Optimum Notch Filtering - I

Transcription

1 Notch Filtering 1

2 Optimum Notch Filtering - I Methods discussed before: filter too much image information. Optimum: minimizes local variances of the restored image f First Step: Etract principal freq. components of the interference pattern. Place a notch pass filter at the location of each spike. N u v = H u v G u v NP ˆ Second Step: Find corresponding pattern in spatial domain. { H u v G u v } 1 n = I Eq.1 NP

3 Optimum Notch Filtering - II Third Step: Subtract weighted noise estimation from nois image. How to get w? f ˆf = g w n Eq. Select w so that the variance of over a neighborhood. f ˆ is minimized Consider: neighborhood size: a+1 b b+1 Average: a b 1 f ˆ = f ˆ + s + t a + 1b + 1 s= at= b Local variance: σ = 1 a + 1b + 1 a b s= at= b fˆ + s + t fˆ 3

4 Optimum Notch Filtering - III [ ] a b t s n w t s g 1 [ ] = = + + = a s b t n w g b a 1 1 σ 0 = w σ To minimize n g n g w = We find Eq.3 n n Get noise reduced image from Eq 1 3 Get noise reduced image from Eq

5 Image Reconstruction : projection Computed Tomograph CT X-Ras from different angles. Back projection Angle varied: 90 Sum of two back projections 5

6 Projection Projection using man angles: more true construction of the original image. Sum of 3 back projections. 6

7 Principles of CT G1: Pencil X-Ra beam; one detector; angle [0 ~ 180 ]. G: Fan X-Ra beam; multiple detectors. G3: Wider X-Ra beam; a bank of detectors G4: Circular positioned detectors G5: No mechanical motion uses electron beams controlled electromagneticall. G6 G7.. 7

8 Color Fundamentals White color: composed of 6 visible colors. 8

9 Primar & Secondar Colors Green Red Blue Yellow Magenta Can 9

10 Characteristics of Colors 1. Brightness: how bright intensit the color is. [Perception.]. Hue: dominant wavelength in a miture of light waves. 3. Saturation: the amount of white light mied with a hue. Inversel proportional Hue + Saturation ti = Chromaticit it 10

11 RGB Color Model Black 000 White bit color o cube. All R G B values are normalized to [0 1] range. 11

12 Safe Color Man sstems in use toda are limited to 56 colors. Fort 40 of these are processed differentl b various operating sstems. The rest 16 are common: de facto standard for safe colors. 6 3 = 16 1

13 CMY and CMYK Color Models = G R M C 1 1 B G Y 1 Used as primar colors in printers. Equal amounts of C M Y should produce Black. But in practice it produces mudd-looking black. Four-color printing: add a fourth color: Black to ou co o p g: dd ou co o : c o CMY producing CMYK model. 13

14 HSI Color Model - I Hue Saturation Intensit. 14

15 HSI Color Model - II Triangular Plane Circular Plane 15

16 Fundamentals of Image Compression Relative data redundanc d Compression ratio: 1 R = 1 C b C = b Bits required in the method Bits required in the base method Average number of bits required to represent each piel: L L = ll r P r Avg 1 k = 0 k nk Pr rk = k = L 1 MN r k 16

17 Variable Length Coding Table 1. For code 1: l 1 r k = 8 bits for al r k. Average = 8. For code : L Avg = = 1.81 bits C = 4.4 R = 1 = % of the data in the original 8-bit intensit arra is redundant. 17

18 Irrelevant Information Spatial redundanc Figure 8.1 Histogram Histogram equalized image 18

19 Entrop Entrop: H = L 1 k =00 p r r k log pr rk Entrop in Table 1.: bits / piel Entrop of Figure 1 b = 8 bits / piel Entrop of Figure 1 c = bits / piel Figure 1 a Higher entrop than Fig. 1 a Little or no information But comparable entrop with Fig. 1a! Entrop of an image is far from intuitive. 19

20 Fidelit Criteria - I Quantifing the nature of the loss after removal of noise. Objective: mathematical ti epression such as rms error SNR etc. e rms 1/ M 1N 1 1 = MN = 0 = 0 [ ] fˆ f Subjective: Human evaluation. 0

21 Fidelit Criteria - II Misleading image rms error: Objective criteria fails. 1

22 Image Compression Models Quantizer: irreversible.

23 Some Compression Standards 3

24 Huffman Coding Encoded: Decoded: a 3a1aaa6 Average length of this code: = L Avg =. bits/piel 4

25 Golomb Coding G m n Step 1: Step : Form the unar code of quotient n/m. Let k = log m c = k m r = n mod m compute r r truncated to k 1 bits 0 r < c r = r + c truncated to k bits otherwise Step 3: Concatenated the results of steps 1 and. Eample: Compute G /4 =.5 5 = Unar code: 110 k = c = 4 = 0 r = After concatenating: G 4 9 = r = 01 truncated to bits. 5

26 Eponential Golomb Coding G k ep n j+ k Step 1: Find an integer i >= 0 such that n < i 1 j= 0 i j= 0 And form the unar code of i. If k = 0 i = log n+1 j+ k Step : Step 3: Truncate the binar representation of n i 1 j= 0 j+ k Concatenated the results of steps 1 and. for k + i least significant bits. Eample: Compute G 0 ep8. i = 3 because k = Check for the equation in Step j+ 0 j= 0 Truncate = 8 7 = 1 = 0001 After concatenating: G 0 ep8 =