Growth of projection ghosts created by iteration

Imants Svalbe a, Shekhar Chandra a, b

a School of Physics, Monash University Australiab ICT, CSIRO, Brisbane, Australia

DGCI Nancy, France 6-8 April, 2011

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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)

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On minimal ghostsMinimal 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

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This work is inspired byHerman 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 largest number of angles

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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

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Equi-spaced angles vs. digital angles via integers pi:qi on

discrete grids

2:31:3

1:√3 60° = tan-1(√3) 17:10 59.53° = tan-1(17/10)

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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.

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Digital angles formed by integer

ratios ±p:±qhave 8-fold symmetry

For x-ray imaging; ≡ 180° +

(blue arrows)

here we use–p:+q and –q:+p+p:+q and +q:+p

(over the range 0 to 180°)

p:q

q:p-q:p

-p:q

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Ghost images and ambiguous image projections

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

Seed ghost

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A minimal ghost: This 79x79 image has zero-sums 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 re-mapped 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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Column iteration: -3 ≤ g ≤ +3

Shifts are [i 1] for 1 ≤ i < N

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.

Row iteration:

-32 ≤ g ≤ +32

Shifts are [1 i] for 1 ≤ i < N

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)

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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 [p i 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

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N = 25, discrete angles around the 1:1 direction1:0 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 footprint

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3D view of previous ghost, ± 128 intensity range

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N = 24 discrete angles around the 0:1 direction1:0 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)

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N = 24 discrete angles around the 0:1 direction1:0 to 5:6, 101x101 image, -80 ≤ g ≤ +80 3D view

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N = 24 discrete angles around the 0:1 direction1:0 to 5:6, 101x101 image, Fourier space view

DFT magnitude (centred, unscaled, 28k max)

DFT magnitude (centred, contrast stretched)

DFT phase (centred, unscaled, ±)

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N= 24; 0° to 180° angle range discrete angles 1:0 to ±3:4

50 x 50 pixel footprintL: -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]

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3D view of previous ghost, ±20 intensity range

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N = 48, discrete angles from 1:0 to ±5:6ghost 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

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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)

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N = 72 (discrete angles 1:0 to ±6:7) ghost footprint: 275 x 275 pixels

L: -131k ≤ g ≤ +131k# R: -55k ≤ g ≤ +55k*

Right image has extra intensity cancelling “redundant” shifts,

[1 0] and [-1 0], [0 1] and [0 -1]

# 131,278 * 55,106

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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

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Log (ghost intensity range, gmax) vs. Number of ghost angles, N

N, Number of zero-sum view angles

Log (abs( gmax))

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Ghost footprint size (pixels) vs. Number of ghost angles, N

N, Number of zero-sum view angles

Ghost footprint (pixels)

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Ghost footprint size vs. Intensity range

Ghost footprint (pixels)

Log (abs (gmax))

The log of the minimum intensity range for iterated ghosts grows almost linearly with the size of the ghost’s footprint

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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:

last added

N gmax footprint vectors48 ±372 149 ±5:6, ±6:552 ±346 165 ±1:7, ±7:156 ±956 183 ±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

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

To be done: construct ghosts on a 3D hexagonal lattice, they may resemble snowflakes