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//web.archive.org/web/20060827184031/http://www.path.unimelb.edu.au/~dersch/interpolator/interpolator.html[10.02.2010 14:20:02]

Testing Interpolator QualityHelmut Dersch

([email protected])

Technical University Furtwangen

Revised June 30, 1999

Manipulation of digital images often requires resampling of the pixel data. Usually, the pixel positions

he transformed image do not coincide with grid positions. Therefore, an interpolation algorithm has to

e applied to somehow derive the pixel data for these in-between positions. The effort spent for this

nterpolation is crucial for the image quality of the result image.

Most common in computer graphics applications is bilinear interpolation. As the name suggests, off-g

oints are linearly interpolated between their four neighboring grid-pixels. Some professional package

ke Photoshop offer cubic interpolation using higher order polynomials and more data points. The mo

ata points are used, and the more sophisticated the interpolation function is, the better is the result an

he slower is the program.

The following is a test performed on a variety of Graphics programs which were available to me. This

est is biased since I used it explicitely to test my own program (the Photoshop/Gimp plug-in Panoram

Tools ) against other solutions.

everal authors have judged interpolator quality by testing enlargements. However, there are quite a fe

roblems associated with that, the most important being that there is no mathematically unique solutio

or digital image enlargements, and, hence, no really objective test. A more practical test is a rotation

n image by some arbitrary angle, which can be easily tested against the original image. The more such

otations the image survives intact, the better is the algorithm. This test is also readily available, sincelmost any Graphics-program has a rotate function.

Two images were used: One real-world photograph shown to the left below, and one line drawing sho

o the right. The images are shown in original size; they are 200x200pixels each.

Both images were rotated in 5 degree steps until a total of 180 degrees was reached. This amounts to

esampling steps. Then the image is flipped back, which is a lossless transformation. To judge and

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ompare image quality, one part of each image is enlarged three times (photo) or six times (line drawi

without any filtering and interpolation, so each source pixel appears as 3x3 (6x6) block in the result

mage. For this publication, the photos were compressed using high-quality JPEG, and the line-drawin

sing GIF-compression. The source images (without being rotated) then look like this:

Ken Turkowski has suggested another test image which gives a more quantitative

estimate of the image sharpness. It is shown below. In this case it was determined how many

of the concentric rings survived the 36 rotations. The count of clearly visible white rings is

indidcated for each image.

Results:

Photoshop Bicubic Interpolation:

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Number of visible bright rings : 8,5

Photoshop Bilinear Interpolation:




Number of visible bright rings : 2

Color It (Macintosh):





GraphicConverter with Error Correction (Macintosh):




Picture Publisher (Windows 95)




Panorama Tools: Cubic Interpolator 'Poly 3'

Number of visible bright rings : 8,5

Panorama Tools: Spline 16 Pixels





Panorama Tools: Spline 36 Pixels





Panorama Tools: Sinc 256 Pixels





The Interpolators in Panorama Tools are available from version 1.8b1. Earlier versions had the two

pline Interpolators, but the other two were different.

Some Comments:

The worst image quality is obtained with nearest neighbor sampling which obviously is implemented i

icture Publisher. Bilinear interpolation leads to significant losses, but is usually acceptable if only onr two sampling steps are involved. The bicubic schemes vary depending on the number of input pixel

Usually 16 pixels are used. Depending on the choosen polynomial, either a very sharp image with som

rtefacts, or a somewhat softer, but more homogeneous image is obtained. The Photoshop polynomial

as a distinct sharpening effect, as can be seen in the brightened edges of dark features. This leads to

rtefacts most pronounced in the structures of the line drawing. The interpolator 'poly 3' of Panorama

Tools has been tuned to closely resemble that of Photoshop.It sharpens slightly more. 'Spline16' is

qually fast, but much smoother, as can be seen in the line drawing. It is also somewhat softer, as judg

rom the rating using Ken's image. Spline36 is identical to Spline16, but uses 36 pixels instead of 16.




This results in better resolution (see the 1pixel wide line in the line drwaing) without significant

rtefacts. The best results are obtained using the sinc-function. This not only is more involved than the

ubic polynomials, but in the present case also requires 256 pixels to interpolate for each input pixel.

inc-function is the theoretical optimum, and can only be improved by using even more pixels.

The Math:

n calculating the pixel value of a particular position in the result image, the interpolator combines

everal pixels of the source image. The closer these pixels are to this position, the more weight the pix

arries. However, this is not just a simple falling function of pixel distance x, but there are certain

ptimum weights.

Linear InterpolationThe weight of the source pixel is calculated by

weight = 1-x

=0 then corresponds to matching source and result pixels. Two pixels are used for 1-dimensionalnterpolation, 4 for 2-dimensional calculations.

Cubic Interpolation

Poly3 of Panorama Tools is a single parameter cubic polynomial with weights

weight = (( A + 2.0 )*x - ( A + 3.0 ))*x*x +1.0; 0<x<1

weight = (( A * x - 5.0 * A ) * x + 8.0 * A ) * x - 4.0 * A; 1<x<2

t uses 4 pixels in one, and 16 pixels in 2 dimensions. The parameter 'A' determines the behavious of t

unction. It is set to -0.75, which makes it very similar to Photoshop's bicubic interpolator.

pline16 of Panorama Tools is a cubic-spline, with derivative set to 0 at the edges (4x4 pixels). The

weight function thus becomes:

weight = ( ( x - 9.0/5.0 ) * x - 1.0/5.0 ) * x + 1.0; 0<x<1

weight = ( ( -1.0/3.0 * (x-1) + 4.0/5.0 ) * (x-1) - 7.0/15.0 ) * (x-1) ; 1<x<2

t also uses 16 pixels in 2 dimensions.

pline36 is like Spline16, except that it uses 6x6=36 pixels. The weight function becomes

weight = ( ( 13.0/11.0 * x - 453.0/ 209.0 ) * x - 3.0/ 209.0 ) * x + 1.0; 0<x<1

weight = ( ( - 6.0/11.0 * (x-1) + 270.0/ 209.0 ) * (x-1) - 156.0/ 209.0 ) *(x-1); 1<x<2

weight = ( ( 1.0/11.0 * (x-2) - 45.0/ 209.0 ) * (x-2) + 26.0/ 209.0 ) *(x-2); 2<x<3

Sinc Interpolation

inc256 of Panorama Tools is a sinc() function windowed to 8 pixels (Lanczos-function). The weights

re calculated

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