1051-361 digital image processing i hw1|solutions · 1051-361 digital image processing i...

1051-361 Digital Image Processing I HW1—Solutions

1. (a) Quantizing from m bits to n bits can be modeled by using

Q{f [x]} =⌊f [x]

2m−n

⌋(1)

where f ∈ Z, 0 ≤ f ≤ 2m − 1, and bxc is the floor of x. The expression

Q{f [x]} =

f [x]fmax − fmin

2m − 1

+12

(2)

is used when quantizing a function to m bits where f is continuos, i.e. f ∈ R, fmin ≤ f ≤ fmax. An example of thistype of function might be:

f [x] =[−1 3.34

√2 π 7

11

](3)

Table 1: Solution to 1a.f [x] 12 12 13 13 10 13 57 54f [x]/8 1.5 1.5 1.75 1.75 1.25 1.75 7.125 6.75bf [x]/8c 1 1 1 1 1 1 7 6

(b) NOTE: When performing IGS quantization from m bits to n bits, if the n most significant bits of f(x) are all “on”[(111) for this problem], the error added is set to 0 [(000) for this problem]. This prevents the output from beinggreater than 2n − 1 which would require n+ 1 bits to store.

Table 2: Solution to 1b.Index Gray Binary Binary + Error Binary IGS Gray IGS−1 N/A (000)(000) N/A N/A N/A

0 12 (001)(100) (001)(100) + (000)(000) = (001)(100) (001) 11 12 (001)(100) (001)(100) + (000)(100) = (010)(000) (010) 22 13 (001)(101) (001)(101) + (000)(000) = (001)(101) (001) 13 13 (001)(101) (001)(101) + (000)(101) = (010)(010) (010) 24 10 (001)(010) (001)(010) + (000)(010) = (001)(100) (001) 15 13 (001)(101) (001)(101) + (000)(100) = (010)(001) (010) 26 57 (111)(001) (111)(001) + (000)(000) = (111)(001) (111) 77 54 (110)(110) (110)(110) + (000)(001) = (110)(111) (110) 6

2. NOTE: The output fq[x] has values of 0 or 2m−1 where m is the bit-depth of the original image. The mean of the output“image” should be the same as the mean of the input “image.”

E [fq[x]] = E [f [x]] = f [x] (4)

It should be further noted that this ideal condition will only occur when the last error is 0.

Table 3: Solution to 2.x f [x] f [x] + ε[x− 1] fq[x] ε[x]−1 N/A N/A N/A 0

0 12 12 + 0 = 12 0 12− 0 = 121 12 12 + 12 = 24 0 24− 0 = 242 13 13 + 24 = 37 63 37− 63 = −263 13 13 +−26 = −13 0 −13− 0 = −134 10 10 +−13 = −3 0 −3− 0 = −35 13 13 +−3 = 10 0 10− 0 = 106 57 57 + 10 = 67 63 67− 63 = 47 54 54 + 4 = 58 63 58− 63 = −5

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3. The following Matlab code can be used to produce the images:

outpath = ’~/Documents/RIT/2010-1/DIP1/HW1/Figures/’;

sx = 128; % set the x sizesy = 128; % set the y sizebd = 8; % set the bit depth

% create an image with half the maximum bit depthim1 = uint8( zeros( sy, sx ) );im1( : ) = uint8( 2^( bd - 1 ) - 1 );imwrite( im1, [ outpath ’image1.png’ ] )figureimshow( im1, [ 0 2^bd-1 ] )

% create an image with one quarter the maximum bit depthim2 = uint8( zeros( sy, sx ) );im2( : ) = uint8( 2^( bd - 2 ) - 1 );imwrite( im2, [ outpath ’image2.png’ ] )figureimshow( im2, [ 0, 2^bd-1 ] )

% create an image with a ramp from 112 to 144

f_max = 144;f_min = 112;

x = ones( sy, 1 ) * ( 1:sx );im3 = uint8( ( x - 1 ) ./ ( sx - 1 ) .* ( f_max - f_min ) + f_min );imwrite( im3, [ outpath ’image3.png’ ] )figureimshow( im3, [ 0, 2^bd-1 ] )

(a) (b) (c)

Figure 1: (a) mid gray, (b) quarter gray, and (c) ramp from 112 to 144.

4. (a) Psuedo-code for one implementation of IGS

input Image Input the imagesx, sy = size(image) Find the size of the image

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newImage = array(sx, sy) Create a new array the same size as the input imagefor i = 1 to sx do

error = 0 Set the error to zerofor j = 1 to sy do

newImage[i, j] = Q(Image[i, j] + error) Run a quantizer on the current location with errorcompute error Compute the new error

endendoutput newImage Output the new image

(b) Psuedo-code for one implementation of 2-D error diffusion

input Image Input the imagesx, sy = size(image) Find the size of the imagenewImage = array(sx, sy) Create a new array the same size as the input imagefor i = j to sy do

for i = 1 to sx donewImage(i, j) = compute output Compute the output value at location (i, j)compute error Compute the new errordistribute error Distribute the error appropriately

endendoutput newImage Output the new image

5. Figures 2, 4, and 6 show the original three images and the following quantizations of the images: uniform quantization(quantized to 1-7 bits), IGS quantization (quantized to 1-7 bits), and 2–D error diffusion (quantized to 1 bit). Note thatyou only had to show images (a) and [(l) or (p)] for credit. The rest of the images are for reference. In all cases IGS isrun with errors propagating from left to right. Histograms of all images are also included for reference and to help withinterpretation. The y-axis scale on histograms vary.

6. Figures 8 and 10 show the original two images and the following quantizations of the images: uniform quantization(quantized to 1-7 bits), IGS quantization (quantized to 1-7 bits), and 2–D error diffusion (quantized to 1 bit). Note thatyou only had to show images (a) and [(l) or (p)] for credit. The rest of the images are for reference. In all cases IGS isrun with errors propagating from left to right. Histograms of all images are also included for reference and to help withinterpretation. The y-axis scale on histograms vary.

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(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 2: Quantizations of image 1 (uniform mid-gray). (a) original, (b) uniform quantization to 7 bits, (c) uniform quantizationto 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g) uniformquantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits, (k)igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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Figure 3: Histograms of quantizations of image 1 (uniform mid-gray). (a) original, (b) uniform quantization to 7 bits, (c) uniformquantization to 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits,(g) uniform quantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits,(k) igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 4: Quantizations of image 2 (uniform quarter-gray). (a) original, (b) uniform quantization to 7 bits, (c) uniform quan-tization to 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g)uniform quantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits,(k) igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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Figure 5: Quantizations of image 2 (uniform quarter-gray). (a) original, (b) uniform quantization to 7 bits, (c) uniform quan-tization to 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g)uniform quantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits,(k) igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 6: Quantizations of image 3 (ramp). (a) original, (b) uniform quantization to 7 bits, (c) uniform quantization to 6 bits,(d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g) uniform quantizationto 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits, (k) igs quantization to5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igs quantization to 1 bits,and (p) 1-bit error diffusion.

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Figure 7: Histograms of quantizations of image 3 (ramp). (a) original, (b) uniform quantization to 7 bits, (c) uniform quantizationto 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g) uniformquantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits, (k)igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 8: Quantizations of image 4 (extended ramp). (a) original, (b) uniform quantization to 7 bits, (c) uniform quantizationto 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g) uniformquantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits, (k)igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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Figure 9: Histograms of quantizations of image 4 (extended ramp). (a) original, (b) uniform quantization to 7 bits, (c) uniformquantization to 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits,(g) uniform quantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits,(k) igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 10: Quantizations of image 5 (lena). (a) original, (b) uniform quantization to 7 bits, (c) uniform quantization to 6 bits,(d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits, (g) uniform quantizationto 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits, (k) igs quantization to5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igs quantization to 1 bits,and (p) 1-bit error diffusion.

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Figure 11: Quantizations of image 2 (uniform quarter-gray). (a) original, (b) uniform quantization to 7 bits, (c) uniformquantization to 6 bits, (d) uniform quantization to 5 bits, (e) uniform quantization to 4 bits, (f) uniform quantization to 3 bits,(g) uniform quantization to 2 bits, (h) uniform quantization to 1 bits, (i) igs quantization to 7 bits, (j) igs quantization to 6 bits,(k) igs quantization to 5 bits, (l) igs quantization to 4 bits, (m) igs quantization to 3 bits, (n) igs quantization to 2 bits, (o) igsquantization to 1 bits, and (p) 1-bit error diffusion.

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