LC NHIU trong min khng gian 9172020 1

LỌC NHIỄU trong miền không gian 9/17/2020 1

Lọc nhiễu trong miền không gian 1. Arithmetic mean Filter 2. Geometric mean filter 3. Harmonic mean filter 4. Contra-harmonic mean filter 5. Meadian filter 6. Max-and-min filter 7. Alpha-trimed mean filter 8. Gausian filter 9/17/2020 2

Mô hình khôi phục ảnh g(x, y) = f(x, y)*h(x, y) + n(x, y), (x, y) f(x, y) - ảnh đầu vào (ảnh gốc) h(x, y) – hàm tác động ảnh n(x, y) – nhiễu cộng g(x, y) - ảnh nhận được Tiêu chí: Ảnh g(x, y) phải gần với ảnh f(x, y) g(x, y) f(x, y), (x, y) 9/17/2020 3

Mô hình khôi phục ảnh Xét mô hình g(x, y) = f(x, y) + n(x, y), (x, y) n(x, y) – nhiễu cộng, tuần hoàn 9/17/2020 4

1. Arithmetic Mean Filters Kí hiệu W(x, y) là một lân cận chữ nhật kích thước a b của điểm (x, y) (điểm (x, y) trùng với tâm hình chữ nhật). (đ Thông thường, W(x, y) được chọn là lân cận 3× 3 của (x, y) 9/17/2020 5

1. Arithmetic Mean Filter (i, j) = 1, (i, j) W(x, y), =9 9/17/2020 Mean Filters (i, j) = 1, (i, j) W(x, y){x, y), (x, y) = 2, =10 (i, j) {1, 2, 4} (i, j) W(x, y) =16 6

1. Arithmetic Mean Filter 9/17/2020 Mean Filters 7

1. Arithmetic Mean Filters For y = 2 To H - 1 For x = 2 To W - 1 ds = Hx(1, 1)*f(y-1, x-1) + Hx(1, 2)*f(y-1, x) + Hx(1, 3)*f(y-1, x+1) Hx(2, 1)*f(y, x-1) + Hx(2, 2)*f(y, x) + Hx(2, 3)*f(y, x+1) Hx(3, 1)*f(y+1, x-1) + Hx(3, 2)*f(y+1, x) + Hx(3, 3)*f(y+1, x+1 Q(y, x) = Round(ds / dt) Next: Next For y = 2 To H - 1 For x = 2 To W - 1 ds = 0 For p = 1 To 3 For q = 1 To 3 ds = ds + Hx(p, q) * f(y + p - 2, x + q - 2) Next: Next Q(y, x) = Round(ds / dt) Next: Next 9/17/2020 8

1. Arithmetic Mean Filters 9/17/2020 9

2. Geometric Mean Filters W(x, y) là lân cận chữ nhật kích thước a b của điểm (x, y) Thông thường, W(x, y) được chọn là lân cận 3× 3 của (x, y) 9/17/2020 10

2. Geometric Mean Filters For y = 2 To H - 1 For x = 2 To W - 1 ds = f(y-1, x-1) * f(y-1, x+1) * f(y, * f(y+1, x-1) * f(y+1, x+1 Q(y, x) = ds^(1/9) Next: Next x-1) * f(y, x) * f(y, x+1) For y = 2 To H - 1 For x = 2 To W - 1 ds = 1 For p = 1 To 3 For q = 1 To 3 ds = ds * f(y + p - 2, x + q - 2) Next: Next Q(y, x) = Round(ds ^(1/9)) Next: Next 9/17/2020 11

2. Geometric Mean Filter 9/17/2020 Mean Filters 12

3. Harmonic Mean Filters W(x, y) là lân cận chữ nhật kích thước a b của điểm (x, y) Thông thường, W(x, y) được chọn là lân cận 3× 3 của (x, y) 9/17/2020 13

3. Harmonic Mean Filters For y = 2 To H - 1 For x = 2 To W - 1 ds = 0 For p = 1 To 3 For q = 1 To 3 ds = ds + (1/ f(y + p - 2, x + q - 2)) Next: Next Q(y, x) = Round (9 / ds) Next: Next 9/17/2020 14

3. Harmonic Mean Filter 9/17/2020 Mean Filters 15

4. Contraharmonic Mean Filter W(x, y) là lân cận chữ nhật kích thước a b của điểm (x, y) Thông thường, W(x, y) được chọn là lân cận 3× 3 của (x, y) Khi Q = -1, contraharmonic trở thành harmonicmean filter. 9/17/2020 16

4. Contraharmonic Mean Filter For y = 2 To H - 1 For x = 2 To W - 1 dt = 0 dm=0 For p = 1 To 3 For q = 1 To 3 dt = dt + (f(y + p - 2, x + q - 2))^(Q+1) dm = dm + (f(y + p - 2, x + q - 2))^Q Next: Next Q(y, x) = Round (dt / dm) Next: Next 9/17/2020 17

4. Contraharmonic Mean Filter 9/17/2020 18

4. Contraharmonic Mean Filter Ảnh gốc Q= -3 Q=-2 Q=-1 9/17/2020 Q= 3 Q= 2 Q= 1 19

4. Contraharmonic Mean Filter abc def a) b) c) d) e) f) Q=-2 Ảnh gốc Q=2 Q=-1 Q=0 Q=1 9/17/2020 20

5. Median filter g(x, y) = median {f(i, j), (i, j) W(x, y)} Let given sequence of numbers: x 1, x 2, . . . , xm+1, xm+2, . . . , x 2 m+1 (1) The value xk {x 1, . . . , x 2 m+1} is called median of sequence (1) if exist: m elements from {x 1, . . . , x 2 m+1}{xk} are not greater than xk, and m others not smaller than xk 9/17/2020 21

5. Median filter Example 1 The median of sequence 1, equal 2, 5, 4 4 because 2 entries: and 2 others: 9/17/2020 9, {1, 2} are not bigger than 4 {9, 5} are not smaller than 4. 22

5. Median filter Example 2 The median of sequence 15, 17, 18, 16, 78, 17, 15, 20 equal 17 because 4 entries are not bigger than 17 and 4 others are not smaller than 17 15, 17, 9/17/2020 18, 16, 78, 17, 15, 20 23

5. Median filter The original image and filtered by median filter 9/17/2020 24

5. Median filter The original image and filtered by median filter Effect: For images corrupted by bipolar or unipolar impulse noise – called salt-and-pepper noise (particularly effective) 9/17/2020 25

5. Median filter 1. Put the windows 3 3 such that center of the windows at the point (x, y). 2. Order the pixels of the windows to a sequence in brightness from smallest to largest 3. Set g(x, y) = centered of a sequence Put to sequence x y 15 17 18 16 78 17 17 15 20 9/17/2020 Odered sequence 15, 17, 18, 16, 78, 17, 15, 20 15, 16, 17, 17, 18, 20, 78 g(x, y) = 17 26

5. Median filter For y = 2 To H - 1 For x = 2 To W - 1 For k = 1 To 3 For q = 1 To 3 D((k-1)*3+q) = f(y+k-2, x+q-2) Next k: Next q Sort: D(1) D(2) . . . D(9) g(x, y) = D(5) Next x: Next y 9/17/2020 27

Max and min filter Max filter For 1 y M, 1 x N: f^(x, y) = max {f(i, j): (i, j) W(x, y) }, Min filter For 1 y M, 1 x N: f^(x, y) = min {f(i, j): (i, j) W(x, y) }, Midpoint Filter For 1 y M, 1 x N: f^(x, y) =[min{f(i, j): (i, j) W(x, y)}+ max{f(i, j): (i, j) W(x, y)}]/2 9/17/2020 28

Max filter For y = 1 To H - 2 For x = 1 To W - 2 1) For k = 1 To 3 For q = 1 To 3 D((k-1)*3+q) = Getpixel(y+k-2, x+q-2) Next k: Next q 2) Sort D: D(1) D(2) . . . D(9) 3) f^(x, y) = D(9) Next x: Next y 9/17/2020 29

Min filter For y = 1 To H - 2 For x = 1 To W - 2 1) For k = 1 To 3 For q = 1 To 3 D((k-1)*3+q) = Getpixel(y+k-2, x+q-2) Next k: Next q 2) Sort D: D(1) D(2) . . . D(9) 3) f^(x, y) = D(1) Next x: Next y 9/17/2020 30

Midpoint filter For y = 1 To H - 2 For x = 1 To W - 2 1) For k = 1 To 3 For q = 1 To 3 D((k-1)*3+q) = Getpixel(y+k-2, x+q-2) Next k: Next q 2) Sort D: D(1) D(2) . . . D(9) 3) f^(x, y) = (D(1)+D(9))/2 Next x: Next y 9/17/2020 31

Alpha-trimmed mean filter Define: • m, n are dimentions of neighborhood W(x, y) • d is an integer value in range [0. . mn-1] • fr(x, y) is remaning of f(x, y) after deletion d/2 lowest and d/2 highest in W(x, y) remark: 1) If d=0 Alpha-trimmed mean filter become arthmetic filter 2) If d=mn-1 Alpha-trimmed mean filter become median filter 9/17/2020 32

Alpha-trimmed mean filter a bcde a) b) c) d) e) 9/17/2020 Origin image d=0 d=2 d=4 d=6 33

Alpha-trimmed mean filter ab cde a) b) c) d) e) 9/17/2020 Origin image d=0 d=2 d=4 d=6 34

Alpha-trimmed mean filter ab cde a) b) c) d) e) 9/17/2020 Origin image d=0 d=2 d=4 d=6 35

Alpha-trimmed mean filter a) Origin image 9/17/2020 d=6 36

Alpha-trimmed mean filter d=? For i = 2 To H - 1 For j = 2 To W - 1 x(1. . 9) = f(i -1. . i+1, j – 1. . j+1) Sắp xếp: x(1) ≤. . ≤x(9) g(i, j) = {x(d/2) +. . . +x(9 -d/2)}/(9 -d) 9/17/2020 37

Alpha-trimmed mean filter d=? For i = 2 To H - 1 For j = 2 To W - 1 For k = 1 To 3 For q = 1 To 3 x((k - 1) * 3 + q) = f(i + k - 2, j + q - 2) For t = 1 To 8 k=t For v = k To 9 If x(v) < x(k) Then k = v tg = x(k): x(k) = xx(t): xx(t) = tg ts = 0 For t = 1 + d/2 to 9 - d/2 ts = ts + x(t) g(i, j) = ts/(9 -d) 9/17/2020 38

Gauusian smoothing where: 1) x is the distance from the origin in the horizontal axis, 2) y is the distance from the origin in the vertical axis, and 3) σ is the standard deviation of the Gaussian distribution. 9/17/2020 39

Gausian smoothing This figure shows a suitable integer-valued convolution kernel that approximates a Gaussian with a of 1. 0 and distance from origin is 2 9/17/2020 40

Gausian smoothing The original and filtered images byte gaussian smoothing operation 9/17/2020 41

Gausian smoothing The original and filtered images byte gaussian smoothing operation 9/17/2020 42

Gausian smoothing The original and filtered images byte gaussian smoothing operation 9/17/2020 43