A review of some pixel image interpolation algorithms

A review of some pixel image interpolation algorithms

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Contents:
A Review of Some Image Pixel Interpolation Algorithms
Don Lancaster Synergetics, Box 809, Thatcher, AZ 85552 copyright c2007 as GuruGram #32 http://www.tinaja.com don@tinaja.com (928) 428-4073

t often may be necessary to resample a bitmap pixel image. Perhaps because it is to be resized, rotated, or have its perspective corrected, or get intentionally distorted, or have its image shape rectified. It turns out there are several newer approaches to this problem that differ markedly in image quality, in processing speed, complexity, language programmability, and ease of understanding. What I thought I would do here is review some pixel interpolation methods that include Nearest Neighbor, Bilineal Interpolation, Bilineal With Lookup, a 3x3 Compromise, and Bicubic Interpolation. I was unable to find any derivation of Bicubic Interpolation that I felt was reasonably complete and understandable, so we will expand upon it in detail here.

I

The Interpolation Problem
Typically, you will have a sampled data system representing your image. With a two dimensional array of samples usually linearly spaced in the x (horizontal) and y (vertical) directions. For one reason or another, you will require a group of new sample points intermediate to your input pixels. Typically, one new pixel will have four nearest neighbors on a rectangular grid...

p(0,1) new p(x,y) p(0,0)

p(1,1)

p(1,0)

-- 83 . 1 --

It is often simplest to assume a unit square between the four nearest pixels. Our new and sought after pixel will then be some x fraction and y fraction into this unit square. Your interpolation problem then consists of finding suitable values for these fractions. Your sought after new pixel position should be the result of some pixel - by - pixel calculation seeking to change the size or distortion of the image. Both the new x and y results will have an integer portion that decides which initial group of four pixels to use. Plus a residue or fractional remainder portion that positions you within the selected four sample square. Your task is to then find a credible intensity value for that new location. Typically, you will have to repeat your interpolation at least three times, once for each of the red, green, and blue pixel bitmap planes. Thus, algorithm speed can easily become a key limiting issue. Especially on megapixel images. A crucial rule...
NO NEW DATA will be added by ANY interpolation scheme! The best you can do is minimize interpolation artifacts.

Before sampling, your image function can be assumed to be some continuous surface. To make the surface smooth, we do require a continuous function and continuous derivatives at each sample point. The surface can be assumed to be bandwidth limited, and the existing samples can be assumed to be dense enough.

Nearest Neighbor
With the Nearest Neighbor scheme, you just grab the nearest pixel and use it. One simple way to do this is to round your x value and add it to a rounded and doubled y value. This will give you four integers 0, 1, 2, and 3 that can use table lookup or case commands to read one of the four corner pixels. We might start with this sample data set array and use it to compare the different interpolation methods...
[ [0.60 0.60 0.48 0.24 0.60 0.60] [0.60 0.60 0.48 0.24 0.60 0.60] [0.00 0.00 0.36 0.12 0.48 0.48] [0.24 0.24 0.48 0.60 0.12 0.12] [0.12 0.12 0.24 0.48 0.36 0.36] [0.12 0.12 0.24 0.48 0.36 0.36] ]

-- 83 . 2 --

Although this is a 6x6 data grid, we will only concern ourselves with the middle 4x4 array representing 3 x 3 = 9 unit squares. The reasons for this will become apparent when we get to the Bicubic Interpolation method. Here is some PostScript code...
dup floor cvi dup /yi exch store sub /yr exch store dup floor cvi dup /xi exch store sub /xr exch store xr round yr round 2 mul add cvi [ {data yi get xi get} {data yi get xi 1 add get} {data yi 1 add get xi get}{data yi 1 add get xi 1 add get} ] exch get exec

Our first lines convert the input position into box and position - in - box values. The middle line converts the position - in - box to an integer 0 to 3. The final lines select the lower left, lower right, upper left, or upper right data point value. Here is what one of your bitmap planes will look like in false color...

As you can see, your results will mostly be in error and the artifacts may get downright ugly. Please note again that this is a false color plot of a single bitmap plane. In this case red = 0 on up through blue = 0.6
-- 83 . 3 --

Nearest Neighbor is by far the fastest and by far the ugliest interpolation scheme. It can introduce major artifacts into your modified bitmap. As a relative speed indicator, on one unoptimized and older PostScript implementation, the time per pixel is just over 10 microseconds. We can use this figure later for a relative baseline comparison of the other interpolation speeds.

Bilineal Interpolation
With Bilineal Interpolation, you proportion your result to the relative distance between your four original points. You could first do an x interpolation and then a y. Or vice versa. Curiously, either way should lead you to this approximate but highly useful final formula...
f(x,y) = f(0,0)(1-x)(1-y) + f(1,0)(x)(1-y) + f(0,1)(1-x)(y) + f(1,1)(x)(y)

This measures the distance to each of the old corner points and provides a weighted average. The results are far better looking than nearest neighbor...

We see that there a very few artifacts remaining, but that bilineal interpolation should be useful for all but the most critical of applications.
-- 83 . 4 --

Here is some bilineal interpolation code done in PostScript...
dup floor cvi dup /yi exch store sub /yr exch store dup floor cvi dup /xi exch store sub /xr exch store data data data data yi yi yi yi get xi get 1 xr sub 1 yr sub mul mul get xi 1 add get xr 1 yr sub mul mul add 1 add get xi get 1 xr sub yr mul mul add 1 add get xi 1 add get xr mul yr mul add

We once again split our input position into a pair of integer positions and two fractional residue values. Each original data point is then converted into a weighted average using the calculations shown. As a relative speed indicator, again on one unoptimized and older PostScript implementation, the execution time per pixel is just over 14 microseconds. Thus, there is only a forty percent penalty for vastly superior results. The surprisingly low speed difference is apparently caused by PostScript being quite fast and adept at on-stack adds and multiplies, but somewhat slower on its proc lookups and executions.

Bilineal via Table Lookup
The speed of calculating any pixel interpolation is highly dependent both on the system speed and the language used. In general, interpreted languages will be slowest and compiled languages faster. Hand crafted machine language will be faster still. Fastest of all, of course, would be dedicated hardware in the form of a FPGA or whatever. It is often faster to look up a value in a table than to calculate it. Which suggests that some implementations of Bilineal Interpolation may be significantly improved by replacing repeated multiply and adds with a quantized table lookup of some predetermined constants. However, this is apparently not the case with PostScript and table lookup interpolation may not be beneficial enough with your language choice. Your results may vary.

The 3x3 Compromise
If extreme speeds are essential, the limiting minimum case of a 3x3 bilineal table lookup might be useful. The results would end up pretty bad, but should still be significantly better than nearest neighbor. Only one lookup is needed 4/9ths of the time, a two lookup average 4/9ths of a time, and the full four lookup average a mere 1/9th of the time.
-- 83 . 5 --

Bicubic Interpolation
This is the Godzilla of pixel interpolation algorithms. It gives absolutely superb results with negligible artifacts. But is very hard to understand and requires an extreme number of complex calculations. Bicubic Interpolation attempts to reconstruct the exact surface between your four initial pixels. It does this by extracting sixteen pieces of information. Based on the values of the samples, the x slopes of those values, the y slopes of those values, and the xy slope cross products of those values. It turns out that any point on a two dimensional unity normalized surface can be represented by a set of sixteen cubic polynomial equations. The key bicubic equations are...
p(x,y) = a00*x^0^y^0 + a01*x^0^y^1 + a02*x^0^y^2 + a03*x^0^y^3 + a10*x^1^y^0 + a11*x^1^y^1 + a12*x^1^y^2 + a13*x^1^y^3 + a20*x^2^y^0 + a21*x^2^y^1 + a22*x^3^y^2 + a23*x^2^y^3 + a30*x^3^y^0 + a31*x^3^y^1 + a32*x^3^y^2 + a33*x^3^y^3

This expression can be simplified somewhat by noting that x^0 = y^0 = 1 and x^1 = x and y^1= y. Our non-trivial problem is to find the sixteen constant coefficients a00 through a33 for an initial four unit square data points and their nearest eight neighbors. We can start by substituting some variables to simplify the notation. So, let w0 = f(0,0) or the pixel value at our lower left point. Let w1 = (f0,1) or the pixel value at our lower right point. Let w2 = f(0,0) or the pixel value at our upper left point. And let w3 = (f1,1) or the pixel value at our upper right point. Similarly, we will let x0 through x3 represent the xslope (or partial derivative with respect to x) at each point. Normally, we do not know the exact x slope value. So, we can approximate it by taking the average of the changes between your previous and next data point. We can also let y0 through y3 represent the yslope (or partial derivative with respect to y) at each point. Finally, to handle the behavior near the middle of our four data point set, we can let z0 through z3 represent the slope product or (partial derivative cross product) at each point.
-- 83 . 6 --

We actually end up using 16 data points for calculation, with the "extras" being needed to find our x and y slopes. This is not a problem interior to your pixel map. But you will have to do something at the edges to either ignore or fake the slopes. Which is why our sample data has "extra" values around its outside edge. The results can be quite impressive...

We see an artifact free surface that we can pick any point off of. The surface is continuous and has continuous slopes at every point. Bicubic calculations are often done using matrix techniques. Since these can be hard to understand, we will instead use ordinary algebra here. Helped along with a partial differential equation or two. All you will need to know about partial differentials here is that (A) A partial differential is the slope in one variable direction, and (B) The differential ( or its limit derivative ) of f(x) = b0 + b1x + b2x^2 + b3x^3 is its slope of b1 + 2b2x + 3b3x^2. Finding a new bicubic interpolated pixel value inside a unit square starts off by finding w0 through z3. These are already known from our available pixel data, or can be easily calculated from it. You then use these w0 through z3 to evaluate expressions for the various ax coefficients. Sadly, we now have w0 through z3 as a function of a00 through a33. We instead need a00 through a33 as a function of w0 through z3. So, we have to solve 16 linear equations in 16 unknowns to isolate the axx values.
-- 83 . 7 --

Finally, from our newly calculated a00 through a33 values, we can find the value of our new pixel using our bicubic formula shown above. Here is how w0 through z3 are related to the a00 through a33 coefficients...
w0 w1 w2 w3 = = = = f(0,0) f(1,0) f(0,1) f(1,1) = = = = a00 a00 + a10 + a20 + a30 a00 + a01 + a02 + a03 a00 + a10 + a20 + a30 + a01 + a11 + a21 + a31 + a02 + a12 + a22 + a32 + a03 + a13 + a23 + a33 a10 a10 + 2a20 + 3a30 a10 + a11 + a12 + a13 1*(a10 + a11 + a12 + a13) + 2*(a20 + a21 + a22 + a23) + 3*(a30 + a31 + a32 + a33) a01 a01 + a11 + a21 + a31 a01 +2Aa02 + 3a03 1*(a01+a11+a21+a31) + 2*(a02+a12+a22+a32) + 3*(a03+a13+a23+a33)

x0 x1 x2 x3

= = = =

fx(0,0) fx(1,0) fx(0,1) fx(1,1)

= = = =

y0 y1 y2 y3

= = = =

fy(0,0) fy(1,0) fy(0,1) fy(1,1)

= = = =

z0 z1 z2 z3

= = = =

fxy(0,0) fxy(1,0) fxy(0,1) fxy(1,1)

= a11 = a11 + 2a21 + 3a31 = a11 + 2a12 + 3a13 = 1*a11 + 2*a12 + 3*a13 + 2*a21 + 4*a22 + 6*a23 + 3*a31 + 6*a32 + 9*a33

In general, these equations are found by substituting 1 and 0 values for x and y and their slope derivatives. Let's look at four typical examples...
� w2: Finds value at x=0, y=1. x=0 drops out all

equations with powers of x=1, x=2, and x=3. Leaving a01*y + a02*y + a03*y. Which at y=1 becomes w2 = a01 + a02 + a03.
� x1: Finds xslope at x=1, y=0. y=0 drops out all

equations with powers of y=1, y=2, and y=3. Leaving a10*x + a20*x^2 + a30*x^3. Whose partial derivative is a10 + 2a20*x + 3a30*x^2.
-- 83 . 8 --

� y3: All equations are initially active. x=1

everywhere. y slopes will be 0 + a01y + ... + a02y + ... + 3a03y^2 + ... Which at y=1 will become the result shown.
� z3: Cross products of the slopes

will be a11(1x*1y) + a12(1x*2y) + a13(1x*3y^2) + a21(2x^2*y) + a22(2x^2*2y^2) + ... Which at x=1 and y=1 becomes 1a11 + 2a12 + 3a13 + 2a21 + 4a22 + ...

Finding the Coefficients
We do now have w0 through z3 as functions of a00 through a33. To solve for individual axx values, we have to solve 16 linear equations in 16 unknowns. This can be done manually, or by inserting the values into any of several math equation programs. The results should be...
a00 a01 a02 a03 a10 a11 a12 a13 a20 a21 a22 = = = = = = = = = = = w0, y0, -3w0 + 3w2 -2y0 - y2 2w0 - 2w2 + y0 + y2 x0 z0 -3x0 + 3x2 - 2z0 - z2 2x0 - 2x2 + z0 + z2 -3w0 + 3w1 - 2x0 - x1 -3y0 + 3y1 - 2z0 - z1 9w0 - 9w1 - 9w2 + 9w3 + 6x0 + 3x1 + -6x2 - 3x3 + 6y0 - 6y1 + 3y2 - 3y3 + 4z0 + 2z1 + 2z2 + z3 = -6w0 + 6w1 + 6w2 - 6w3 -4x0 - 2x1 + 4x2 + 2x3 -3y0 + 3y1 - 3y2 + 3y3 + -2z0 - z1 - 2z2 - z3 = 2w0 - 2w1 + x0 + x1 = 2y0 - 2y1 + z0 + z1 = -6w0 + 6w1 + 6w2 -6 w3 -3x0 - 3x1 + 3x2 + 3x3 -4y0 + 4y1 - 2y2 + 2y3 + -2z0 - 2z1 - z2 - z3 = 4w0 - 4w1 - 4w2 + 4w3 + 2x0 + 2x1 + -2x2 - 2x3 + 2y0 - 2y1 + 2y2 - 2y3 + z0 + z1 + z2 + z3

a23

a30 a31 a32

a33

-- 83 . 9 --

It is tedious and highly error prone to try and do this by hand. There are some obvious simplifications. Four values are available by inspection. Eight of these can be solved with pairs of equations. Leaving four equations in four unknowns. Ferinstance, if you want to do a manual solution, you can grab a00, a01, a10, and a11 by inspection. w1 and x1 can then be simultaneously solved for a20 and a30. Then x2 and z2 can be simultaneously solved for a12 and a13. And w2 and y2 can be simultaneously solved for a02 and a03. And y1 and z1 can in turn get simultaneously solved for a21 and a31. If you save them for last, the remaining variables can end up as four equations in four unknowns. With all previous calculated values being reducible down into four constants, one for each equation. Two of these equations can then relate a22 and a33, while another two can relate a32 and a33. And substituted back to solve for a33. The final three variables follow by simple arithmetic.

A Summary
An important warning: Bicubic calculations might rarely end up slightly above unity or slightly below zero. The rest of your code must be able to clip these values or otherwise deal with them. Again on one unoptimized and older PostScript implementation, the execution time per pixel for a bicubic interpolation was just over 100 microseconds. Thus bicubic speed in this instance is seven times worse than bilineal and is ten times worse than nearest neighbor. I suspect that when further optimized on a newer Distiller and a faster machine, the results will end up something like three seconds per megabyte per color plane for bilineal and twenty seconds per megabyte per color plane for bicubic. Thus, bicubic can most certainly be used for typical PostScript bitmap image manipulations but will likely forever remain somewhat on the slow side. Your results will vary with your language and implementation choices. Again, compiled languages will usually beat out interpreted ones. Hand crafted machine code will often be much faster still, and custom hardware may even be faster. Reducing bilineal to somewhat coarser table lookups may or may not prove time effective or worthwhile on some implementations. In some cases, doubling your sample rate can significantly reduce interpolation problems. But only with 4X speed and 4x storage penalties that may not end up being cost effective. I originally thought a 2X sampled 3x3 compromise interpolation solution would be useful for such things as eBay images . But it appears that a genuine bicubic interpolation can end up nearly speed competitive. With far better results.
-- 83 . 10 --

Some Improvements
Finding "real" derivative slopes at each data point could possibly further improve bicubic. This would tend to sharpen the transitions. But would take some sort of more content specific and wider area processing. Once any method and language is selected, second or third passes can be made to tweak and improve your code's operating speed. Some general techniques for PostScript speedups appear here. This slightly obtuse rework of our bilinear code is 34 percent faster...
dup cvi dup /yi exch store sub /yr exch store dup cvi dup /xi exch store sub /xr exch store data yi get dup xi get 1 xr sub mul exch xi 1 add get xr mul add 1 yr sub mul data yi 1 add get dup xi get 1 xr sub mul exch xi 1 add get xr mul add yr mul add

For More Help
Complete bicubic PostScript code detailed examples can be extracted from the sourcecode for this GuruGram. Working PostScript code for bilineal and bicubic has also been extracted to this utility and this demo. Much of our analysis and review done here was based on the two Wikipedia entries found here and here. Additional insight into single dimensional cubic spline image interpolation appears here. Similar tutorials and additional support materials are found on our Cubic Spline, our PostScript and our GurGram library pages. As always, Custom Consulting is available on a cash and carry or contract basis. As are seminars. For details, you can email don@tinaja.com. Or call (928) 428-4073.

-- 83 . 11 --

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