Bilinear interpolation

From Rosetta Code
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Bilinear interpolation is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Bilinear interpolation is linear interpolation in 2 dimensions, and is typically used for image scaling and for 2D finite element analysis.

C

<lang c>#include <stdint.h> typedef struct { 

   uint32_t *pixels;
   unsigned int w;
   unsigned int h;

} image_t;

  1. define getByte(value, n) (value >> (n*8) & 0xFF)

uint32_t getpixel(image_t *image, unsigned int x, unsigned int y){ 

   return image->pixels[(y*image->w)+x];

} float lerp(float s, float e, float t){return s+(e-s)*t;} float blerp(float c00, float c10, float c01, float c11, float tx, float ty){ 

   return lerp(lerp(c00, c10, tx), lerp(c01, c11, tx), ty);

} void putpixel(image_t *image, unsigned int x, unsigned int y, uint32_t color){ 

   image->pixels[(y*image->w) + x] = color;

} void scale(image_t *src, image_t *dst, float scalex, float scaley){ 

   int newWidth = (int)src->w*scalex;
   int newHeight= (int)src->h*scaley;
   int x, y;
   for(x= 0, y=0; y < newHeight; x++){
       if(x > newWidth){
           x = 0; y++;
       }
       float gx = x / (float)(newWidth) * (src->w-1);
       float gy = y / (float)(newHeight) * (src->h-1);
       int gxi = (int)gx;
       int gyi = (int)gy;
       uint32_t result=0;
       uint32_t c00 = getpixel(src, gxi, gyi);
       uint32_t c10 = getpixel(src, gxi+1, gyi);
       uint32_t c01 = getpixel(src, gxi, gyi+1);
       uint32_t c11 = getpixel(src, gxi+1, gyi+1);
       uint8_t i;
       for(i = 0; i < 3; i++){
           //((uint8_t*)&result)[i] = blerp( ((uint8_t*)&c00)[i], ((uint8_t*)&c10)[i], ((uint8_t*)&c01)[i], ((uint8_t*)&c11)[i], gxi - gx, gyi - gy); // this is shady
           result |= (uint8_t)blerp(getByte(c00, i), getByte(c10, i), getByte(c01, i), getByte(c11, i), gx - gxi, gy -gyi) << (8*i);
       }
       putpixel(dst,x, y, result);
   }

}</lang>

D

This uses the module from the Grayscale Image task.

Translation of: C

<lang d>import grayscale_image;

/// Currently this accepts only a Grayscale image, for simplicity. Image!Gray rescaleGray(in Image!Gray src, in float scaleX, in float scaleY) pure nothrow @safe in { 

   assert(src !is null, "Input Image is null.");
   assert(src.nx > 1 && src.ny > 1, "Minimal input image size is 2x2.");
   assert(cast(uint)(src.nx * scaleX) > 0, "Output image width must be > 0.");
   assert(cast(uint)(src.ny * scaleY) > 0, "Output image height must be > 0.");

} body { 

   alias FP = float;
   static FP lerp(in FP s, in FP e, in FP t) pure nothrow @safe @nogc {
       return s + (e - s) * t;
   }

   static FP blerp(in FP c00, in FP c10, in FP c01, in FP c11,
                   in FP tx, in FP ty) pure nothrow @safe @nogc {
       return lerp(lerp(c00, c10, tx), lerp(c01, c11, tx), ty);
   }

   immutable newWidth = cast(uint)(src.nx * scaleX);
   immutable newHeight = cast(uint)(src.ny * scaleY);
   auto result = new Image!Gray(newWidth, newHeight, true);

   foreach (immutable y; 0 .. newHeight)
       foreach (immutable x; 0 .. newWidth) {
           immutable FP gx = x / FP(newWidth) * (src.nx - 1);
           immutable FP gy = y / FP(newHeight) * (src.ny - 1);
           immutable gxi = cast(uint)gx;
           immutable gyi = cast(uint)gy;

           immutable c00 = src[gxi,     gyi    ];
           immutable c10 = src[gxi + 1, gyi    ];
           immutable c01 = src[gxi,     gyi + 1];
           immutable c11 = src[gxi + 1, gyi + 1];

           immutable pixel = blerp(c00, c10, c01, c11, gx - gxi, gy - gyi);
           result[x, y] = Gray(cast(ubyte)pixel);
       }

   return result;

}

void main() { 

   const im = loadPGM!Gray(null, "lena.pgm");
   im.rescaleGray(0.3, 0.1).savePGM("lena_smaller.pgm");
   im.rescaleGray(1.3, 1.8).savePGM("lena_larger.pgm");

}</lang>

J

<lang J> Note 'FEA' 

  Here we develop a general method to generate isoparametric interpolants.

  The interpolant is the dot product of the four shape function values evaluated
  at the coordinates within the element with the known values at the nodes.
  The sum of four shape functions of two variables (xi, eta) is 1 at each of four nodes.
  Let the base element have nodal coordinates (xi, eta) of +/-1.



   2               3 (1,1)
  +---------------+
  |               |
  |               |
  |        (0,0)  |
  |       *       |
  |               |
  |               |
  |               |
  +---------------+
   0               1

  determine f0(xi,eta), ..., f3(xi,eta).
  f0(-1,-1) = 1, f0(all other corners) is 0.
  f1( 1,-1) = 1, f1(all other corners) is 0.
  ...

  Choose a shape function.
  Use shape functions C0 + C1*xi + C2*eta + C3*xi*eta .
  Given (xi,eta) as the vector y form a vector of the
  coefficients of the constants (1, xi, eta, and their product)

     shape_function =: 1 , {. , {: , */

     CORNERS NB. are the ordered coordinates of the corners
  _1 _1
   1 _1
  _1  1
   1  1
  
     (=i.4)  NB. rows of the identity matrix are the values of each shape functions at each corner
  1 0 0 0
  0 1 0 0
  0 0 1 0
  0 0 0 1
  
     (=i.4x) %. shape_function"1 x: CORNERS  NB. Compute the values of the constants as rational numbers.
   1r4  1r4  1r4 1r4
  _1r4  1r4 _1r4 1r4
  _1r4 _1r4  1r4 1r4
   1r4 _1r4 _1r4 1r4

  This method extends to higher order interpolants having more nodes or to other dimensions.

)

mp =: +/ .* NB. matrix product

CORNERS =: 21 A.-.+:#:i.4 shape_function =: 1 , ] , */ COEFFICIENTS =: (=i.4) %. shape_function"1 CORNERS shape_functions =: COEFFICIENTS mp~ shape_function interpolate =: mp shape_functions </lang> 

Note 'demonstrate the interpolant with a saddle'
   lower left has value 1,
   lower right: 2
   upper left: 2.2
   upper right: 0.7
)

require'viewmat'
GRID =: |.,~"0/~(%~i:)100
SADDLE =: 1 2 2.2 0.7 interpolate"_ 1 GRID
viewmat SADDLE
assert 0.7 2.2 -: (<./ , >./) , SADDLE

File:J bilinear interpolant.jpg

Let n mean shape function, C mean constants, i mean interpolant, and the three digits meaning dimensionality, number of corners, and (in base 36) the number of nodes we construct various linear and quadratic interpolants in 1, 2, and 3 dimensions as <lang J> Note 'Some elemental information' 

  Node order
  1D:

  0   2   1



  2D:

  2   7   3

  5   8   6   Node 8 at origin, Node 3 at (1,1)

  0   4   1

  Names for shape functions and constants:
  n249: n means shape function, 2 dimensions, 4 corners (quadrilateral), 9 nodes
  C244: C       constants for   2 dimensions, 4 corners (quadrilateral), 4 nodes



  3D
  At z = _1           z = 1            z = 0
  2   b   3           6   j   7        e   o   f

  9   k   a           h   p   i        m   q   n

  0   8   1           4   g   5        c   l   d

) mp =: ($: |:) : (+/ .*) NB. A Atranspose : matrix product A B identity =: =@:i. NB. generate identity matrix


NB. 1D NB. master nodes N1 =: ,._1 1 0x NB. form of shape functions n122 =: 1 , ] n123 =: [: , ^/&(i.3) NB. constants C122 =: x:inv@:(x:@:identity@:# %. n122"1)2{.N1 C123 =: x:inv@:(x:@:identity@:# %. n123"1)3{.N1 NB. interpolants i122 =: mp (C122 mp~ n122) i123 =: mp (C123 mp~ n123)


NB. 2D NB. nodes are arranged 4&{. are the corners, 8&{. the corners and edges, ] include the center. N2 =: 336330 A.-.3x#.inv i.*:3 NB. 336330 (-: A.) 8 2 6 0 5 7 1 3 4

NB. terms of shape functions n244 =: [: , [: *// ^/&(i.2) NB. all linear combinations n248 =: }:@:n249 NB. exclude (xi eta)^2 n249 =: [: , [: *// ^/&(i.3) NB. all quadratic combinations

NB. constants C244 =: x:inv@:(x:@:identity@:# %. n244"1)4{.N2 NB. serendipity C248 =: x:inv@:(x:@:identity@:# %. n248"1)8{.N2 NB. serendipity C249 =: x:inv@:(x:@:identity@:# %. n249"1)9{.N2 NB. non-serendipity

NB. interpolants i244 =: mp (C244 mp~ n244) i248 =: mp (C248 mp~ n248) i249 =: mp (C249 mp~ n249)

NB. 3D N3 =: 267337661061030402017459663x A.<:3#.inv i.3^3 NB. 267337661061030402017459663x (-: A.) 0 18 6 24 2 20 8 26 9 3 21 15 1 19 7 25 11 5 23 17 12 10 4 22 16 14 13 NB. corners n388 =: [: , [: *// 1 , ] NB. all linear combinations

Note 'simplification not yet apparent to me' 

  combinations =: 4 : 0
    if. x e. 0 1 do. z=.<((x!y),x)$ i.y
    else. t=. |.(<.@-:)^:(i.<. 2^.x)x
      z=.({.t) ([:(,.&.><@;\.)/ >:@-~[\i.@]) ({.t)+y-x
      for_j. 2[\t do.
        z=.([ ;@:(<"1@[ (,"1 ({.j)+])&.> ])&.> <@;\.({&.><)~ (1+({.j)-~{:"1)&.>) z
        if. 2|{:j do. z=.(i.1+y-x)(,.>:)&.> <@;\.z end.
      end.
    end.
    ;z
  NB.)
  n38k =: 1 , ] , */"1@:((2 combinations 3)&{) , *: , (1&, * */) , ,@:(*:@:|. (*"0 1) (2 combinations 3)&{) NB. include mid-edge nodes

) n38q =: }:@:n38r NB. include mid-face nodes, all quadratic combinations but (xyz)^2 n38r =: [: , [: *// ^/&(i.3) NB. now this is simple! 3*3*3 nodal grid. C388 =: x:inv@:(x:@:identity@:# %. n388"1)8{.N3 NB.C38k =: x:inv@:(x:@:identity@:# %. n38k"1)36bk{.N3 C38q =: x:inv@:(x:@:identity@:# %. x:@:n38q"1)36bq{.N3 C38r =: x:inv@:(x:@:identity@:# %. x:@:n38r"1)36br{.N3 i388 =: mp (C388 mp~ n388) NB.i38k =: mp (C38k mp~ n38k) i38q =: mp (C38r mp~ n38r) i38r =: mp (C38r mp~ n38r) </lang>

Racket

This mimics the Wikipedia example. <lang racket>#lang racket (require images/flomap)

(define fm 

 (draw-flomap
  (λ (dc)
    (define (pixel x y color)
      (send dc set-pen color 1 'solid)
      (send dc draw-point (+ x .5) (+ y 0.5)))  
    (send dc set-alpha 1)
    (pixel 0 0 "blue")
    (pixel 0 1 "red")
    (pixel 1 0 "red")
    (pixel 1 1 "green"))
  2 2))

(flomap->bitmap 

(build-flomap
 4 250 250
 (λ (k x y)
   (flomap-bilinear-ref 
    fm k (+ 1/2 (/ x 250)) (+ 1/2 (/ y 250))))))</lang>
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