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 fˆ x First Step: Extract 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 Second Step: Find corresponding pattern in spatial domain. n x 1 H NP u v G u v Eq.1 2

3 Optimum Notch Filtering - II Third Step: Subtract weighted noise estimation from nois image. fˆ x g x w x n x Eq.2 How to get wx? Select wx so that the variance of over a neighborhood. fˆ x is minimized Consider: neighborhood size: 2a+1 b 2b+1 Average: a b 1 f ˆ x fˆ x s t 2a 12b 1 satb Local variance: 2 x 1 2a 12b 1 a b satb fˆ x s t fˆ x 2 3

4 4 Optimum Notch Filtering - III a a s b b t x n w x x g t s x n w x t s x g b a x w x x 2 2 x n x n x n x g x n x g x w To minimize We find Eq.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 32 back projections. 6

7 Principles of CT G1: Pencil X-Ra beam; one detector; angle [0 ~ 180]. G2: 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.] 2. Hue: dominant wavelength in a mixture of light waves. 3. Saturation: the amount of white light mixed with a hue. Inversel proportional Hue + Saturation = Chromaticit 10

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

12 Safe Color Man sstems in use toda are limited to 256 colors. Fort 40 of these are processed differentl b various operating sstems. The rest 216 are common: de facto standard for safe colors. 6 3 =

13 13 CMY and CMYK Color Models B G R Y M C 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 CMY producing CMYK model.

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 R 1 1 C Compression ratio: C b b Bits required in the method Bits required in the base method Average number of bits required to represent each pixel: L 1 LAvg l rk Pr rk k 0 nk Pr rk k L 1 MN 16

17 Variable Length Coding Table 1. For code 1: l 1 r k = 8 bits for al r k. Average = 8. For code 2: L Avg bits C 4.42 R % 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 k0 p r r k log 2 pr rk Entrop in Table 1.: bits / pixel Entrop of Figure 1 b = 8 bits / pixel Entrop of Figure 1 c = bits / pixel 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 expression such as rms error SNR etc. e rms 1 MN M 1N 1 x0 0 fˆ x f x 2 1/ 2 Subjective: Human evaluation. 20

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

22 Image Compression Models Quantizer: irreversible. 22

23 Some Compression Standards 23

24 Huffman Coding Encoded: Decoded: a a a a a6 Average length of this code: L Avg 2.2 bits/pixel 24

25 Golomb Coding G m n Step 1: Step 2: Step 3: Form the unar code of quotient n/m. Let k = log 2 m c = 2 k m r = n mod m compute r r truncatedto k 1 bits r r c truncat edtok bits 0 r c otherwis e Concatenated the results of steps 1 and 2. Example: Compute G /4 = 2.25 = 2 Unar code: 110 k = 2 c = = 0 r = After concatenating: G 4 9 = r = 01 truncated to 2 bits. 25

26 Exponential Golomb Coding G k exp n Step 1: Find an integer i >= 0 such that And form the unar code of i. i1 j0 2 jk n i j0 2 jk If k = 0 i = log 2 n+1 Step 2: Step 3: Truncate the binar representation of n i 1 j0 2 jk for k i Concatenated the results of steps 1 and 2. least significan t bits. Example: Compute G 0 exp8. i = 3 because k = Check for the equation in Step j0 2 j0 Truncate After concatenating: G 0 exp8 =