Image Processing. Transforms. Mylène Christine Queiroz de Farias
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1 Image Processing Transforms Mylène Christine Queiroz de Farias Departamento de Engenharia Elétrica Universidade de Brasília (UnB) Brasília, DF de Março de 2017 Class 03: Chapter 3 Spatial Transforms
2 Spatial Transformation Fucntions g(x, y) = T [f (x, y)] Linear Operations: H(a f +b g) = a H(f )+b H(g) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
3 Spatial Transformations g(x, y) = T [f (x, y)] Global: all image Neighborhood: regions (squares, circles, etc.) Pixel-to-pixel: 1 1 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
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5 Transformations s = T (r) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
6 Transformations Mylène Farias (ENE-UnB) IP 13 de Março de / 64
7 Negative Transformations s = L 1 r Mylène Farias (ENE-UnB) IP 13 de Março de / 64
8 Transformations Mylène Farias (ENE-UnB) IP 13 de Março de / 64
9 Logarithm Transformations s = c log(1 + r) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
10 Transformation Function Mylène Farias (ENE-UnB) IP 13 de Março de / 64
11 Gamma Transformation Function s = cr γ Mylène Farias (ENE-UnB) IP 13 de Março de / 64
12 Gamma Transformation Function Mylène Farias (ENE-UnB) IP 13 de Março de / 64
13 Gamma Transformation Function Mylène Farias (ENE-UnB) IP 13 de Março de / 64
14 Gamma Transformation Function Image is corrected before visualization Mylène Farias (ENE-UnB) IP 13 de Março de / 64
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18 Piecewise-Linear Transformation Fucntions Mylène Farias (ENE-UnB) IP 13 de Março de / 64
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20 Bit-plane slicing Mylène Farias (ENE-UnB) IP 13 de Março de / 64
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24 Contrast Mylène Farias (ENE-UnB) IP 13 de Março de / 64
25 Discrete Function: h(r k ) = n k r k intensity level n k number of pixels with intensity level equal to r k Normalized total number of pixels probability sum = 1
26 Histogram Transformation The histogram transformation function transforms a pixel distribution into a more interesting distribution; A function T(r) is used, which must satisfy the following conditions: 1 Be monotonically increasing in the interval 0 r (L 1) 2 0 T (r) L 1 for 0 r (L 1) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
27 Histogram Transformation Mylène Farias (ENE-UnB) IP 13 de Março de / 64
28 Histogram Transformation p r e p s are probability density functions (PDF); T (r) is the function that transforms a variable (r) into another (s); Mylène Farias (ENE-UnB) IP 13 de Março de / 64
29 Histogram Transformation p r e p s are probability density functions (PDF); T (r) is the function that transforms a variable (r) into another (s); If pr(r) and T (r) are known, with T (r) continuous and differentiable, p s (s) can be obtained by: p s (s) = p r (r) dr ds Mylène Farias (ENE-UnB) IP 13 de Março de / 64
30 Histogram Transformation p r e p s are probability density functions (PDF); T (r) is the function that transforms a variable (r) into another (s); If pr(r) and T (r) are known, with T (r) continuous and differentiable, p s (s) can be obtained by: p s (s) = p r (r) dr ds One example of T (r): s = T (r) = (L 1) r 0 p r (w)dw Mylène Farias (ENE-UnB) IP 13 de Março de / 64
31 Histogram Transformation Considering: r s = T (r) = (L 1) p r (w)dw 0 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
32 Histogram Transformation Considering: r s = T (r) = (L 1) p r (w)dw 0 ds dr = dt (r) dr Mylène Farias (ENE-UnB) IP 13 de Março de / 64
33 Histogram Transformation Considering: r s = T (r) = (L 1) p r (w)dw 0 ds dr = dt (r) dr ds dr = (L 1) d r p r (w)dw = (L 1)p r (r) dr 0 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
34 Histogram Transformation Considering: r s = T (r) = (L 1) p r (w)dw 0 ds dr = dt (r) dr r ds dr = (L 1) d p r (w)dw = (L 1)p r (r) dr 0 p s (s) = p r (r) dr ds = p r (r) 1 p r (r)(l 1) = 1 L 1 Uniform Distribution Mylène Farias (ENE-UnB) IP 13 de Março de / 64
35 Histogram Transformation r s = T (r) = (L 1) p r (w)dw 0 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
36 Histogram Transformation Continuous Case: s = T (r) = (L 1) r 0 p r (w)dw Mylène Farias (ENE-UnB) IP 13 de Março de / 64
37 Histogram Transformation Continuous Case: Discrete Case: For k = 0, 1,..., L 1. s = T (r) = (L 1) s k = T (r k ) = (L 1) = (L 1) = (L 1) M N r 0 p r (w)dw k p r (r j )s k j=0 k j=0 k j=0 n j M N n j Mylène Farias (ENE-UnB) IP 13 de Março de / 64
38 Histogram Transformation Ex: 3 bits Image ( L = 8) of size pixels (M N = 4096) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
39 Histogram Transformation Ex.: 3 bits Image ( L = 8) of size pixels (M N = 4096) s 0 = T (r 0 ) = 7 s 1 = T (r 1 ) = 7 0 p r (r j ) = 7 0, 19 = 1, p r (r j ) = 7 (0, , 25) = 3, s 2 = 4, 55 5 s 3 = 5, 67 6 s 4 = 6, 23 6 s 5 = 6, 65 7 s 6 = 6, 86 7 s 7 = 7, 00 7 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
40 Histogram Transformation Mylène Farias (ENE-UnB) IP 13 de Março de / 64
41 Histogram Transformation s k = T (r k ) = (L 1) M N k j=0 n j Mylène Farias (ENE-UnB) IP 13 de Março de / 64
42 Histogram Transformation Mylène Farias (ENE-UnB) IP 13 de Março de / 64
43 Histogram Transformation Mylène Farias (ENE-UnB) IP 13 de Março de / 64
44 Histogram Transformation Question: Does the histogram equalization always gives good results? Mylène Farias (ENE-UnB) IP 13 de Março de / 64
45 Histogram Transformation Question: Does the histogram equalization always gives good results? NO! Mylène Farias (ENE-UnB) IP 13 de Março de / 64
46 Histogram Specification An image has a specific histogram; Let p r (r) and p z (z) be the probability density functions of the variables r e z. p z (z) is the specified probability density function; Let s be a random variable s = T (r) = (L 1) r 0 p r (w)dw Mylène Farias (ENE-UnB) IP 13 de Março de / 64
47 Histogram Specification We obtain the function G: r s = T (r) = (L 1) G(z) = (L 1) z 0 0 p z (t)dt = s z = G 1 (s) = G 1 [T (r)] p r (w)dw Mylène Farias (ENE-UnB) IP 13 de Março de / 64
48 Histogram Specification Mylène Farias (ENE-UnB) IP 13 de Março de / 64
49 Histogram Specification 1 Obtain the p r (r) of the input image and calculate the s values: r s = T (r) = (L 1) p r (w)dw 0 2 Use the specified PDF to obtain the function G(z): 3 Map s to z z G(z) = (L 1) p z (t)dt = s 0 z = G 1 (s) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
50 Histogram Specification Ex.: If we assume continuous values: p r (r) = { 2r, (L 1) 2 para0 r L 1 0, caso contrário (1) Find the transformations thah produces: p z (z) = { 3z 2, (L 1) 3 para0 z L 1 0, caso contrário (2) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
51 Histogram Specification Steps: Find the equalization transformation: s = T (r) = (L 1) r = (L 1) 0 r 0 p r (w)dw 2w (L 1) 2 dw = r 2 L 1 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
52 Histogram Specification Steps: Find the equalization transformation: s = T (r) = (L 1) r = (L 1) 0 r 0 p r (w)dw 2w (L 1) 2 dw = r 2 L 1 The transformation for the histogram: G(z) = (L 1) = (L 1) z 0 z 0 p z (t)dt 3t 2 (L 1) 3 dt = z 3 (L 1) 2 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
53 Histogram Specification Transformation Function: z = [ (L 1) 2 s ] 1/3 = [ (L 1) 2 r 2 = [ (L 1)r 2] 1/3 L 1 ] 1/3 So, to obtain the transformed image, we apply the above function in the r variable. Mylène Farias (ENE-UnB) IP 13 de Março de / 64
54 Histogram Specification Discrete case: Obtain p r (r j ) of the input image and, later, the s k values, rounding the values to integers in the interval [0, L 1]: s k = T (r k ) = (L 1) M N k j=0 n j Use the specified PDF to obtain the function G(z q ), rounding the values to integers in the interval [0, L 1]: q G(z k ) = (L 1) p z (z i ) = s k i=0 Map values of s k to z q : z q = G 1 (s k ) Mylène Farias (ENE-UnB) IP 13 de Março de / 64
55 Histogram Specification In practice, we do not calculate the inverse... We calculate all possible values of G to every q q G(z k ) = (L 1) p z (z i ) = s k i=0 This values are scaled and rounded to the nearest integers in the interval [0, L 1] and saved (table); For a given value s k, we find the closest value in this table. Mylène Farias (ENE-UnB) IP 13 de Março de / 64
56 Histogram Specification Discrete Case: 1 Calculate the histogram of the input image, p r (r), and use it to calculate the equalization transformation: s k = T (r k ) = (L 1) M N k j=0 n j Round the results, s k, to integers in the interval [0, L 1 2 Calculate all transformation values G for q = 0, 1,..., L 1 G(z k ) = (L 1) q p z (z i ) = s k i=0 Round the results of G to integers in the interval [0, L 1 e save these results in a table. Mylène Farias (ENE-UnB) IP 13 de Março de / 64
57 Histogram Specification Discrete Case: 1 For each value of s k, k = 0, 1,..., L 1, use the values of G (step 2) to find the corresponding values of z q, in a way that G(z q ) is the closest to s k. Save this mapping values form s to z. 2 Build an image with a specified histogram, performing first the equalization and, then, the mapping from s k to z q. Mylène Farias (ENE-UnB) IP 13 de Março de / 64
58 Histogram Specification Example: 3 bits image (L = 8), (M N = 4096) pixels Mylène Farias (ENE-UnB) IP 13 de Março de / 64
59 Histogram Specification Example: Mylène Farias (ENE-UnB) IP 13 de Março de / 64
60 Especificação de Histogramas Exemplo: s 0 = 1, 33 1 s 1 = 3, 08 3 s Obtain equalized samples: 2 = 4, 55 5 s 3 = 5, 67 6 s 4 = 6, 23 6 s 5 = 6, 65 7 s 6 = 6, 86 7 s 7 = 7, 00 7 Calculate the transformation value: G(z 0 ) = 7 0 p z (z j ) = 0, 00 0 j=0 G(z 1 ) = 0, 00 G(z 2 ) = 0, 00 G(z 3 ) = 1, 05 G(z 4 ) = 2, 45 G(z 5 ) = 4, 55 G(z 6 ) = 5, 95 G(z 7 ) = 7, 00 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
61 Histogram Specification Example: s 0 = 1, 33 1 s 1 = 3, 08 3 s Obtain equalized samples: 2 = 4, 55 5 s 3 = 5, 67 6 s 4 = 6, 23 6 s 5 = 6, 65 7 s 6 = 6, 86 7 s 7 = 7, 00 7 Calculate the transformation values and round it to the closest integers: 0 G(z 0 ) = 7 p z (z j ) = 0, 00 0 j=0 G(z 1 ) = 0, 00 0 G(z 2 ) = 0, 00 0 G(z 3 ) = 1, 05 1 G(z 4 ) = 2, 45 2 G(z 5 ) = 4, 55 5 G(z 6 ) = 5, 95 6 G(z 7 ) = 7, 00 7 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
62 Histogram Specification Find the smallest value of z q in a way that G(z q ) is the closest value to s k. Mylène Farias (ENE-UnB) IP 13 de Março de / 64
63 Histogram Specification Mylène Farias (ENE-UnB) IP 13 de Março de / 64
64 Original Mylène Farias (ENE-UnB) IP 13 de Março de / 64
65 Equalized Mylène Farias (ENE-UnB) IP 13 de Março de / 64
66 Specified Mylène Farias (ENE-UnB) IP 13 de Março de / 64
67 Local Processing of Histograms Define a neighborhood and move the the central position pixel by pixel; In each position, the histograms of the neighborhood is calculated. We can perform a histogram equalization or a histogram specification; Map the intensity of the central neighborhood pixel; Move to the next positions and repeat the procedure. Mylène Farias (ENE-UnB) IP 13 de Março de / 64
68 Local Processing of Histograms Mylène Farias (ENE-UnB) IP 13 de Março de / 64
69 Histogram Statistics Intensity average Intensity Variance L 1 m = L 1 σ 2 = u 2 (r) = i=0 i=0 r i p(r i ) = 1 MN M 1 N 1 x=0 y=0 (r i m) 2 p(r i ) = 1 MN M 1 f (x, y) N 1 [f (x, y) m] 2 x=0 y=0 n-th Moment L 1 u n (r) = (r i m) n p(r i ) i=0 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
70 Histogram Statistics Local Intensity Average: L 1 m Sxy = r i p Sxy (r i ) i=0 Local Intensity Variance L 1 σs 2 xy = (r i m Sxy ) 2 p Sxy (r i ) S xy neighborhood i=0 Mylène Farias (ENE-UnB) IP 13 de Março de / 64
71 Local Histogram Processing Mylène Farias (ENE-UnB) IP 13 de Março de / 64
72 Local Histogram Processing Mylène Farias (ENE-UnB) IP 13 de Março de / 64
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