Cultivos de coca en coca en la cuenca de Inambari – Tambopata

SMPVC was first designed as a voxel-based method. The concept was to extend the RBV correction by iteratively modifying region boundaries and calculating changes to regional variability based on estimates of PV-corrected voxel values. The technique operates as follows: first, RBV correction is performed using the initial MRI and PET data. Regional values for the mean and variance are calculated based on the RBV-corrected PET data. All voxels that lie on GM to GM boundaries are then subjected to analysis.

5.2.1.1 Label swapping

Each edge voxel is treated in turn and becomes thetarget voxel. The label of the target voxel is swapped for the label of the neighbouring voxel. Where there exists more than one neigh- bouring GM region, the assigned swap label becomes the label which is the mode of GM voxels in the 3×3×3 neighbourhood surrounding the target voxel. When a tie exists between

5.2. Algorithm development 125 candidate neighbours, a swap label is chosen at random.

Once the label has been swapped, the neighbourhood surrounding the target voxel is evaluated to ensure that the label change has not resulted in a disconnection of any GM re- gions. If a disconnection has occured, the label of the target voxel is returned to its original value and no further evaluation of that voxel is performed. A furtherconnectedness constraint is placed on the target voxel, which defines the number of voxels of a particular label that must exist in the 3 × 3 × 3 neighbourhood. The number of voxels belonging to the same label as the chosen swap label (m) is calculated. If m is less than the connectedness constraint, the original voxel label is restored and processing of that voxels is stopped.

The connectedness constraint is used to ensure that region surfaces remain intact, with the constraint being relaxed from 18 voxels down to 6 voxels as the algorithm iterates. The values for constraint were found experimentally. Strictly speaking, the initial value for the connectedness constraint of 18 could have been set to 25 (3 ×3×3 - target - neighbour). How- ever, this results in several iterations of the algorithm where very few voxels can be changed and is therefore computationally inefficient. The lower value of 6 was chosen as a minimum connectedness (m_{mi n}) to ensure that all voxels were at least6-connected to their region. This is to prevent regions ‘breaking-up’ as the algorithm iterates, although it restricts the changing of voxels in areas where the anatomy is very thin. Them_{mi n} value of 6 is therefore a trade-off between maintaining region integrity and allowing the flexibility to modify regions.

5.2.1.2 Voxel value estimation

All voxels that successfully meet the criteria for swapping in terms of their label are then evaluated according to an estimate of their PV-corrected PET value. To calculate the PV- corrected value of a voxel, it is necessary to perform PVC. However, to calculate a complete RBV correction in order to evaluate a single voxel (or set of voxels), over multiple iterations, is very computationally expensive. With thousands of edge voxels to evaluate, as is the case with a parcellated brain image, directly calculating the PV-corrected voxel value by performing a complete PVC becomes intractable. Therefore, an approximation of the voxel value, given its new label, is made. The calculation of the new value is performed using the estimates of the RBV-corrected regional mean values. The process is described below.

For the purposes of explanation, the equation for RBV correction (equation 3.1, page 56) is given here: f_C(x) ≈ f_O(x) – s(x) s(x) ⊗ h(x) ™ , (5.1)

5.2. Algorithm development 126 s(x) = X

i=1..N

[Tipi(x)].

where T_i is the regional mean value ofi calculated by the GTM, p_iis the mask of regioni, s is a piece-wise constant image of the regional mean values, h is the PSF of the scanner, f_O is the observed PET image and f_C is the corrected image. Equation 5.1 is then re-written as:

f_C(x) = f_O(x)– s(x) b(x)

™

, (5.2)

b(x) = s(x) ⊗ h(x).

For a voxelx, let u be the index of the voxel’s original label and v the index of the new label. The imagess and b are then updated with an approximation of their values if the membership of the target voxel had actually been swapped:

s(x)0 = s(x) + (T_v− Tu)δ(x), (5.3)

b(x)0 = b(x) + (T_v− Tu)δ(x) ⊗ h.

where T_u and T_v are the regional mean values calculated by the GTM. The approximated PV-corrected voxel value can then be computed by:

f_C0(x) ≈ f_O(x)– s 0_(x) b0(x) ™ (5.4) 5.2.1.3 Acceptance criterion

A label swap is accepted for a particular voxelx if the value f_C0(x) reduces the global variance over the set of regions. This is calculated using a region-based cost function, using the global variance before and after f_C0(x) is applied. The measure of global variance (GV) for an image

f is given by: GV(f ) = X i=1..N   σ2 i(f )/µi(f )
p_n i(f )   (5.5)

wherei is an index to the region label, N is the number of regions,σ_i2 is the variance,µi is

the mean andn_iis the total number of voxels belonging to regioni. The difference made to the global variance by swapping voxelx is then calculated by:

∆GV = GV (f0

5.2. Algorithm development 127 A negative value for∆GV indicates that swapping voxel x from label u to label v reduces the variance across the set of regions. Voxel changes that result in a negative∆GV value are therefore accepted as valid voxel changes. To reduce computation time required to calculate ∆GV , the mean, variance and size of the regions are computed using an online variance calculation [Knuth, 1998, p. 232]. This means that the region properties are updated as voxels are changed rather than being recomputed every time. However, the value of∆GV is likely to be very small, especially when the regions being modified are large and is sensitive to noise. The noise sensitivity will be discussed later in the section.

5.2.1.4 Iterative scheme

Once all edge voxels have been evaluated, a full RBV correction is performed using the modified mask. ∆GV is calculated between the original f_C and the modified image f_C0. If ∆GV < 0 then the changes to the mask are accepted and fC is set to equal fC0. The voxel eval-

uation process starts again based on the new corrected image and the new mask. If∆GV ≥ 0, the changes are rejected. In this case, the connectedness criterion is decremented by 1. The process is terminated if either the connectedness criterion becomes less than mmi n or 200

iterations have been performed. Once the stopping criterion has been met, a final RBV cor- rection is performed on the original PET image, using the current mask.

5.2.1.5 Preliminary evaluation

Initial tests were carried out by performing voxel-based SMPVC on digital phantom data. These data were generated using the image generator described in section 3.2.3 (page 67). A full description of the phantom dataset is given in section 5.3 as the same dataset was later used to evaluate the surface-based version of SMPVC. In brief, a parcellation error was introduced into the mask image used for the purposes of PV-correcting the PET.

Ten noise realisations of an AD-like[11C]PIB distribution were evaluated. For some realisations, the voxel-based SMPVC approach would correctly modify the region mask until it was very similar to the ground truth. The boundaries tended be ragged, but this was to be expected given that the correction is based on the changing individual pixels. However, in four of the ten realisations, the voxel-based technique failed to correct the boundary error. Of these, three exhibited a ‘tearing’ where two regions move against each other, resulting in long strands at the region boundary. The tearing can be seen in figure 5.1.

The other realisation that failed to correct properly was due to an infinite loop caused by a pattern of voxel changes that would repeat until the maximum number of iterations was exceeded. Both the tearing and looping can be attributed to voxel noise sensitivity. The same phantom was tested without noise (other than the resolution blurring) and these effects were

5.2. Algorithm development 128

Error Truth SMPVC

Figure 5.1: An example of tearing observed when applying the voxel-based SMPVC. Images are of the parcellated brain mask. The mask error (left), the ground truth (centre) and result of voxel-based SMPVC (right) are shown. The frontal region is shown in green, central region in purple, WM in red and parietal in yellow.

not observed. It was felt that modifying region surfaces, rather than individual voxels along the surface, would be less sensitive to noise. This led to the development of the surface-based approach detailed in the next section.