (10.30). - For more details of the DFT, see the book by Ambradar (1995). - In Figure 10.11(a) this signal is sampled with N = 8, but only over one half-period, which gives a sampling frequency well above the Nyquist frequency. - Figure 10.11(b) shows the same signal with N = 16 sampled over one period. - Figure 10.11(c) shows the same signal with N = 24 sampled over 1.5 periods and the resulting spectrum. - Finally, Figure 10.11(d) shows the same signal with N = 82 sampled over 10.25 periods. - For details of the FFT we refer to Ambardar (1995) and Morrison (1994).. - PE of 1 kHz sine: D = 10.25, N = 82. - Figure 10.11 The DFT of sampled periodic signals. - 10.1 A common measure for the transmission of digital data is the baud rate, defined as the number of bits transmitted per second. - 10.2 How does contrast stretching influence edge detection by a gradient-based operator?. - PROBLEMS 267 10.3 Obtain the Fourier transforms of (a) df (x)/dx, (b) [∂f (x, y)/∂x + ∂f (x, y)/∂y. - 10.4 The basic approach used to compute the digital gradient involves taking differences of the form f (x, y. - 10.5 The result obtained by a single pass through an image of some two-dimensional masks can also be achieved by two passes using one-dimensional masks.. - (a) Show that the result obtained by a single pass of the mask given in Figure 10.7(a) also can be obtained by first passing the mask [1 1 1] through the image followed by a vertical counterpart.. - (b) Show that the Sobel operator can be implemented by one pass of a differencing mask of the form. - mask of the form. - 10.6 Write down the 3 × 3 mask representing the Laplacian operator given in Equation (10.21).. - 10.7 The two-dimensional Gaussian function is given by G(x, y
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