Quantization

1
Quantization

• If too few levels of gray, (e.g. decrease
  halftone spot size to increase spatial
  resolution), then boundaries between adjacent
  gray levels become apparent. This can happen in
  color halftoning also.
• See demo at http//www.ctr.columbia.edu/sfchang/c
  ourse/dip/demos/Quat.html

2
Saturation

• Distance from white point
• Adding white desaturates but does not change hue
  or perceptual brightness.
• HSB model is approximate representative of this.
  See PhotoShop

3
Device Independence

• Calibration to standard space
• typically CIE XYZ
• Coordinate transforms through standard space
• Gamut mapping

4
Device independence

• Stone et. al. Color Gamut Mapping and the
  Printing of Digital Color Images, ACM
  Transactions on Graphics, 7(4) October 1998, pp.
  249-292.
• The following slides refer to their techniques.

5
Device to XYZ

• Sample gamut in device space on 8x8x8 mesh (7x7x7
  343 cubes).
• Measure (or model) device on mesh.
• Interpolate with trilinear interpolation
• for small mesh and reasonable function
  XYZf(device1, device2, device3) this
  approximates interpolating to tangent.

6
XYZ to Device

• Invert function XYZf(device1, device2, device3)
• hard to do in general if f is ill behaved
• At least make f monotonic by throwing out
  distinct points with same XYZ.
• e.g. CMY device
• (continued)

7
XYZ to CMY

• Invert function XYZf(c,m,y)
• Given XYZx,y,z want to find CMYc,m,y such
  that f(CMY)XYZ
• Consider X(c,m,y), Y(c,m,y), Z(c,m,y)
• A continuous function on a closed region has max
  and min on the region boundaries, here the cube
  vertices. Also, if a continuous function has
  opposite signs on two boundary points, it is zero
  somewhere in between.

8
XYZ to CMY

• Given X0, find c,m,y such that f(c,m,y) X0
• if ci,mi,yi cj,mj,yj are vertices on a given
  cube, and UX(c,m,y)- X0 has opposite sign on
  them, then it is zero in the cube. Similarly Y,
  Z. If find such vertices for all of X0,Y0,Z0,
  then the found cube contains the desired point.
  (and use interpolation). Doing this recursively
  will find the desired point if there is one.

9
Gamut Mapping

• Criteria
• preserve gray axis of original image
• maximum luminance contrast
• few colors map outside destination gamut
• hue, saturation shifts minimized
• increase, rather than decrease saturation
• do not violate color knowledge, e.g. sky is blue,
  fruit colors, skin colors

10
Gamut Mapping

• Special colors and problems
• Highlights this is a luminance issue so is about
  the gray axis
• Colors near black locus of these colors in image
  gamut must map into something reasonably similar
  shape else contrast and saturation is wrong

11
Gamut Mapping

• Special colors and problems
• Highly saturated colors (far from white point)
  printers often incapable.
• Colors on the image gamut boundary occupying
  large parts of the image. Should map inside
  target gamut else have to project them all on
  target boundary.

12
Gamuts
CRT
Printer
13
Gamut Mapping

• First try map black points and fill destination
  gamut.

14
device gamut
image gamut
15
device gamut
translate Bito Bd
image gamut
bs (black shift)
16
device gamut
translate Bito Bd
image gamut
scale by csf
17
device gamut
translate Bito Bd
image gamut
scale by csf
rotate
18
Gamut Mapping

• Xd Bd csf R (Xi - Bi)
• Bi image black, Bd destination black
• R rotation matrix
• csf contrast scaling factor
• Xi image color, Xd destination color
• Problems
• Image colors near black outside of destination
  are especially bad loss of detail, hue shifts
  due to quantization error, ...

19
Xd Bd csf R (Xi - Bi) bs (Wd - Bd)
shift and scale alongdestination gray
20
Fig 14a, bs0, csf small, image gamut maps
entirelyinto printer gamut, but contrast is low.
Fig 14b, bs0, csf large, more contrast, more
colors inside printer gamut, butalso more
outside.
21
Saturation control

• Umbrella transformation
• Rs Gs Bs monitor whitepoint
• Rn Gn Bn new RGB coordinates such that Rs
  Gs Bs Rn Gn Bnand Rn Gn Bn maps
  inside destination gamut
• First map R RsG GsB Bs to R RnG GnB Bn
• Then map into printer coordinates
• Makes minor hue changes, but relative colors
  preserved. Achromatic remain achromatic.

22
Projective Clipping

• After all, some colors remain outside printer
  gamut
• Project these onto the gamut surface
• Try a perpendicular projection to nearest
  triangular face in printer gamut surface.
• If none, find a perpendicular projection to the
  nearest edge on the surface
• If none, use closest vertex

23
Projective Clipping

• This is the closest point on the surface to the
  given color
• Result is continuous projection if gamut is
  convex, but not else.
• Bad want nearby image colors to be nearby in
  destination gamut.

24
Projective Clipping

• Problems
• Printer gamuts have worst concavities near black
  point, giving quantization errors.
• Nearest point projection uses Euclidean distance
  in XYZ space, but that is not perceptually
  uniform.
• Try CIELAB? SCIELAB?
• Keep out of gamut distances small at cost of use
  of less than full printer gamut use.