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# bicubic interpolation on image enlargements

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16 replies to this topic

### #1Basiror  Members

Posted 03 August 2005 - 09:36 PM

Hi I am trying to implement a few image resizing algorithms into my library I know how bicubic reducing works you calculate the destination pixel from the bicubic weights of a 4x4 pixel field around the nearest pixel coordinate in the source image [a href="http://astronomy.swin.edu.au/~pbourke/colour/bicubic/"]becubic interpolation[/a] my problem is how can i enlarge the image e.g.: having a 32*32 image and scale it to 256*256 in order to enlarge i would calculate the 4x4 pixel values out of the 32*32 image and write them to the 256*256 image but in order to do this i hade to interatively rescale 32*32->64*64->128*128->256*256 or is there a simpler solution e.g.: when i downscale i use a 4x4 field so 64*64->32*32 using 4x4 and 256x256->32*32 using a 64x64 field? so i had to distribute one pixel of the 32*32 image to a 64x64 field in the 256*256 image? any idea on how to do this correctly?

### #2dragongame  Members

Posted 03 August 2005 - 10:01 PM

Hi,

you "just" oversample the pickture.

so to go from 32*32 to 256*128 (this is to show you it could be done this way)

dx = 32 / 256;dy = 32 / 128;for(int y=0; y < 128; y++){  for(int x=0; x < 256; x++)  {     float ix = x*dx;     float iy = y*dy;     float s = ix - (float)((int)(ix+.5));     float t = iy - (float)((int)(iy+.5));     int ox = (int)ix;     int oy = (int)oy;     new_pic[y*256+x] = orig_pic[oy*32+ox]*(1-s)*(1-t)                      + orig_pic[(oy+1)*32+ox]*(1-s)*t                      + orig_pic[oy*32+ox+1]*s*(1-t)                      + orig_pic[(oy+1)*32+ox+1]*s*t;     }}

### #3Basiror  Members

Posted 03 August 2005 - 10:09 PM

isn t this the bilinear approach?

http://www.inq7.net/inf/2003/jun/10/inf_37-2.htm

there they mention SI(staired interpolation which is basically what I mentioned above with 32*32->64*64->128*128->256*256

so maybe I should implement it like that

thx

### #4dragongame  Members

Posted 03 August 2005 - 10:16 PM

Sorry, my error!

### #5timw  Members

Posted 03 August 2005 - 10:19 PM

just think of it as a texture mapping problem, use bilinear interpolation to smooth it out, if this is giving to blocky results you can smooth further by bluring the input image b4 you do it.

### #6Basiror  Members

Posted 03 August 2005 - 10:22 PM

the problem with bluring is that I want to use it for a heightfield generator
and too blurry heightmaps don t look that good

i just try the SI approach now maybe it delivers acceptable results

### #7Basiror  Members

Posted 04 August 2005 - 11:32 PM

I am still having problems to get this implementation of cubic interpolation to run

http://astronomy.swin.edu.au/~pbourke/colour/bicubic/

What I am doing:
1. I calculate the factor sourcesize/destinationsize (images are powers of 2)
2. I loop through the destination coordinates and calculate the corresponding source coordinates "i*fact" & "j*fact"
3.then I find the nearest neightbour pixel and calculate the deltas dx & dy

4. I map the is and js to the source coordinates
5. I loop through the 4x4 field of neighbouring pixes in the source image and try to add them to the destination pixel
the cubicweight function should return the blend factors for each neighbouring pixel but all it does is returning 0s i have already tries to use
"dy - n" and "n - dy" nothing changed though

any idea what's wrong here?
In the time gamedev.net has been down I was able to get it to work for downsampling and in comparsion to photoshop the results are at least 10-20x sharper :)

Now all I can t get to work is up sampling, can I use this method for upsampling at all?
or do you just take the source pixels calculate the destination coordinate and write to the current source pixel (i,j) to the destinationpixel (id,jd) +=(i,j)*bicubicweight(parameters...)?

[source lang ="cpp"]
//sample image up( powers of 2)
template<class U, class W> void samplebicubic(const U& s, W& d)
{
uint32	i,j;
int32	is,js,m,n;
float x,y,fact,dx,dy;
float test;

fact = U::MAP_SIZE/ W::MAP_SIZE;

for(i=0;i<W::MAP_SIZE;i++)
{
//calculate source coordinates
x = i * fact;
is = static_cast<int32>(x);
dx = x - is;
is = (is + W::MAP_SIZE)%W::MAP_SIZE;

for(j=0;j<W::MAP_SIZE;j++)
{
//calculate source coordinates
y = j * fact;
js = static_cast<int32>(y);
dy = y - js;
js = (js + W::MAP_SIZE)%W::MAP_SIZE;

d.m_Map[i][j] = 0.0f;
for(m=-1;m<3;m++)
{
for(n=-1;n<3;n++)
{
test = cubicweight(m-dx) * cubicweight(n-dy);
d.m_Map[i][j] += s.m_Map[(is + m + U::MAP_SIZE)%U::MAP_SIZE][(js + n + U::MAP_SIZE)%U::MAP_SIZE]
*test;
}
}
}
}
};



cubic weight function from the article


//cubic weight function
float	cubicweight(float x)
{
float result;
float tmp;

//P(x+2)^3
tmp = (x+2)>0?x+2:0;
result = tmp*tmp*tmp;

//-4*P(x+1)^3
tmp = (x+1)>0?x+1:0;
result += -4*tmp*tmp*tmp;

//6*P(x)^3
tmp = x>0?x:0;
result += 6*tmp*tmp*tmp;

//-4P(x-1)^3
tmp = (x-1)>0?x-1:0;
result += -4*tmp*tmp*tmp;

result*=0.1666666f;
return result;
};



[Edited by - Basiror on August 5, 2005 6:32:38 AM]

### #8outRider  Members

Posted 05 August 2005 - 04:59 PM

If you know how to linearly interpolate from 32,32 to 256,256 then you can do it using cubic, cosine, any other method that uses t from 0 to 1 as a parameter.

For linear interpolation you have

lerp(x,y,t)

to return a value between x and y based on t.

For cubic you have

cubic(a,b,c,d,t)

to return a value between b and c based on t.

### #9Nypyren  Members

Posted 05 August 2005 - 05:54 PM

If you're using it for heightfield resizing, and you want "more detail" and don't care whether it's perfect or not, you could try introducing fractal noise to generate some bumpy looking terrain in the expansion gaps.

You can probably look up "Fractal random heightfield generator" in google and find the algorithm that produces random mountains without any source bitmap... and just modify it so that it works to fill in expansion gaps instead.

### #10Basiror  Members

Posted 06 August 2005 - 08:58 AM

I am still having trouble with the bicubic interpolation

i have 2 functions now
one with cubic interpolation and one with b spline cubic interpolation
the first function delivers results comparable with bilinear interpolation
and the second function which uses b splines delivers smoother results than photoshops implementation

here the cubic function:
i get the coordinates in the source image

then i calculate the weights of 4 pixels in a column

and once this is done i calculate the weighted value of the 4 weights
illustrated here

http://www.olympusmicro.com/primer/java/digitalimaging/processing/geometricaltransformation/
any idea what i am doing wrong here?

do i need another interpolation factor or whats wrong here?


//sample image up( powers of 2)
template<class U, class W> void sample_bicubic2(const U& s, W& d)
{
uint32	i,j;
int32	is,js;
float x,y,fact,dx,dy;
int32	XCoords[4];
int32	YCoords[4];
float	weights[4];

fact = static_cast<float>(U::MAP_SIZE)/ static_cast<float>(W::MAP_SIZE);

for(i=0;i<W::MAP_SIZE;i++)
{
//calculate source coordinates
x = i * fact;
is = static_cast<int32>(x);
dx = x - is;

XCoords[0] = is-1;
XCoords[1] = is;
XCoords[2] = is+1;
XCoords[3] = is+2;

for(j=0;j<W::MAP_SIZE;j++)
{
//calculate source coordinates
y = j * fact;
js = static_cast<int32>(y);
dy = y - js;

YCoords[0] = js-1;
YCoords[1] = js;
YCoords[2] = js+1;
YCoords[3] = js+2;

weights[0] = cubicweight2(
s.m_Map[(XCoords[0] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[0] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[0] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[1] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[0] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[2] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[0] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[3] + U::MAP_SIZE)%U::MAP_SIZE],
0.5f);

weights[1] = cubicweight2(
s.m_Map[(XCoords[1] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[0] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[1] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[1] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[1] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[2] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[1] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[3] + U::MAP_SIZE)%U::MAP_SIZE],
0.5f);

weights[2] = cubicweight2(
s.m_Map[(XCoords[2] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[0] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[2] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[1] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[2] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[2] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[2] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[3] + U::MAP_SIZE)%U::MAP_SIZE],
0.5f);
weights[3] = cubicweight2(
s.m_Map[(XCoords[3] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[0] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[3] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[1] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[3] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[2] + U::MAP_SIZE)%U::MAP_SIZE],
s.m_Map[(XCoords[3] + U::MAP_SIZE)%U::MAP_SIZE][(YCoords[3] + U::MAP_SIZE)%U::MAP_SIZE],
0.5f);

d.m_Map[i][j] = cubicweight2(weights[0],weights[1],weights[2],weights[3],0.5f);
}
}
};


and here with b splines


//sample image up( powers of 2)
template<class U, class W> void sample_bicubic(const U& s, W& d)
{
uint32	i,j;
int32	is,js,m,n;
float x,y,fact,dx,dy;
float test;

fact = static_cast<float>(U::MAP_SIZE)/ static_cast<float>(W::MAP_SIZE);

for(i=0;i<W::MAP_SIZE;i++)
{
//calculate source coordinates
x = i * fact;
is = static_cast<int32>(x);
dx = x - is;
//is = (is + W::MAP_SIZE)%W::MAP_SIZE;

for(j=0;j<W::MAP_SIZE;j++)
{
//calculate source coordinates
y = j * fact;
js = static_cast<int32>(y);
dy = y - js;
//js = (js + W::MAP_SIZE)%W::MAP_SIZE;

d.m_Map[i][j] = 0.0f;
for(m=-1;m<3;m++)
{
test = cubicweight(dx-m);
for(n=-1;n<3;n++)
{

d.m_Map[i][j] += s.m_Map[(is + m + U::MAP_SIZE)%U::MAP_SIZE][(js + n + U::MAP_SIZE)%U::MAP_SIZE]
*test* cubicweight(dy-n);;
//std::cout<<std::endl;
}

}
}
}
};
//cubic weight function
float	cubicweight(float x)
{
float result;
float tmp;

//P(x+2)^3
tmp = (x+2)>0?x+2:0;
result = tmp*tmp*tmp;

//-4*P(x+1)^3
tmp = (x+1)>0?x+1:0;
result += -4*tmp*tmp*tmp;

//6*P(x)^3
tmp = x>0?x:0;
result += 6*tmp*tmp*tmp;

//-4P(x-1)^3
tmp = (x-1)>0?x-1:0;
result += -4*tmp*tmp*tmp;

result*=0.1666666f;
return result;
};



### #11outRider  Members

Posted 06 August 2005 - 02:03 PM

Your t=0.5 is not right, the value of t depends on the source size, dest size, and the current dest pixel, so most definitely you shouldn't hard code it to 0.5.

I'll post some code that works a little later if you can't figure it out, don't have time now.

### #12Basiror  Members

Posted 06 August 2005 - 09:44 PM

0.5f would actually be the same as bilinear interpolation

i could find anything about the correct blend factors :(

i have search on google for hours and couldn t find any decent article that doesn t only give your the formula

the only think i found was the b spline approach which works but delivers too smooth results

### #13outRider  Members

Posted 07 August 2005 - 04:43 PM

Alright, here goes

// x,y coords for pixels in old and new imagesfloat ox,oy;int   nx,ny;// temppixel t[4];dx = oldwidth / newwidth;dy = oldheight / newheight;for (ny = 0; ny < newheight; ++ny){  oy = ny * dy;  for (nx = 0; nx < newwidth; ++nx)  {    ox = nx * dx;    // ox.frac, oy.frac represent fractional portions of ox,oy    // i.e. for ox=4.15, ox.frac=0.15    // oimg, nimg are image buffers    t[0] = interpolate(oimg[ox - 1,oy - 1], oimg[ox,oy - 1], oimg[ox + 1,oy - 1], oimg[ox + 2,oy - 1], ox.frac);    t[1] = interpolate(oimg[ox - 1,oy], oimg[ox,oy], oimg[ox + 1,oy], oimg[ox + 2,oy], ox.frac);    t[2] = interpolate(oimg[ox - 1,oy + 1], oimg[ox,oy + 1], oimg[ox + 1,oy + 1], oimg[ox + 2,oy + 1], ox.frac);    t[3] = interpolate(oimg[ox - 1,oy + 2], oimg[ox,oy + 2], oimg[ox + 1,oy + 2], oimg[ox + 2,oy + 2], ox.frac);    nimg[nx,ny] = interpolate(t[0],t[1],t[2],[3], oy.frac);    // Sorry, should have been interpolate() not cubic(), in case anyone else is looking at this.  }}pixel interpolate(pixel a,b,c,d, t){  return pixel(           cubic(a.r, b.r, c.r, d.r, t),           cubic(a.g, b.g, c.g, d.g, t),           cubic(a.b, b.b, c.b, d.b, t),           );}cubic(v0,v1,v2,v3, t){  int p = (v3 - v2) - (v0 - v1);  int q = (v0 - v1) - p;  int r = v2 - v0;  int s = v1;  float tSqrd = t * t;  return (p * (tSqrd * t)) + (q * tSqrd) + (r * t) + s;}

[Edited by - outRider on August 8, 2005 11:43:19 AM]

### #14Basiror  Members

Posted 07 August 2005 - 07:45 PM

thx it works now

I searched on google for some information on cubic interpolation and especially the weight factors and there s absolutely no useful information maybe one should write a little tutorial about it

this is when I swap the interpolation factors, e.g.: xfrac for y axis
it uses SI (stepping interpolation) size*2 per step
after the cubic filter I applied 4 smooths to get this result

here are the results
1. my cubic filter

2. photoshop's cubic filter

without swapping it get results a little bit sharper then photoshop's

### #15outRider  Members

Posted 08 August 2005 - 05:22 AM

Using xfrac on the y-axis and vice versa will give you incorrect results, your result looks a little stretched to me, but if that's what you prefer it won't matter as much for something like a greyscale heightmap, but for enlarging images I think it won't look good at all.

And if you use something like what I posted you don't need to size by stepping, you can directly go from 32,32 to 256,256. It should be much faster.

### #16Basiror  Members

Posted 08 August 2005 - 06:45 AM

yeah right it looks a little bit streched, the next thing ill do is blend it with several octaves as descriped in the simple clouds tutorial this should result in a decent heightmap for terrains and with exponential filter i can use it for clouds

### #17ddyer  Members

Posted 12 October 2012 - 03:39 PM

The quoted code for "cubic" cannot be correct. cubic(255,255,255,85,0.811) = 276
no way can a correct interpolated result be greater than any of the inputs.

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