1. Image Processing and Sampling
Aaron Bloomfield
CS 445: Introduction to Graphics
Fall 2006

2. Overview Image representation Halftoning and dithering
What is an image?
Halftoning and dithering
Trade spatial resolution for intensity resolution
Reduce visual artifacts due to quantization

3. What is an Image? An image is a 2D rectilinear array of pixels
Continuous image
Digital image

4. What is an Image? An image is a 2D rectilinear array of pixels
Continuous image
Digital image
A pixel is a sample, not a little square!

5. What is an Image? An image is a 2D rectilinear array of pixels
Continuous image
Digital image
A pixel is a sample, not a little square!

6. Image Acquisition Pixels are samples from continuous function
Photoreceptors in eye
CCD cells in digital camera
Rays in virtual camera

7. Image is reconstructed
Image Display
Re-create continuous function from samples
Example: cathode ray tube
Image is reconstructed
by displaying pixels
with finite area
(Gaussian)

8. Image Resolution Intensity resolution Spatial resolution
Each pixel has only “Depth” bits for colors/intensities
Spatial resolution
Image has only “Width” x “Height” pixels
Temporal resolution
Monitor refreshes images at only “Rate” Hz
Width x Height Depth Rate
NTSC 640 x
Workstation x
Film 3000 x
Laser Printer 6600 x
Resolutions
Typical

9. Sources of Error Intensity quantization Spatial aliasing
Not enough intensity resolution
Spatial aliasing
Not enough spatial resolution
Temporal aliasing
Not enough temporal resolution

10. Overview Image representation Halftoning and dithering
What is an image?
Halftoning and dithering
Reduce visual artifacts due to quantization

11. Quantization Artifacts due to limited intensity resolution
Frame buffers have limited number of bits per pixel
Physical devices have limited dynamic range

12. Uniform Quantization P(x, y) = trunc(I(x, y) + 0.5) I(x,y) P(x,y)
(2 bits per pixel)

13. Uniform Quantization Images with decreasing bits per pixel:
256 bpp bpp bpp 2 bpp
Notice the quantization error, most noticeable in the 8 and 2 bpp images – large flat areas with no detail
8 bits
4 bits
2 bits
1 bit

14. Reducing Effects of Quantization
Halftoning
Classical halftoning
Dithering
Error diffusion dither
Random dither
Ordered dither

15. Classical Halftoning Use dots of varying size to represent intensities
Area of dots proportional to intensity in image
I(x,y)
P(x,y)

16. Classical Halftoning
Newspaper image from North American Bridge Championships Bulletin, Summer 2003

17. Halftone patterns Use cluster of pixels to represent intensity
Trade spatial resolution for intensity resolution
Figure from H&B

18. Dithering Distribute errors among pixels
Exploit spatial integration in our eye
Display greater range of perceptible intensities
Uniform quantization discards all errors
i.e. all “rounding” errors
Dithering is also used in audio, by the way

19. Floyd-Steinberg dithering
Any “rounding” errors are distributed to other pixels
Specifically to the pixels below and to the right
7/16 of the error to the pixel to the right
3/16 of the error to the pixel to the lower left
5/16 of the error to the pixel below
1/16 of the error to the pixel to the lower right
Assume the 1 in the middle gets “rounded” to 0

20. Error Diffusion Dither
Spread quantization error over neighbor pixels
Error dispersed to pixels right and below    
 +  +  +  = 1.0
Figure from H&B

21. Floyd-Steinberg dithering
Floyd-Steinberg dithering is a specific error dithering algorithm
Uniform
Quantization
(1 bit)
Floyd-Steinberg
Dither (1 bit)
Floyd-Steinberg
Dither (1 bit)
(pure B&W)
Original
(8 bits)

22. Random Dither P(x, y) = trunc(I(x, y) + noise(x,y) + 0.5)
Randomize quantization errors
Errors appear as noise
I(x,y)
P(x,y)
I(x,y)
P(x,y)
P(x, y) = trunc(I(x, y) + noise(x,y) + 0.5)

23. Random Dither Original (8 bits) Uniform Quantization (1 bit) Random

24. Ordered Dither i = x mod n j = y mod n e = I(x,y) - trunc(I(x,y))
Pseudo-random quantization errors
Matrix stores pattern of thresholds
i = x mod n
j = y mod n
e = I(x,y) - trunc(I(x,y))
if (e > D(i,j))
P(x,y) = ceil(I(x, y))
else
P(x,y) = floor(I(x,y))

25. Ordered Dither Bayer’s ordered dither matrices
Reflections and rotations of these are used as well

26. Ordered Dither Original (8 bits) Uniform Quantization (1 bit)
4x4 Ordered
Dither
(1 bit)

27. Dither Comparison Original (8 bits) Random Dither (1 bit) Ordered
Floyd-Steinberg
Dither
(1 bit)

28. Color dithering comparison
Original image Web-safe palette, no dithering Web-safe palette, FS dithering
Optimized 256 color palette Optimized 16 color palette Optimized 16 color palette
FS dithering No dithering FS dithering

29. Summary Image representation Halftoning and dithering
A pixel is a sample, not a little square
Images have limited resolution
Halftoning and dithering
Reduce visual artifacts due to quantization
Distribute errors among pixels
Exploit spatial integration in our eye
Sampling and reconstruction
Reduce visual artifacts due to aliasing
Filter to avoid undersampling
Blurring is better than aliasing