Answers to Selected Problems and Questions

Chapter 1 Introduction to Image Understanding

1-9{Hint} In general, most high-level tasks have this feature, one of the reasons is that it is determined by the characteristics of the current computer system, Von Neumann machine.

Chapter 2 Stereo Vision

2-1According to the formula D = A · B/(A − Z), for point (1m, 0m, 2m), there is D = 0.0010256m; for point (2m, 0m, 3m), there is D = 0.00067796m. When D = 0.002 m, there is Z = 0.95 m.

2-9Here, the measurement accuracy of two binocular systems with the same baseline length is compared. According to eq. (2.16), the parallax is inversely proportional to the distance, and the distance between the point P and the camera R is closer than the distance between the point P and the camera T, so that the stereoscopic image composed of the left image L and the right image R can produce more accurate results when measuring the depth value of the point P.

Chapter 3 Image Enhancement

$\begin{array}{l}E\left(x,y\right)=R\left(p,q\right)=\frac{1+p_{s}p+q_{s}q}{\sqrt{1+p^2+q^2}\sqrt{1+p_s^2+q_s^2}}\\ =\frac{1+ax/z+by/z}{\sqrt{1+{\left(x/z\right)}^2+{\left(y/z\right)}^2\sqrt{1+a^2+b^2}}}=\frac{1+ax/z+by/z}{\sqrt{r^2\sqrt{1+a^2+b^2}}}=\frac{1+ax/z+by/z}{\sqrt{r\sqrt{1+a^2+b^2}}}\end{array}$

3-9The orientation of the object plane is determined by two angles, where the ratio of the major axis length over the minor axis length gives the inclination angle of the plane normal to the camera axis, and it is arccos(3/4) = 41.41° here; and the angle between the major axis and the X-axis 135° gives the rotation angle of the plane normal to the camera axis. In terms of symmetry, there are two possible orientations of the object plane, namely (inclination angle, rotation angle) = (41.41°, 135°) and (inclination angle, rotation angle) = (−41.41°, 135°).

Chapter 4 Image Restoration

4-7Use the method of counter-evidence. Suppose the two graphs are isomorphic, then the parallel edges a and c should correspond to the parallel edges x and z (the only parallel edges), so the vertices A and B should correspond to vertices X and Y, but both A and B are the endpoints of the three edges, and X and Y are not all the endpoints of three edges, contradiction.

Chapter 5 Scene Analysis and Semantic Interpretation

5-2g = 170.

5-11{Hint} Refer to Figure 5.1.

Chapter 6 Multi-Sensor Image Fusion

6-8(1)  Since $X\cap Y\subseteq X,X\cap Y\subseteq Y,\text{so }R\text{*}\left(X\right),R*\left(X\cap Y\right)\subseteq R*\left(Y\right)\cdot$

(2)Since $X\subseteq X\cup Y,X\cap Y,\text{so}{R}_{*}\left(X\cup Y\right),R_{*}\left(Y\right)\subseteq R_{*}\left(X\cap Y\right)\cdot$

6-12Assume X is R-rough and R-definable, then R*(X) ≠ , R*(X) ≠ L.
$R_{*}\left(X\right)\ne \varnothing \iff \forall x\in X,\text{so}{\left[x\right]}_R\subseteq X\iff {\left[x\right]}_R\subseteq \cap \overline{X}=\varnothing \iff R*\left(\overline{X}\right)\ne L\cdot$
$R*\left(X\right)\ne L\iff \forall y\in L,so{\left[y\right]}_R\cap R^{*}\left(X\right)=\cancel{0}\iff {\left[y\right]}_R\subseteq R_{*}\left(\overline{X}\right)\iff R_{*}\left(\overline{X}\right)\ne \cancel{0.}$

Chapter 7 Content-Based Image Retrieval

7-1The match value between Figure Problem 7-1(a, b) is 7/16. The match value between Figure Problem 7-1(b, c) is 9/16.

7-6For example, the ratio of height and width of Feret box (see Section 3.3.2 in Volume II) as well as form factor (see Section 6.3.1 in Volume II). The appearance of the trousers in the three images is mainly different in the bifurcation of trousers legs. Thus, their ratios of height and width of Feret box would be significantly different; while their shape factors would be quite close (the contour length and the silhouette area are similar).

Chapter 8 Color Image Processing

8-3{Hint} One possible approach is to resample (or interpolate) the raw data, so as to transform the anisotropic data into isotropic data.