Growth of projection ghosts
created by iteration
Imants Svalbe a, Shekhar Chandra a, b
a
School of Physics, Monash University Australia
b ICT, CSIRO, Brisbane, Australia
DGCI Nancy, France 6-8 April, 2011
Background:
Ghosts are discrete objects that are invisible in a
projected view taken at any of N discrete angles
Ghosts play an important role in the reconstruction of
images from projected views of an object*
Different images that contain one or more (scaled and
translated) ghosts will have exactly the same set of
projections in these N directions
*Unique image reconstruction is possible when the set of projections
excludes all ghosts from the image space (Katz, 1978)
2
On minimal ghosts
Minimal ghosts have the smallest possible pixel footprint in
the image space and are constructed from the smallest
possible range of image intensities
Truly minimal ghosts span a pixel size of about 2N x 2N (for
N zero-sum directions) and require only +1 or -1 image
intensities*
Truly minimal ghosts exploit the periodic wrapping of rays
to produce zero sums. However they cease being ghosts
with N zero-sum views whenever they are embedded
into an image of larger size
* Imants Svalbe and Nicolas Normand, DGCI 2011, Thursday pm poster session
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On almost minimal ghosts
We can construct ghosts using iterated convolution
(adding successive vector displacements of a seed
ghost)
However iterated ghosts grow to be large in size
and span a large range of intensities quite quickly
The aim of this talk is: to produce iterated
ghosts of minimal spatial extent and that utilise
the smallest possible range of intensities
4
This work is inspired by
Herman and Davidi: Inverse Problems 24 2008
The tumor added to image (a) to obtain reconstructed image (b) is invisible in 22
projected views. The arrowed structure is a ghost. Our aim is to construct the
simplest ghost structures that will remain invisible when viewed from the 5
largest number of angles
Discrete projection transforms are pixel-oriented:
Arbitrary sets of discrete
projections of variable
length are called Mojette
Transforms.
See the book: The Mojette
Transform: Theory and
Applications, JP Guédon
(Ed.), Wiley, 2009
6
Equi-spaced angles vs.
digital angles via integers pi:qi on
discrete grids
1:3
1:√3
60° = tan-1(√3)
17:10
2:3
59.53° = tan-1(17/10)
7
Left:
The Farey set selects relatively prime integer ratios p:q to
represent discrete angles
Centre: Digital angles are symmetric, p:q and q:p are
complementary angles
Right:
We can restrict the directions to the set of shortest
discrete vectors lying within a cut-off-radius, R.
8
Digital angles
formed by integer
ratios ±p:±q
have 8-fold
symmetry
For x-ray imaging;
q ≡ 180° + q
(blue arrows)
here we use
–p:+q and –q:+p
+p:+q and +q:+p
p:q
-p:q
-q:p
q:p
(over the range 0 to 180°)
9
Ghost images and ambiguous image projections
Seed ghost
The ghost images are constructed cumulatively, the right image sums to zero
over all five directions {0:1, 1:0, 1:1, -1:1 and 1:2} (red lines)
ALL 6x6 or larger sized images that are reconstructed using these 5
projection directions cannot be unique. Scaled and translated versions of this
ghost function can be added to the image without changing its projections 10
A minimal ghost: This 79x79 image has zerosums at 40 angles and is constructed using 40
(+1) pixels (white) and 40 (-1) pixels (black)
Minimal ghosts on a p
x p array are invariant
to translation and
other affine
transformations
(…provided the remapped image
coordinates are
wrapped modulus p…)
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Minimal ghosts: N = 10, 79*79
Pixel values are ±1, shown are affine down-scaled ghosts
(using successive discrete 45° rotations*). They each remain
as N = 10 ghosts, but the image size MUST be kept the same
Minimal ghosts rely on the wrapping of rays around a
period image to create zero-sums. Embedding minimal
ghosts in a larger image array breaks the pattern of
wrapping and the rays no longer sum to zero
* Svalbe DGCI 2009
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Iterated (almost minimal) ghosts: N = 11; discrete angles
1:0 to 1:10; 79*79 array. The ghost size and intensity levels
grow with the number of zero-sum angles.
Alternating Row and Column iteration: -3 ≤ g ≤ +3
Shifts are [1 i] then [i+1, 1] 1 ≤ i < N, i += 2
The seed ghost here is
+1 at (0,0) -1 at (0,1)
Row iteration:
-32 ≤ g ≤ +32
Shifts are [1 i]
for 1 ≤ i < N
Column iteration: -3 ≤ g ≤ +3
Shifts are [i 1] for 1 ≤ i < N
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Constructing iterated ghosts:
Initialise a single +1 pixel, located at, for example,
(0, 0) in an otherwise empty image space I(x, y)
To generate N ghosts:
for i= 1:N
I(x, y) = I(x, y) – {I(x, y) circularly shifted by [pi qi]}
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Constructing iterated ghosts:
The set of shifts, [pi qi], 0 ≤ i ≤ N, should use the shortest
discrete angle vectors
This keeps the ghost footprint as small as possible
Extra “redundant” shifts of [1 0] and [-1 0], [0 1] and [0 1] are used to reduce the total ghost intensity
This makes large + and – ghost values destructively
interfere by folding the signed values along vertical and
horizontal symmetry axes
15
N = 25, discrete angles around the 1:1 direction
1:0 to 3:4, 79x79 image
L: -128 ≤ g ≤ +128
R: -50 ≤ g ≤ +50
The right image, a sum of the left image and the two arrow-shifted versions of
itself, has a narrower range of grey levels and occupies almost the same
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footprint
3D view of previous ghost, ± 128 intensity range
17
N = 24 discrete angles around the 0:1 direction
1:0 to 5:6, 101x101 image
L: -124 ≤ g ≤ +124
R: -80 ≤ g ≤ +80
The right image, a sum of the left image and a left/right flipped version of
itself, has a narrower range of grey levels and occupies almost the same
footprint (98 x 51 pixels)
18
N = 24 discrete angles around the 0:1 direction
1:0 to 5:6, 101x101 image, -80 ≤ g ≤ +80 3D view
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N = 24 discrete angles around the 0:1 direction
1:0 to 5:6, 101x101 image, Fourier space view
DFT magnitude
(centred, unscaled,
28k max)
DFT magnitude
(centred, contrast
stretched)
DFT phase (centred,
unscaled, ±p)
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N= 24; 0° to 180° angle range
discrete angles 1:0 to ±3:4
50 x 50 pixel footprint
L: -20 ≤ g ≤ +20
R: -8 ≤ g ≤ +8
Right image includes the effect of the extra “intensity cancelling” through
adding redundant convolution shifts at
[1 0] and [-1 0], [0 1] and [0 -1]
21
3D view of previous ghost, ±20 intensity range
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N = 48, discrete angles from 1:0 to ±5:6
ghost footprint: 151 x 151, -372 ≤ g ≤ +372*
*The intensity range can
be further reduced (to
span from -236 to +290
for N = 48) by summing
this image with shifted
versions of itself. This is
at the expense, however,
of increasing the size of
the ghost’s footprint and
a loss of 4-fold symmetry
23
A composite
image of 25
copies of the
previous ghost
embedded into a
757 x 757 image
space remains
an N = 48 ghost
(iterated
ghosts are
invariant to
changes in the
array size)
24
N = 72 (discrete angles 1:0 to ±6:7)
ghost footprint: 275 x 275 pixels
L: -131k ≤ g ≤ +131k#
R: -55k ≤ g ≤ +55k*
#
131,278
Right image has extra intensity cancelling “redundant” shifts,
[1 0] and [-1 0], [0 1] and [0 -1]
*
55,106
25
Same ghosts as on previous slide; displayed using
higher contrast settings
Right image has extra intensity cancelling “redundant” shifts,
[1 0] and [-1 0], [0 1] and [0 -1]
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N = 88 (discrete angles 1:0 to ±7:8)
ghost footprint: 371 x 371 pixels
-3.3 M ≤ g ≤ +3.3 M*
* 3,333,030
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N = 112 (discrete angles 1:0 to ±8:9)
ghost footprint: 533 x 533 pixels
-39 M ≤ g ≤ +39 M*
* 39,810,211
28
Log (ghost intensity range, gmax) vs. Number of
ghost angles, N
Log (abs( gmax))
N, Number of zero-sum view angles
29
Ghost footprint size (pixels) vs. Number of ghost
angles, N
Ghost footprint (pixels)
N, Number of zero-sum view angles
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Ghost footprint size vs. Intensity range
Log (abs (gmax))
The log of the
minimum intensity
range for iterated
ghosts grows
almost linearly with
the size of the
ghost’s footprint
Ghost footprint (pixels)
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Intensity fluctuations in ghosts and the symmetry
of the p:q angle set
Adding extra ghosts for ±1:(n+1) to an existing
symmetric ghost built from angles 1:0 to (n-1):n
often reduces the intensity range.
For example:
N
48
52
56
gmax footprint
±372
149
±346
165
±956
183
last added
vectors
±5:6, ±6:5
±1:7, ±7:1
±2:7, ±7:2
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Choosing the p:q vectors
Ghosts at N angles made by using all of the
shortest vectors have the smallest footprint and
smallest dynamic range of intensities
Ghosts made by choosing N* “random” pi:qi
vectors have larger footprints and gmax values
*Random vector sums often have N’ < N angles. The translated pixels
may partially cancel he existing ghost and loose the zero-sum property
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Conclusion
To be done: construct
ghosts on a 3D hexagonal
lattice, they may resemble
snowflakes
We described simple iterative convolution methods to produce discrete ghosts;
images whose discrete projections vanish at N angles
The aim was to produce ghosts that have compact size and grey levels, yet remain
invisible over a large range of projection angles. We achieved this construction
using symmetric sets of N shortest discrete vector angles.
These iterated ghosts resemble realistic image artefacts. They can be inserted into
different sized images and retain their ghostliness
We examined how ghosts of increasing N grow in grey level intensity and pixel size
We showed that the log of their intensity scales with the size of their footprint. Ghost
size and depth are important to guarantee that images reconstructed from
projections will be artefact-free
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