Astronomy

68 ASTRONOMY • DECEMBER 2015

Representing images in ways

other than pixels allows for

powerful processing. This

subject, based on signal pro-

cessing, requires a mathemati-

cal f luency to understand the

concepts fully. Many software

programs include tools such

as fast-Fourier transform fil-

ters and wavelet-based filters.

Luckily we don’t need to derive

the math to understand how

to use these utilities. However,

some background informa-

tion can make parameters a bit

more understandable.

We can deconstruct images

as the sum of periodic varia-

tions of brightness and repre-

sent the frequencies we get by

sine and cosine functions. This

process is a Fourier transform.

In fact, you can represent

images by transforming them

from brightness at any pixel

position to a frequency with a

particular amplitude.

Image #1 shows an example

for a repeating pattern. In this

domain, just a few dots charac-

terize the image. If you modify

one of the dots by, say, erasing

it (making it black) and trans-

form back to the original

image, you’ll remove the signal

that repeats at that frequency.

Note, however, that this modi-

fication will affect all struc-

tures because functions that

model the image are infinite

and range the whole image.

Wavelets get around this

restriction because these func-

tions have shapes that limit

their oscillations. You can cre-

ate a new image in a wavelet

domain instead of the strict

frequency domain by using

these shapes.

The wavelet domain can

isolate structures by their sizes.

So when you use a “Wavelet

Transform,” the image decon-

structs using the same wavelet

function at different scales,

rather than the same sine wave

at different frequencies.

Using a particular wavelet

function, I’ve broken an image

of the Lagoon Nebula (M8)

into small-scale features

around four pixels in size,

large-scale features 32 pixels in

size, and a residual image that

is everything else (Image #2).

Adding these “layers” back

together gives me my original

image. But now I can modify

any layer to enhance or dimin-

ish the information there.

PixInsight software has a

“Wavelet Transform” tool

(Image #3) with many param-

eters. The “Scaling Function” is

the wavelet function you select

from the pull-down menu in

the form of a kernel filter (the

discrete representation of a

wavelet). You choose how many

layers (scale sizes) to decon-

struct the image into.

You can set the layers up as

a doubling scheme (a scale of

one pixel for the first, two for

the second, etc.). Turning off

(removing) the first layer and

combining the remaining ones

will remove features on the

order of one pixel in size. This

example shows how you might

remove noise in your image.

Changing the “Bias” of a

layer gives it more or less

weight in the reconstruction,

which makes information at a

particular scale size more or

less obvious. To de-emphasize

an image’s small stars, just

decrease that layer’s weight.

Another popular program

that uses wavelets for planetary

processing is RegiStax. The

layer adjustments are similar

with the sliders being like the

“Bias” setting in PixInsight. In

fact, here’s a secret many of the

top planetary imagers know:

They modify the wavelet (filter

kernel) and increase the weight

of the center of the matrix

(Image #4). Changing the

shape of the wavelet in this way

(ma k ing it more “pea ked ”)

better probes small features

when you adjust the scale size.

In my next column, I’ll show

you how I used wavelet layers

in combination with high

dynamic range processing to

handle the M8 image.

COSMICIMAGING

BY ADAM BLOCK

Understanding

wavelets

BROWSE THE “COSMIC IMAGING” ARCHIVE AND FIND VIDEO TUTORIALS AT http://www.Astronomy.com/Block.

Image #1. In this repeating pattern, the

dots represent the frequency values of

the variation in the original image.

Image #2. The author deconstructed this image of the Lagoon Nebula (M8) using wave-

lets. It contains layers that correspond to four pixels, 32 pixels, and all residual pixels.

You’ll find this image online at http://skycenter.arizona.edu/gallery/Nebulae/M8_32in.

Image #3. This screen shot

shows PixInsight’s “Wavelet

Trans form” to ol.

Image #4. This screen shot

from RegiStax shows the

screen that allows you to

modify wavelets.

ALL IMAGES: ADAM BLOCK