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Created: 17 April 2003
extendedopacity − theory of netpbm interpolation and extrapolation
This page is a copy of http://www.sgi.com/grafica/interp/ on April 17, 2003, with some slight formatting changes, included in the Netpbm documentation for convenience.
Paul Haeberli and Douglas Voorhies
Introduction
Interpolation and extrapolation between two images offers a
general, unifying approach to many common point and area
image processing operations. Brightness, contrast,
saturation, tint, and sharpness can all be controlled with
one formula, separately or simultaneously. In several cases,
there are also performance benefits.
Linear interpolation is often used to blend two images. Blend fractions (alpha) and (1 - alpha) are used in a weighted average of each component of each pixel:
out = (1 - alpha)*in0 + alpha*in1
Typically alpha is a number in the range 0.0 to 1.0. This is commonly used to linearly interpolate two images. What is less often considered is that alpha may range beyond the interval 0.0 to 1.0. Values above one subtract a portion of in0 while scaling in1. Values below 0.0 have the opposite effect.
Extrapolation is particularly useful if a degenerate version of the image is used as the image to get "away from." Extrapolating away from a black-and-white image increases saturation. Extrapolating away from a blurred image increases sharpness. The interpolation/extrapolation formula offers one-parameter control, making display of a series of images, each differing in brightness, contrast, sharpness, color, or saturation, particularly easy to compute, and inviting hardware acceleration.
In the following examples, a single alpha value is used per image. However other processing is possible, for example where alpha is a function of X and Y, or where a brush footprint controls alpha near the cursor.
Changing
Brightness
To control image brightness, we use pure black as the
degenerate (zero alpha) image. Interpolation darkens the
image, and extrapolation brightens it. In both cases,
brighter pixels are affected more.
brightness
Changing
Contrast
Contrast can be controlled using a constant gray image with
the average image luminance. Interpolation reduces contrast
and extrapolation boosts it. Negative alpha generates
inverted images with varying contrast. In all cases, the
average image luminance is constant.
contrast
If middle gray or the average pixel color is used instead, contrast is again altered, but with middle gray or the average color left unaffected. Shades and colors far away from the chosen value are most affected.
Changing
Saturation
To alter saturation, pixel components must move towards or
away from the pixel’s luminance value. By using a
black-and-white image as the degenerate version, saturation
can be decreased using interpolation, and increased using
extrapolation. This avoids computationally more expensive
conversions to and from HSV space. Repeated update in an
interactive application is especially fast, since the
luminance of each pixel need not be recomputed. Negative
alpha preserves luminance but inverts the hue of the input
image.
saturation
Sharpening
an Image
Any convolution, such as sharpening or blurring, can be
adjusted by this approach. If a blurred image is used as the
degenerate image, interpolation attenuates high frequencies
to varying degrees, and extrapolation boosts them,
sharpening the image by unsharp masking. Varying alpha acts
as a kernel scale factor, so a series of convolutions
differing only in scale can be done easily, independent of
the size of the kernel. Since blurring, unlike sharpening,
is often a separable operation, sharpening by extrapolation
may be far more efficient for large kernels.
sharpening
Note that global contrast control, local contrast control, and sharpening form a continuum. Global contrast pushes pixel components towards or away from the average image luminance. Local contrast is similar, but uses local area luminance. Unsharp masking is the extreme case, using only the color of nearby pixels.
Combined
Processing
An unusual property of this interpolation/extrapolation
approach is that all of these image parameters may be
altered simultaneously. Here sharpness, tint, and saturation
are all altered.
combined
Conclusion
Image applications frequently need to produce multiple
degrees of manipulation interactively. Image applications
frequently need to interactively manipulate an image by
continuously changing a single parameter. The best hardware
mechanisms employ a single "inner loop" to achieve
a wide variety of effects. Interpolation and extrapolation
of images can be a unifying approach, providing a single
function that supports many common image processing
operations.
Since a degenerate image is sometimes easier to calculate, extrapolation may offer a more efficient method to achieve effects such as sharpening or saturation. Blending is a linear operation, and so it must be performed in linear, not gamma-warped space. Component range must also be monitored, since clamping, especially of the degenerate image, causes inaccuracy.
These image manipulation techniques can be used in paint programs to easily implement brushes that saturate, sharpen, lighten, darken, or modify contrast and color. The only major change needed is to support alpha values outside the range 0.0 to 1.0.
It is surprising and unfortunate how many graphics software packages needlessly limit interpolant values to the range 0.0 to 1.0. Application developers should allow users to extrapolate parameters when practical.
References
For a slightly extended version of this article, see: P.
Haeberli and D. Voorhies. Image Processing by Linear
Interpolation and Extrapolation. IRIS Universe Magazine
No. 28, Silicon Graphics, Aug, 1994.
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