Imaging tests o f 2D:3D mismatch compensation using difference data

Assuming there is no structure out of the plane, a 2D mesh can be used to reconstruct

difference measurements between a homogenous reference phantom and the object of interest

if we assume that [ 2.3.4 ] holds. The effectiveness of the 2D:3D compensation should be

equivalent to the correction applied to the absolute images shown in Figure 2.3.4. However in

addition, other model errors and systematic errors on data should be better accounted for in

difference measurements.

In addition to the data acquired on the 3D Multi-level phantom used for the absolute

images above data were also acquired, using exactly the same acquisition protocol, on a

homogenous cylindrical phantom with similar but not identical background optical properties.

‘Raw’ mean-time data were extracted from the TPSFs collected on both phantoms, without

calibration measurements being applied (in fact the calibration measurements were utilised,

but purely to determine the optimal window over which to calculate datatypes see section

2.1.3.2.3). The raw data were read-in to TOAST as ‘FILE’ and ‘REFFILE’, effectively

providing TOAST with data in the form of [ 2.3.3 ]. Figure 2.3.5 shows the images produced

Ë. M. C. H illm an . PhD thesis 2002 Chapter 2 . 3 — 12 5

by T O A S T (the 11th iterations) w hen d ata are reconstructed (top) on a 2D m esh and (b o tto m ) u sing a cylindrical 3D m esh. T he 3D im age show n rep resen ts the reco n stru cted values in the plane o f the fibre holder. For the 3D im age, 11 iterations required 23 hours on a 700 M H z Pentium III co m p u ter. T he 2D im ages took 6 m inutes to perform 11 iterations.

Pa p's Data: ' ---' 5xp, / 5xp\ _ 2D reconstruction 0 .0 0 9 9 m m ' 0 .0 1 H m m ' 0 .7 5 2 m m ' 1 .0 8 3 m m ' 3D reconstruction 0 .(X )9 9 m m '' 0 .0 1 1 1 m m '0 .7 3 1 m n V ' 1 .1 3 9 m m

Figure 2.3.5 Images produ ced from the difference in mean-tim e betw een the 3D M u lti-level phantom and the homogenous phantom, (top) im ages reconstructed using a sim ple 2D mesh, (bottom ) the .same data reconstru cted

on a correct 3D cylindrical mesh.

2 . 3 . 1 . 3 . 1 I m a g e s u m m a r y ( d i f f e r e n c e )

T he 2D :3D com parison in F igure 2.3.5 was published in (H illm an et al, 2001c)

(subm itted Jan 2001 ).

Phantom 3D Multi-level - Homogenous 3D M ulti-level - Homogenous

Mesh Circular 3781 nodes 7392 linear

elements (2D)

(3D) Cylindrical 140 mm tall, 31525 nodes. 21600 quadratic elements. (24 layers, 950 nodes in each)

Basis Pixel 24x24 Pixel 16x16x5

Starting parameters: Pa 0.01 m m ' 0.01 m m '

Starting parameters: p% 1 mm ' 1 m m '

Iteration 11 1 1

Sources 32 32

Detectors per source 22 22

Datatypes (raw) Mean (raw) Mean

Calibration Difference (homog reference) Difference (homog reference)

2d 3d correction applied? No No

Simultaneous Pa and p's Pa and p's

Acquisition time (per source) 10 secs 10 secs

Wavelength 8(X) nm 800 nm

Table 2.3.2 P roperties o f itnages shosvn in Figure 2.3.5

2.3.1.4 Discussion of 2D:3D corrections

C learly the ad-hoc 2D :3D correction applied to ab so lu te d ata red u ces the artefacts relating to 2D :3D m ism atch seen in u n co rrected im ages (F ig u re 2.3.4). H ow ever, m anipulating d ataty p es prior to im age reco n stru ctio n m ay have rep ercu ssio n s on im age accuracy and con v erg en ce particularly if d atatypes are not necessarily q u an titativ ely co rrect anyw ay (e.g. variance - see section 2.2.6.2.1 ).

E. M . C . H illm a n . P h D th e sis 2 0 0 2 _____________________________________________________________________ C h a p te r 2 . 3 — 1 2 6

Difference imaging was quite successful at producing 2D images from measured data

(Figure 2.3.5). Note however that whether absolute corrections or difference data are used to

compensate for 2D:3D mismatches, both are strictly limited in usefulness since they require

the regions above and below the plane of interest to be homogenous. As a method of re­

constructing clinical data, particularly brain data, using a 2D model is likely to yield

unacceptable results regardless of attempts to perform a 2D:3D correction. Arm images

reconstructed from data where a 2D-3D correction has been applied are shown in chapter 2.5.

The 2D and 3D difference images shown in Figure 2.3.5 agree remarkably well with

each-other, both qualitatively and (especially for Pa) quantitatively. The likely reasons why

this method has been so effective are:

• that there is no structure out of the plane (within the areas probed by the PMDFs),

• that systematic and modelling errors have been very well cancelled out by using

difference data (as suggested in section 2.1.2.3).

• that the effect of the 3D propagation of photons through the homogenous layers above

and below the plane of interest has somehow cancelled, since it is approximately the same

for both data sets despite the slight differences in the optical properties of the phantoms.

The quantitative solution is far from the target. However, low iterations are shown, and

peak Pa and p% values do not necessarily indicate the quantitative accuracy of an image, since

any blurring will always reduce the peak value in the image. These effects are explored in

appendix A (2.9.8 ) and chapter 2.5.

The fact that the difference imaging results for 2D and 3D imaging (shown in Figure

2.3.5) were so similar suggested that subsequent phantom images could be reconstructed on

both 2D and 3D meshes. In general 2D reconstructions are used for speed and evaluating

image-data quality. Convergence of 2D reconstructions is also often better than for 3D. This

might be due to the fact that the number of unknowns (given by the number of nodes in the

mesh) is a lot smaller for 2D problems. Comparison between 2D and 3D results for a number

o f phantoms is presented below.

2 .3 .2 Multi-level phantom 2D and 3D im age reconstructions

Figure 2.3.1 showed plots of data acquired at 14 levels on the multi-level phantom

(which contains 3 inclusions at different heights). Data were also acquired on a homogenous

reference phantom. These data can be reconstructed in a number of ways:

E. M . C . H illm a n . P h D th esis 2 0 0 2 ______________________________________________________________________ C h a p te r 2 . 3 — 1 2 7

• 2D:3D corrected absolute data can be used to reconstruct 14 images, each

representing a 2D slice through the phantom (or use difference data relative to the

homogeneous phantom to compensate for 2D:3D mismatches).

• All of the absolute data (or a subset) can be reconstructed on a cylindrical 3D mesh.

• Difference data corresponding to each level subtracted from the homogenous

phantom measurement can be reconstructed on a 3D cylindrical mesh.