CS 445 HW#6 Solutions

Transcription

1 CS 445 HW#6 Solutions Text problem 6.1 From the figure, x = 0.43 and y = 0.4. Since x + y + z = 1, it follows that z = These are the trichromatic coefficients. We are interested in tristimulus values X, Y, and Z, which are related to the trichromatic coefficients by Eqs. (6.1-1) through (6.1-3). We note however, that all the tristimulus coefficients are divided by the same constant, so their percentages relative to the trichromatic coefficients are the same as those of the coefficients. Thus, the answer is X = 0.43, Y = 0.40, and Z = Text problem 6.2 Denote by c the given color, and let its coordinates be denoted by between c and c 1 is. The distance Similarly the distance between c 1 and c 2 The percentage p 1 of c 1 in c is The percentage of p 2 of c 2 is simply p 2 = p 1. In the preceding equation we see, for example, that when c = c 1, then d(c, c 1 ) = 0 and if follows that p 1 = 100% and p 2 = 0%. Similarly, when d(c, c 1 ) = d(c 1, c 2 ), it follows that p 1 = 0% and p 2 = 100%. Values in between are easily seen to follow from these simple relations. Text problem 6.3 Consider Fig P6.3, in which c 1, c 2 and c 3 are the given vertices of the color triangle and c is an arbitrary color point contained within the triangle or on its boundary. The key to solving this problem is to realize that any color on the border of the triangle is made up of proportions from the two vertices defining the line segment that contains the point. The contribution to a point on the line by the color vertex opposite this line is 0%. The line segment connecting points c 3 and c is shown extended (dashed segment) until it intersects the line segment connecting c 1 and c 2. The point of intersection is denoted c 0. Because we have the values of c 1 and c 2, if we knew c 0, we could compute the percentages of c 1 and c 2 contained in c 0 by using the method described in Problem 6.2. Denote the ratio of the content of c 1 and c 2 in c 0 be denoted by R 12. If we now add color c 3 to c 0, we know from Problem 6.2 that the point will start to move toward c 3 along the line shown. For any position of a point along this line we could determine the percentage of c 3 and c 0, again, by using the method described in Problem 6.2. What is important to keep in mind is that the ratio R 12 will remain the same for any point along the segment connecting c 3 and c 0. The color of the points along this line is different for each position, but the ration of c 1 to c 2 will remain constant.

2 So, if we can obtain c 0, we can then determine the ratio R 12, and the percentage of c 3, in color c. The point c 0 is not difficult to obtain. Let y = a 12 x + b 12 be the straight line containing points c 1 and c 2, and y = a 3c x + b 3c the line containing c 3 and c. The intersection of these two lines gives the coordinates of c 0. The lines can be determined uniquely because we know the coordinates of the two point pairs needed to determine the line coefficients. Solving for the intersection in terms of these coordinates is straightforward, but tedious. Our interest here is in the fundamental method, not the mechanics of manipulation simple equations so we do not give the details. At this juncture we have the percentage of c 3 and the ration between c 1 and c 2. Let the percentages of these three colors composing c be denoted by p 1, p 2 and p 3 respectively. Since we know that p 1 + p 2 = 100 p 3, and that p 1 / p 2 = R 12, we can solve for p 1 and p 2. Finally, note that this problem could have been solved the same way by intersecting one of the other two sides of the triangle. Going to another side would be necessary, for example, if the line we used in the preceding discussion had an infinite slope. A simple test to determine if the color of c is equal to any of the vertices should be the first step in the procedure; in this case no additional calculations would be required. Text problem 6.4 Use color filters sharply tuned to the wavelengths of the colors of the three objects. Thus, with a specific filter in place, only the objects whose color corresponds to that wavelength will produce a predominant response on the monochrome camera. A motorized filter wheel can be used to control filter position from a computer. If one of the colors is white, then the response of the three filters will be approximately equal and high. If one of the colors is black, the response of the three filters will be approximately equal and low. Text problem 6.5 At the center point we have

3 which looks to a viewer like pure green with a boot in intensity due to the additive gray component. Text problem 6.6 For the image give, the maximum intensity and saturation requirement means that the RGB component values are 0 or 1. We can create the following table with 0 and 255 representing black and white, respectively: Thus, we get the monochrome displays shown in Fig. P6.6. Text problem 6.9 (a) For the image given, the maximum intensity and saturation requirement means that the RGB component values are 0 or 1. We can create Table P6.9 using Eq. (6.2-1):

4 Thus, we get the monochrome displays shown in Fig. P6.9(a). (b) The resulting display is the complement of the starting RGB image. From left to right, the color bars are (in accordance with Fig. 6.32) white, cyan, blue, magenta, red, yellow, green, and black. The middle gray background is unchanged. Text problem 6.10 Equation (6.2-1) reveals that each component of the CMY image is a function of a single component of the corresponding RGB image C is a function or R, M of G, and Y of B. For clarity, we will use a prime to denote the CMY components. From Eq. (6.5-6), we know that for I = 1,2,3 (for the R, G, and B components). And from Eq. (6.2-1), we know that the CMY components corresponding to the r i and s i (which we are denoting with primes) are and Thus, and

5 so that Text problem 6.11 (a) The purest green is 00FF00, which corresponds to cell (7, 18). (b) The purest blue is 0000FF, which corresponds to cell (12, 13). Text problem 6.12 Using Eqs. (6.2-2) through (6.2-4), we get the result shown in Table P6.12. Note that, in accordance with Eq. (6.2-2), hue is undefined when R = G = B since addition, saturation is undefined when R = G = B = 0 since Eq. (6.2-3) yields Thus, we get the monochrome display shown in Fig. P In Text problem 6.15 The hue, saturation, and intensity images are shown in Fig. P6.15, from left to right.

6 Text problem 6.16 (a) It is given that the colors in Fig. 6.16(a) are primary spectrum colors. It also is given that the gray-level images in the problem statement are 8-bit images. The latter condition means that hue (angel) can only be divided into a maximum number of 256 values. Since hue values are represented in the interval from 0 o to 360 o this means that for an 8-bit image the entire the increments between contiguous hue values are now 255/360. Another way of looking at this is that the entire [0, 360] hue scale is compressed to the range [0, 255]. Thus for example, yellow (the first primary color we encounter, which is 60 o now becomes 43 (the closest integer) in the integer scale of the 8-bit image shown in the problem statement. Similarly, green, which is 120 o becomes 85 in this image. From this we easily compute the values of the other two regions as being 170 and 213. The region in the middle is pure white [equal proportions of red green and blue in Fig. 6.61(a)] so its hue by definition is 0. This also is true of the black background. (b) The colors are spectrum colors, so they are fully saturated. Therefore, the values shown of 255 applies to all circles regions. The region in the center of the color image is white, so its saturation is 0. (c) The key to getting the values in this figure is to realize that the center portion of the color image is white, which means equal intensities of fully saturated red, green, and blue. Therefore, the value of both darker gray regions in the intensity image have value 85 (i.e., the same value as the other corresponding region). Similarly, equal proportions of secondaries yellow, cyan, and magenta produce white, so the two lighter gray regions have the same value (170) as the region shown in the figure. The center of the image is white, so its value is 255. Text problem 6.17 (a) Because the infrared image which was used in place of the red component image has very high gray-level values. (b) The water appears as solid black (0) in the near infrared image [Fig. 6.27(d)]. Threshold the image with a threshold value slightly larger than 0. The result is

7 shown in Fig. P6.17. It is clear that coloring all the black points in the desired shade of blue presents no difficulties. (c) Note that the predominant color of natural terrain is in various shades of red. We already know how to take out the water from (b). Thus a method that actually removes the background of red and black would leave predominantly the other man-made structures, which appear mostly in a bluish light color. Removal of the red [and the black if you do not want to use the method as in (b)] can be done by using the technique discussed in Section Text problem 6.20 The RBG transformations for a complement [from Fig. 6.33(b)] are: where i = 1, 2, 3 (for the R, G, and B components). But from the definition of the CMY space in Eq. (6.2-1), we know that the CMY components corresponding to r i and s i, which we will denote using primes, are Thus, and so that Text problem 6.24 The conceptually simplest approach is to transform every input image to the HIS color space, perform histogram specification per the discussion in Section on the intensity

8 ( I ) component only (leaving H and S alone), and convert the resulting intensity component with the original hue and saturation components back to the starting color space. Text problem 6.25 (a) The boundary between red and green becomes thickened and yellow as a result of blurring between the red and green primaries (recall that yellow is the color between green and red in, for example, Fig. 6.14). The boundary between green and blue is similarly blurred into a cyan color. The result is shown in Fig. P6.25. (b) Blurring has no effect in this case. The intensity image is constant (at its maximum value) because the pure colors are fully saturated.