# Spectral Extraction of Extended Sources using Wavelet Interpolation Paul Barrett

```Spectral Extraction of Extended
Sources using Wavelet
Interpolation
Paul Barrett
Linda Dressel
STIS Calibration Group
2005 July 21
Raw Spectral Image
Interpolated Spectral Image
Interpolating Subdivision
Construct a polynomial
p of degree N-1:
p(xj,k+n) = yj,k+n for -D &lt;
nD
Calculate value at
midpoint:
yj+1,2k+1 = p(xj+1,2k+1)
Average Interpolation Subdivision
Construct a polynomial
p of degree N-1:
int p(x)dx = yj,k+n for D&lt;n&lt;D
Calculate two coeff. at
k and k+0.5
Note: the polynomial
fits the cumulative
values.
Wavelet Properties
Compact Support: w(x) is exactly zero outside
the interval [-N+1, N].
Average-interpolation: w(x) is averageinterpolating in the sense that int
k+1
k
f(x)dx = yk,0.
Symmetry: w(x) is symmetric about x = .
Polynomial reproduction: w(x) reproduces
polynomials up to degree N-1.
Wavelet Properties
Smoothness: R(x) is continuous of order R, with
R = R(N) &gt; 0.
Refinability: (x) satisfies a refinement relation of
the form (x) = Nl=-N+1hl (2x-l).
The construction implies that h0 = h1 = 1 and h2 = h2l+1 if l0.
This last property distinguishes wavelets from filter
functions.
Average Interpolation Algorithm
Step 1: Subdivide pixel into 2 subpixels using an
N-order (=7) polynomial to partition the counts in.
Step 2: Apply inverse Haar transform (wavelet).
Step 3: Repeat j times.
Step 4: Convolve subpixels using instrumental