Spline Functions – An Elegant View of Interpolation
Bruce Cohen
bic@cgl.ucsf.edu
http://www.cgl.ucsf.edu/home/bic
David Sklar
dsklar46@yahoo.com
Start with xv intensity controls
Some Goals
To present a concrete introduction to a widely used class of
methods for creating continuous functions that interpolate
discrete data
To apply straightforward geometric and algebraic operations with
functions and their graphs that may revolutionize your views of
interpolation and approximation
To present a concrete example of a general method for extending
a 1-dimensional scheme into higher dimensions
To stimulate some curricular or student project ideas
To see how elementary themes can lead to some beautiful
pictures and a lingering vision
Game Plan
Connecting the dots – continuous piecewise linear interpolation
using the data to compute a linear equation for each subinterval
using a linear combination of “Spline basis functions”
Connecting the dashes – smooth, piecewise cubic, interpolation
using slope and function data to compute
a cubic equation for each subinterval
using a linear combination of “Cubic spline basis functions”
As time allows:
Interpolation using 2-D splines
Bilinear spline basis functions
Bicubic spline basis functions
Overview – “elementary mathematics from an advanced standpoint”
linear algebra, finite dimensional function spaces, inner products,
small support, almost orthogonal bases, tensor products, finite
element methods, wavelets, …
Connecting the Dots
Write a formula for a piecewise linear function
that interpolates five given data points
 3,2.5
 2, 2 
5,2
 4,1.5
1,1
0
1
f  x
2
 x
1 x 1
2
 
11
 x  2
 12 x  12
3
4
if 1  x  2 
if 2  x  3 

if 3  x  4 
if 4  x  5 
5
6
Write a formula for a piecewise linear function
that interpolates n given data points
 x2 , y2 
 x3 , y3 
 xn , yn 
 xn1, yn1 
 x1, y1 
x1
f  x
x2
x3
y1 x2  y2 x1
 y2  y1
x

 x x
x2  x1
 2 1
y2 x3  y3 x2
 y3  y 2
x


x3  x2
  x3  x2


yn 1 xn  yn xn 1
 yn  yn 1
x

x x
xn  xn 1
 n
n 1
xn 1
xn

if x1  x  x2 


if x2  x  x3 




if xn 1  x  xn 

Interpolation using a Linear Spline Basis
The “linear spline function” approach involves carefully choosing a set of “basis
functions” 1 ,  2 , ,  n such that the interpolating function f can be
written as a simple linear combination:
f  x   y11  x   y2  2  x  
 yn  n  x  
n
 y   x
i 1
i
i
, for all x1  x  xn
 xi , yi 
 xi1, yi1 
 xi2 , yi2 
 xi1, yi1 
 xi2 , yi2 
i
xi  2
xi 1
 xi ,1
xi
xi 1
xi  2
A closer look at a linear combination of basis functions
f  x   y11  x   y2  2  x  
yi 1i 1  yi i
 yn  n  x  
yi 1i 1
i
xi 1
i 1
i
 xi2 , yi2 
yi  i
xi  2
 y   x
 xi , yi 
 xi1, yi1 
 xi2 , yi2 
n
 xi1, yi1 
 xi ,1
xi
xi 1
xi  2
i
The linear spline basis functions can be constructed as sums of
translations and horizontal scalings of two “elementary basis functions”
0
1
if x  0,1 
1  x
1  x   

0
if
x

0,1




0

 x

 0
2  x   
1
if x  0,1 


if x  0,1 

 x  xi 
 x  xi 1 
i  x    2 

  1 
x

x
x

x
 i 1 i 
i 1 
 i
xi 1
xi
xi 1
A summary description of the linear spline basis
1. Elementary basis functions – basically constructed on the unit interval
1  x
 0
1  x   
if x  0,1 

if x  0,1 

x
0

2  x   
if x  0,1 


if x  0,1 

1
0
2. A set of nodes -- x1  x2 
0
1
 xn
3. Spline basis functions – sums of (usually) two translated and scaled
elementary basis functions
interior: i  2,
,n 1
 x  xi 1 
 x  xi 
i  x    2 


 1

 xi  xi 1 
 xi 1  xi 
xi 1
endpoints: i  1 and n
 x  x1 
1  x   1 

 x2  x1 
xi
xi 1
 x  xn1 
n  x    2 

 xn  xn1 
x1
x2
xn 1
xn
Points from sin(x)
Points
Points from sin(x)
Linear Interpolation
Points from sin(x)
Linear Interpolation & “retail”
1
e
Points from g  x  
2
x2

2
Points from g(x)
Linear Interpolation
Points from g(x)
Linear Interpolation & “retail”
Find a smooth piecewise cubic function that has specified
function values and slopes at n given data points
 x3 , y3 
y 3
 x2 , y2 
 xn , yn 
y n
y 2
 xn1, yn1 
yn 1
 x1, y1 
y1
x1
x2
x3
xn 1
xn
where yi denotes the slope or first derivative at the data point
Why cubic ?
 xi , yi 
Find a cubic function that has specified function values and
slopes at 2 given data points
 x2 , y2 
y 2
 x1, y1 
y1
x1
3
2
Let p  x   ax  bx  cx  d
Using the data
we have
x2
so
p  x   3ax 2  2bx  c
p  x1   ax13  bx12  cx1  d  y1
p  x2   ax23  bx22  cx2  d  y2
p  x1   3ax12  2bx1  c
 y1
p  x2   3ax22  2bx2  c
 y2
We have 4 equations in the four
unknown coefficients. So we
can solve for a, b, c, and d.
Geometer’s sketch pad demonstration
The cubic spline basis functions can be constructed as sums of
translations and scalings of four “elementary basis functions”
0
1
2  0  0, 2 1  1
2  0  0, 2 1  0
0
1
 2  0  0, 2 1  0
 2  0  0, 2 1  1
0
1
1  0  1 , 1 1  0
1  0  0 , 1 1  0
0
1
1  0  0, 1 1  0
1  0  1 , 1 1  0
Construction of the cubic elementary basis functions
A somewhat more interesting way is to
note that the most general cubic with a
zero value and a zero derivative at 0 is
2  x   x 2  ax  b so 2  x   3ax 2  2bx
0
Now
1
and
2  0  0, 2 1  1
2  0  0, 2 1  0
2 1  1 
2 1  0  3a  2b  0
a  2, b  3
Hence
and
There are several reasonable ways to find the
cubic function with the desired properties. A
straightforward way is to write a general cubic,
compute its derivative, substitute the desired
conditions and solve the resulting system of
four linear equations for the coefficients.
a  b 1
2  x   x 2  3  2 x 
We can see by symmetry that
1  x   2 1  x 
 1  x  3  2 1  x 
2
0
1
1  0  1, 1 1  0
1  0  0, 1 1  0
 1  x   2 x  1
2
  x  1  2 x  1
2
Construction of the cubic elementary basis functions
0
1
2  0  0, 2 1  0
2  0  0, 2 1  1
In this case we note that the most general
cubic with a zero value and a zero
derivative at 0 and a zero value at 1 is
2  x   ax  x  1 so  2  x   a  3x 2  2 x 
2
Now 2 1  1  a  1
hence
2  x   x 2  x  1
0
1
1  0  0, 1 1  0
1  0  1, 1 1  0
Again using symmetry we see that
1  x   2 1  x 
  1  x  1  x   1
2
 1  x  x
2
 x  x  1
2
The Cubic Spline Elementary Basis Functions
0
2

 x 3  2 x 
2  x   
0


0
2

 x  x  1
2  x   
 0

1
if x  0,1 


if x  0,1 

1
if x  0,1 


if x  0,1 

1
0
if x  0,1 

if x  0,1 
 2 x  1 x  12
1  x    2 1  x   
0

0
2

 x  x  1
1  x    2 1  x   

 0
1
if x  0,1 


if x  0,1 

Construction of the cubic spline basis functions as translations and
scalings of four “elementary basis functions”
0
1
0
1
2  x 
1  x 
 x  xi 1 
 x  xi 
i  x    2 




1
 xi  xi 1 
 xi 1  xi 
used to match function values
xi 1
xi
xi 1
Scaling the derivative basis functions
1  x 
0
1  2x 
1 x
1  1 
2 2
0
or
x
1
1  1 
2
A horizontal scaling by a factor of ½
changes the slopes by a factor of 2.
1
2
A vertical scaling by a factor of ½ returns
the slopes to their corresponding values.
0
1
2
x
x
In general the function h g   has the same slope at
as g  x  has at x
h
h
Construction of the cubic spline “derivative” basis functions from the
“derivative” elementary basis functions
0
0
1
2  x 
1
1  x 
xi 1
xi
xi 1
 x  xi 1 
 x  xi 
i  x    xi  xi 1   2 

x

x

  i 1 i  1 

x

x
x

x
i 1 
 i
 i 1 i 
used to match slope values
A summary description of the Cubic Spline Basis
1. Elementary basis functions – basically constructed on the unit
interval four for the cubics
0
2

 x 3  2 x 
0


2  x   
0
2

 x  x  1
2  x   
 0

if x  0,1 


if x  0,1 

 2 x  1 x  1
1  x    2 1  x   
0

0
1
if x  0,1 


if x  0,1 

2. A set of nodes -- x1  x2 
1
0
1
2

 x  x  1
1  x    2 1  x   
 0

 xn
if x  0,1 

if x  0,1 
2
1
if x  0,1 


if x  0,1 

3. Spline basis functions – sums of (usually) two translated and scaled
elementary basis functions
interior: i  2,
 x  xi 1 
 x  xi 
i  x    2 


 1

 xi  xi 1 
 xi 1  xi 
,n 1
 x  xi 1 
 x  xi 
i  x    xi  xi 1   2 
   xi 1  xi  1 

 xi  xi 1 
 xi 1  xi 
endpoints: i  1 and n
 x  x1 
1  x   1 

 x2  x1 
x1
 x  x1 
1  x    x2  x1  1 

 x2  x1 
x1
x2
xi 1
xi
xi 1
xi 1
xi
xi 1
 x  xn 1 
n  x   2 

 xn  xn 1 
x2
 x  xn1 
 n  x    xn  xn 1   2 

 xn  xn1 
xn 1
xn 1
xn
xn
A closer look at a linear combination of cubic basis functions
f  x   y11  x   y11  x   y22  x   y22  x  

n

i 1
 ynn  x   ynn  x 
yi  i  x   yi  i  x 
 xi , yi 
 xi1, yi1 
yi
 xi1, yi1 
yi1
yi1
yi i  x   yi i  x 
 xi2 , yi2 
yi 2
i
yi2
xi  2
i
yi  i
 xi2 , yi2 
xi 1
xi
yii
xi 1
xi  2
Points from sin(x)
Linear Interpolation & slopes
Points from sin(x)
Cubic Interpolation
Points from sin(x)
Interpolations and “Retail”
1
e
Points from g  x  
2
x2

2
Points from g(x)
Linear Interpolation
Points from g(x)
Linear Interpolation & slopes
Points from g(x)
Cubic Interpolation
Points from g(x)
Interpolations and “Retail”
Geometer’s sketch pad demonstration
Some nice properties of cubic splines
Localization – changing the data at a point only changes the interpolant on
the two adjacent subintervals
Simple construction - starts with really only two elementary basis
functions; the rest follows from symmetry, scaling,
translation and linear combinations
Easy “synthesis” - since the interpolating function is a finite linear
combination with the data as the coefficients it is easy
to construct
Easy “analysis” - if we are given a cubic splines function we can find the
coefficients by evaluating function and its derivative at
the nodes
Smoothness – good but not great. We have continuous first derivatives, but
generally discontinuous second derivatives.
Easy implementation - it is straight forward to program a computer to set
up and evaluate cubic splines efficiently.
Find a two-dimensional analog of linear interpolation for
specified function values at mn given data points
 xn , ym 
 x1, ym 
x , y 
i
 x1, y1 
x , y ,z 
j
i
 xn , y1 
 x1, y1, z11 
j
ij
 xn , ym , znm 
 x1, ym , z1m 
 xn , y1, zn1 
A closer view of four data points
There a are four “bilinear” 2-D elementary basis functions.
They are constructed from the 1-D elementary basis functions
1,1,1
0
y
1  t 
1
0
1
2 t 
We define the bilinear 2-D elementary
basis functions as follows:
11  x, y   1  x 1  y 
12  x, y   1  x 2  y 
x
21  x, y   2  x 1  y 
22  x, y   2  x 2  y 
The two 1-D elementary basis functions
0
1  t 
1
0
2 t 
The four bilinear 2-D elementary
basis functions
1
y
y
21  x, y   2  x 1  y 
x
x
11  x, y   1  x 1  y 
22  x, y   2  x 2  y 
y
y
12  x, y   1  x 2  y 
x
x
We can construct the 2-D bilinear spline basis functions
directly from the 1-D linear spline basis functions
We define the ijth 2-D bilinear spline
basis function as follows:
i  x 
xi 1
xi
i j  x, y   i  x   j  y 
xi 1
 x , y ,1
 j  y
y j 1
i
yj
j
y j 1
As in the 1-D case we want our
basis functions to be zero at all
but one of the data points.
x
i 1
, y j 1 ,0 
We can construct the 2-D bicubic spline basis functions
directly from the 1-D bicubic spline basis functions
At each data point we define four 2-D
bicubic spline basis functions as follows:
i  x 
xi 1
xi
xi 1
i j  x, y   i  x   j  y 
i  x 
xi 1
i j  x, y   i  x   j  y 
xi
xi 1
 i j  x, y   i  x   j  y 
 i j  x, y   i  x   j  y 
 x , y ,1
i
 j  y
y j 1
yj
y j 1
yj
y j 1
j
 j  x
y j 1
x
i 1
, y j 1 ,0 
The 2-D bicubic spline basis functions


y j 1
y j 1
yj
yj
y j 1
xi 1
y j 1
xi 1
xi
i j  x, y   i  x   j  y 
xi 1
xi
i j  x, y   i  x   j  y 
xi 1
y j 1
y j 1
yj
yj
y j 1
xi 1
y j 1
xi 1
xi
xi
 i j  x, y   i  x   j  y 
xi 1
 i j  x, y   i  x   j  y 
xi 1
What have we done?
Connecting the dots – continuous piecewise linear interpolation
using the data to compute a linear equation for each subinterval
using a linear combination of “Spline basis functions”
Connecting the dashes – smooth, piecewise cubic, interpolation
using slope and function data to compute
a cubic equation for each subinterval
using a linear combination of “Cubic spline basis functions”
Interpolation using 2-D splines
Bilinear spline basis functions
Bicubic spline basis functions
Overview – “elementary mathematics from an advanced standpoint”
linear algebra, finite dimensional function spaces, inner products,
small support, almost orthogonal bases, tensor products, finite
element methods, wavelets, …
The end