Image Acquisition Image Digitization Spatial domain Intensity domain Image Characteristics

Transcription

1 Image Acquisition Image Digitization Spatial domain Intensity domain Image Characteristics 1

2 What is an Image? An image is a projection of a 3D scene into a 2D projection plane. An image can be defined as a 2 variable function f(x,y): R 2 R, where for each position (x,y) in the projection plane, f(x,y) defines the light intensity at this point. 2

3 Image as a function 3

4 Image Acquisition i(x,y) g(i,j) f(x,y)=i(x,y) r(x,y) r(x,y) pixel=picture element 4

5 Acquisition System World Camera Digitizer Digital Image CMOS sensor 5

6 Image Types Three types of images: Binary images g(x,y) {0, 1} Gray-scale images Color Images g(x,y) C typically c={0,,255} three channels: g R (x,y) C g G (x,y) C g B (x,y) C 6

7 Gray Scale Image y = x =

8 Color Image 8

9 Notations Image Intensity - Light energy emitted from a unit area in the image Device dependence Image Brightness - The subjective appearance of a unit area in the image Context dependence Subjective Image Gray-Level - The relative intensity at each unit area Between the lowest intensity (Black value) and the highest intensity (White value) Device independent 9

10 Intensity vs. Brightness 10

11 Intensity vs. Brightness 11

12 Intensity vs. Brightness f 1 < f 2, f 1 = f 2 Intensity f 1 f1 f 2 f2 Equal intensity steps: Equal brightness steps: 12

13 Weber Law Describe the relationship between the physical magnitudes of stimuli and the perceived intensity of the stimuli. In general, f needed for just noticeable difference (JND) over background f was found to satisfy: f = f const Brightness log(f) 13

14 What about Color Space? JND in XYZ color space was measured by Wright and Pitt, and MacAdam in the thirties MacAdam ellipses: JND plotted at the CIE-xy diagram Conclusion: measuring perceptual distances in the cie-xyz space is not a good idea

15 Perceptually Uniform Color Space v * u * MacAdam Ellipses of JND plotted in CIE- L * u * v * Coordinates:

16 Acquisition System World Camera Digitizer Digital Image CMOS sensor 16

17 Digitization Two stages in the digitization process: Spatial sampling: Spatial domain Quantization: Gray level f x j y Continuous Image f(x,y) x,y R f(x,y) R i Digital Image g(i,j) C g(i,j) in finite set of values i,j = 1,2,3,...,c 17

18 Spatial Sampling When a continuous scene is imaged on the sensor, the continuous image is divided into discrete elements - picture elements (pixels)

19 Spatial Sampling Two principles: coverage of the image plane uniform sampling (pixels are same size and shape)

20 Spatial Sampling N = 4 N = 8 N = 16 N = 32 N = 64 N = 128

21 Sampling - Image Resolution The density of the sampling denotes the separation capability of the resulting image Image resolution defines the finest details that are still visible by the image Cyclic patterns test separation capability of an image Frequency = Wavelength number of cycles unit length = 1 frequency 0 x

22 Sampling Rate

23 Nyquist Frequency Nyquist Rule: To observe details at frequency f (wavelength d) one must sample at frequency > 2f (sampling intervals < d/2) The Frequency 2f is the Nyquist Frequency. Aliasing: If the pattern wavelength is less than 2d erroneous patterns may be produced. 1D Example: 0

24 Aliasing - Moiré Patterns

25 Aliasing - Moiré Patterns +

26 Temporal Aliasing

27 Temporal Aliasing Example

28 Image De-mosaicing Can we do better than Nyquist?

29 Image De-mosaicing Basic idea: use correlations between color bands A joint Histogram of r x v.s. g x Green derivative Red derivative

30 Non Uniform Sampling

31 Quantization Choose number of gray levels (according to number of assigned bits) Divide continuous range of intensity values

32 Quantization Number of Gray Levels

33 Quantization Low freq. areas are more sensitive to quantization 8 bits image 4 bits image

34 How should we quantize an image? Simplest approach: uniform quantization 10 Z i+ 1 q Z i i = Z Z k Z K + Z 2 i 1 i = + 0 Gray-Level Sensor Voltage q 0 q 1 q 2 q q k-1 Z 0 Z 1 Z 2 Z 3 Z Z k-1 Z k quantization level sensor voltage

35 Non-uniform Quantization Quantize according to visual sensitivity (Weber s Law) q 0 q 1 q 2 q 3 q 4 q 5 q 6 Z 0 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 High Visual Sensitivity Low Visual Sensitivity

36 Non-uniform Quantization Non uniform sensor voltage distribution sensor voltage Z 0 q 1 q 2 q 3 q 4 q 5 Z k quantization level

37 Optimal Quantization (Lloyd-Max) Content dependant Minimize quantization error q 0 q 1 q 2 q 3 quantization level sensor voltage Z 0 Z 1 Z 2 Z 3 Z 4

38 Optimal Quantization (Lloyd-Max) Also known as Loyd-Max quantizer Denote P(z) the probability of sensor voltage The quantization error is : E = k 1 i= 0 z i+ 1 z i P ( )( ) 2 z z q dz i Solution: demo q i = z i+ 1 z z i i+ 1 z i zp P ( z) ( z) dz dz z i q + q 2 i 1 i = Iterate until convergence (but optimal minimum is not guaranteed).

39 Example 8 bits image 4 bits image Uniform quantization 4 bits image Optimal quantization

40 Color Quantization Typically 256 levels for each Red, Greed, Blue channels, or = colors. How can an image be displayed with fewer colors than it contains? Select a subset of colors (the colormap or pallet) and map the rest of the colors to them. 24 bit 4 bit (16 colors) from: Daniel Cohen-Or

41 Digital Grayscale Image Digitization: Sampling Quantization y = x =