IMAGE UNDERSTANDING

9.15 IMAGE UNDERSTANDING

Image understanding (IU) refers to a body of knowledge that transforms pictorial inputs into commonly understood descriptions or symbols. Image pattern­

recognition techniques we have studied classify an input into one of several categories. Interpretation to a class is provided by a priori knowledge, or super­ vision. Such pattern-recognition systems are the simplest of IU systems. In more­ advanced systems (Fig. 9 . 62), the features are first mapped into symbols; for exam-

-0. 0 (b)

(a)

(c)

Figure 9.63 A rule-based approach for printed circuit board inspection. (a) Pre­

processed image; (b) image after thinning and identifying tracks and pads; (c) segmented image (obtained by region growing). Rules can be applied to the image in (c) and violations can be detected.

Sec. 9. 1 5 Image U nderstanding 421 Sec. 9. 1 5 Image U nderstanding 421

visual models and practical rules are adopted. For example, syntactic techniques provide grammars for strings of symbols. Other relational models provide rules for

describing relations and interconnections between symbols. For example, pro­ jections at different angles of a spherical object may be symbolically represented as several circles. A relational model would provide the interpretation of a sphere or a ball. Figure

9.63 shows an example of image understanding applied to inspection of printed circuit boards [73, 74]. Much work remains to be done in formulation of problems and development

of techniques for image understanding. Although the closing topic for this chapter, it offers a new beginning to a researcher interested in computer vision.

PROBLEMS

9.1 Calculate the means, autocorrelation, covariance, and inertia [see Eq. (9.116)] of the second-order histogram considered in Example 9.1.

9.2* Display the following features measured over 3 x 3, 5 x 5, 9 x 9, and 16 x 16 windows of a 512 x 512 image: (a) mean, (b) median, (c) dispersion, (d) standard deviation, (e) entropy, (f) skewness, and (g) kurtosis. Repeat the experiment for different images and draw conclusions about the possible use of these features in image processing

applications. 9.3* From an image of your choice, extract the horizontal, vertical, 30°, 45°, and 60° edges, using the DFT and extract texture using the Haar or any other transform. 9.4* Compare the performances of the gradient operators of Table 9.2 and the 5 x 5 stochastic gradient of Table 9.5 on a noisy ideal edge model (Fig. 9.11) image with SNR =

9. Use the performance criteria of (9.25) and (9.26). Repeat the results at different noise levels and plot performance index versus SNR. 9.5* Evaluate the performance of zero-crossing operators on suitable noiseless and noisy images. Compare results with the gradient operators. 9.6 Consider a linear filter whose impulse response is the second derivative of the Gaussian kernel exp( -x 2/2a 2). Show that, regardless of the value of a, the response of this filter to an edge modeled by a step function, is a signal whose zero-crossing is at the location of the edge. Generalize this result in two dimensions by considering the Laplacian of the Gaussian kernel exp[ - ( x 2+ y 2)!2a 2]. 9.7 The gradient magnitude and contour directions of a 4 x 6 image are shown in Fig. P9. 7. Using the linkage rules of Fig. 9.16b, sketch the graph interpretation and find the edge path if the evaluation function represents the sum of edge gradient magnitudes.

Apply dynamic programming to Fig. P9.7 to determine the edge curve using the criterion of Eq. (9.27) with a= 4hr, 13 = 1 and d(x, y) = Euclidean distance between x and y.

9.8 a. Find the Hough transforms of the figures shown below in Figure P9.8.

b. (Generalized Hough transform) Suppose it is desired to detect a curve defined 422

Image Analysis and Computer Vision Chap. 9

/ - '\.._ '\.._

5 3 4 3 5 - 5 - --

- - - --

4 ""'

4 ""' / / / 3 3 3

Figure P9.7

* (a) (b) (c)

DO

Figure P9.8

parametrically by <!>(x, y,

a) = 0, where a is a p x 1 vector of parameters, from a set of edge point (x;, y;), i = 1, . . . , N. Run a counter C(a) as follows:

Initialize: C(a) = 0 Do i = l,N: C(a) = C(a) + 1,

where a is such that <!>(x;, y;, a ) = 0 Then the local maxima of C(a) gives the particular curve(s) that pass through the

given edge points. If each element of a is quantized to L different levels, the dimension of vector C(a) will be LP. Write the algorithm for detecting elliptical segments described by

(x - Xo)2 (y - Yo)2

a b 2 -1

If x0, y0, a, and b are represented by 8-bit words each, what is the dimension of C(a)? c. If the gradient angles 0; at each edge point are given, then show how the relation

for (x, y, 0) = ax (x;, y; 0;) ay might be used to reduce the dimensionality of the search problem.

aQ> a<j> + tan

9.9 a. Show that the normalized uniform periodic B -splines satisfy

Bo,k (t) dt = l and k-1 L Bo,k (t +j) = l,

O :s t < l

;-o

b. If an object of uniform density is approximated by the polygon obtained by joining the adjacent control points by straight lines, find the expressions for center of mass,

perimeter, area, and moments in terms of the control points. Problems

Chap. 9 423

9.10 (Cubic B-splines) Show that the control points and the cubic B-splines sampled at uniformly spaced nodes are related via the matrices 84 as follows:

B, = l

where the first matrix is for the periodic case and the second is for the nonperiodic case.

9. 11 (Properties of FDs)

a. Prove the properties of the Fourier descriptors summarized in Table 9.8. b. Using Fig. 9.26, show that the reflection of Xi,X2 is given by

i1 =

2 B2 [(B A + 2 -A 2)x1 - 2ABx2 - 2AC]

A +B 2 [-2ABx1 + (A2 - B2)x2 - 2BC] From these relations prove (9.53).

i2 = 2

c. Show how the size, location, orientation, and symmetry of an object might be determined if its FDs and those of a prototype are given.

d. Given the FDs of u (n ), find the FDs of X1 (n) and X2 (n) and list their properties with respect to the geometrical transformations considered in the text.

9.12 (Additional properties of FDs [32])