Adrien Le Pogam

Mathieu Hatt
email: <hatt@univ-brest.fr

Patrice Descourt

Nicolas Boussion

Charalampos Tsoumpas

Federico E Turkheimer

Caroline Prunier-Aesch

Jean-Louis Baulieu

Denis Guilloteau

Dimitris Visvikis

Adrien Le Pogam

Pet Methodology

Evaluation of a 3D local multiresolution algorithm for the correction of partial volume effects in positron

Keywords: Algorithms, Artifacts, Brain, radionuclide imaging, Humans, Imaging, Three-Dimensional, methods, Linear Models, Positron-Emission Tomography, methods, Reproducibility of Results, Whole Body Imaging emission tomography, partial volume effects, resolution and intensity recovery, wavelet transform, multi-modality

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INTRODUCTION

Partial volume effects (PVE) refers to two distinct phenomena leading in under-or over-estimation of the tissues uptake. The first results from the limited spatial resolution of the imaging device, leading to a 3-dimensional (3D) blurring and a loss of signal in tissues of size similar to the system s point spread function (PSF) Full Width at Half Maximum (FWHM), as well as activity cross-contamination (

' "
spillover ) between structures with different uptakes . The second phenomenon arises from the discrete representation on a grid with " [START_REF] Aston | Positron emission tomography partial volume correction: estimation and algorithms[END_REF] voxel sizes from 1 to 5 mm for the reconstruction of images. The voxel values at the edges are consequently a mixture of different tissues, an effect commonly known as tissue fraction effect . This effect exists on all image modalities but its magnitude is lower on anatomical " " datasets such as MRI or CT, and the introduction of higher resolution details from anatomical images in the functional images could reduce its impact. In this study we considered both the spatial resolution of the scanner and the voxel grid sampling, which are specific to the low resolution images obtained with PET.

Most of the previously proposed approaches for PVE correction consist in using anatomical information provided by either a priori computed tomography (CT) or magnetic resonance (MR) images. Although these methodologies usually aim at the recovery of accurate uptakes in specific regions of interest (ROIs) requiring either a co-registered atlas or a segmentation step, such as the ROI based [1 ȓ4 ] geometric transfer matrix (GTM) by Rousset et al , some voxel-based implementations have been recently proposed. They could be [START_REF] Rousset | Correction for partial volume effects in PET: principle and validation[END_REF] classified in two groups; namely deconvolution without the use of anatomical information or voxel-wise correction based on the use [5 ȓ8 ] of anatomical and functional images available from multi-modality devices .

[9 ȓ11 ] Teo considered the use of iterative deconvolution restricted to a ROI to avoid noise increase in the overall image, while a et al. [START_REF] Teo | Partial-volume correction in PET: validation of an iterative postreconstruction method with phantom and patient data[END_REF] more recent study applied iterative deconvolution to brain images . Boussion proposed the use of wavelet-based denoising at [START_REF] Tohka | Deconvolution-based partial volume correction in Raclopride-PET and Monte Carlo comparison to MR-based method[END_REF] et al. [START_REF] Boussion | Incorporation of wavelet-based denoising in iterative deconvolution for partial volume correction in whole-body PET imaging[END_REF] the iterative level of the deconvolution process to avoid noise amplification. The efficacy of this denoising greatly depends on the choice of the wavelet filtering algorithm and associated threshold values. Kirov suggested the use of regularized iterative deconvolution et al. [START_REF] Kirov | Partial volume effect correction in PET using regularized iterative deconvolution with variance control based on local topology[END_REF] with variance control based on local topology as an alternative solution for noise reduction. This latter approach is not fully automatic because the determination and optimisation of the regularization parameters are dependent on the properties of the image.

One methodology based on the use of anatomical information to correct functional data was proposed by Boussion and is et al. [START_REF] Boussion | A multiresolution image based approach for correction of partial volume effects in emission tomography[END_REF] referred to as mutual multi-resolution analysis (MMA). It uses discrete wavelet transforms of a low resolution PET image L and a [START_REF] Dutilleux | An implementation of the algorithme a trous to compute the wavelet transform[END_REF] corresponding co-registered high resolution anatomical MRI (or CT) image H. The method extracts the spatial frequencies like details, edges, and textures from wavelet decompositions at a level of resolution common to H and L (a specific decomposition layer in which both wavelet images have the same PSF FWHM). A global linear model is then inferred to build the lacking details of L from these found in H.

The method demonstrated accurate quantitative correction comparable with the methodologies considered as the current state of the art[2 ȓ but limited to ROI analyses. This approach has the advantage of generating PVE corrected images, allowing for an accurate activity 4 ] recovery, without any segmentation or other pre-processing steps. Despite these advantages, the original MMA approach suffers from two limitations. First, it is based on a global correlation between anatomical and functional structures, as a linear and global link is assumed in order to model the relation between the wavelet coefficients of both modality transforms at the same level of resolution. Consequently, where there is little to no correlation between these structures, artifacts may appear in the corrected emission image as a result of the incorporation of anatomical structures with no functional significance. An alternative approach based on MMA has been recently published restricted to the brain domain and making use of [START_REF] Shidahara | Functional and structural synergy for resolution recovery and partial volume correction in brain PET[END_REF] atlases for the provision of the anatomical information in an attempt to reduce the impact of functional and anatomical image mismatches.

However such approach is dependent on the use of atlases and is therefore only applicable to brain imaging. The second limitation of the original MMA algorithm is due to the use of a 2D modeling as the correction is applied independently slice by slice, whereas PVE is a 3D effect for which it is important to consider interactions in all three spatial directions.

The goal of the present study was to design a new model in order to overcome the shortcomings associated with the original MMA algorithm as highlighted above. More specifically, a new 3D wavelet decomposition scheme was designed and the global linear relation model was replaced with an improved local analysis, in order to process limited image areas at a time and adapt this model to each image part based on local information. This new approach was evaluated on synthetic and simulated images, and applied to brain and whole body patient datasets, and its performance was compared to the results obtained with both the original 2D global MMA algorithm described by Boussion and the GTM reference methodology . et al. [START_REF] Boussion | A multiresolution image based approach for correction of partial volume effects in emission tomography[END_REF] [2 ]

MATERIALS AND METHODS

Multiresolution analysis in the wavelet domain

Performing multiresolution image analysis using wavelet transforms consists in analyzing details across different levels of resolution or scale , in order to extract consecutive layers of details from large structures to small edges, by separating the spatial frequencies they [START_REF] Daubechies | The wavelet transform, time-frequency localization and signal analysis[END_REF] contain. Among the many algorithms developed to perform wavelet transform of an image, the most common approach is the multi-resolution pyramidal methodology consisting in iteratively reducing the resolution of the image. Such sub-sampling may cause a [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF] loss of linear continuity in spatial features such as edges and the appearance of artifacts in those structures . Therefore, the undecimated [START_REF] Yocky | Artifacts in wavelet image merging[END_REF] algorithms are often more appropriate, for instance for the image fusion purpose . The trous ( with holes ) algorithm was used here [START_REF] Krista | Wavelet based image fusion techniques -An introduction, review and comparison[END_REF] "à " " "

and extended to 3D to perform the wavelet decomposition based on the initial work of Boussion . This algorithm presents [START_REF] Dutilleux | An implementation of the algorithme a trous to compute the wavelet transform[END_REF] et al. [START_REF] Boussion | A multiresolution image based approach for correction of partial volume effects in emission tomography[END_REF] several advantages such as a straightforward implementation, a reconstruction without any loss of information and an isotropic process (i.e. no specific directions selected). This algorithm is related to the standard discrete wavelet transform decomposition scheme as it [START_REF] Shensa | The discrete wavelet transform: wedding the a trous and Mallat algorithms[END_REF] performs the sub-sampling of the filtered image by up-sampling the low-pass filter; inserting zeros between each of the filter s coefficient ' at each level. The detail coefficients images w are then obtained as the difference I I between the low-pass filtered images from

{ j } { j 1 + -j }
two consecutive levels. At each iteration j, the resolution of the image I is reduced to obtain the smoothed image I (called residual) j j 1 + using:

(1

)
where h is a 3D low-pass filter defined by:

(2)

with the convolution operator and h a binomial filter of order 4.

* 1D [START_REF] Starck | The undecimated wavelet decomposition and its reconstruction[END_REF] The inverse transform can be computed by adding the detail layers w from all levels to the final low-resolution image I :

{ j } J (3)
where, J is the number of iterations from the initial image I to the final approximation I . algorithm to an initial point source of different sizes (from 0.6 to 1.4 mm) using the Mathematica software. This relationship was found to be linear (see ). In order to change the voxel size of the initial image I and to obtain a residual I with a specific spatial resolution table 1 0 j according to the Mathematica software analysis, trilinear interpolation and 3 order B-spline resampling were used for anatomical and rd functional images respectively.

Using both and this resampling, the spatial frequencies w of the anatomical and functional images are extracted in 3D equation 3

{ j }
and the spatial resolution of the final residual I is accurately determined. We define the spatial resolution of the anatomical image H as q, J (initial image H referred to as H ) and that of the functional image L as r q p, (initial image L referred to as L ), with r>q and p the q = + r number of decompositions that have to be performed to reach a common level of resolution between the two modalities. This number is deduced from the PSF FWHM and the voxel sizes of both initial images. We can then perform the extraction of the spatial frequencies at a level of resolution common to H and L (q p 1), using the trous algorithm:

+ + "à " (4)
(5)

where, W and W are the wavelet coefficients from the wavelet decomposition of the functional and anatomical images respectively.

L H

Local mutual multiresolution analysis

Similarly to previously developed methodologies using wavelets for PVE correction , , our method is based on the exploitation of [9 11 ] existing correlations in the wavelet domain. Such correlation is defined by establishing a relationship between the anatomical and functional wavelet coefficients at a common level of resolution, independently of the original images content (in terms of units). In our proposed approach, we assume that the residual L can be estimated by a space dependent scaling of H , for which the scaling factor can be the original functional image value (L) the wavelet coefficient. Each voxel is processed individually however the parameter is computed α for each voxel using a 3D sliding cube, simultaneously applied within the L and H wavelet layers to obtain the MRM ( ). eq.7 [START_REF] Boussion | Incorporation of wavelet-based denoising in iterative deconvolution for partial volume correction in whole-body PET imaging[END_REF] where WIND is a window in 3D (i.e. a cube) centered on (x,y,z) with a fixed size of 3 3 3 voxels (i 1 3) and T is a threshold

× × = …
operator. therefore computes for each voxel of coordinates x, y, z, as the median (within a cube centered on this voxel) of the Equation 7α ratios between the wavelets coefficients of each transforms. The cubic moving window was introduced instead of a simple voxel by-voxel ratio to account for local variations in the anatomical and functional images, contrary to the global model and to reduce the noise and [START_REF] Boussion | A multiresolution image based approach for correction of partial volume effects in emission tomography[END_REF] mis-registration sensitivity associated with a voxel by-voxel analysis. The choice of using the median instead of the mean within the cubic sliding window, as well as the actual size of the cube was made based on the results obtained on simulations (see results section). Different sizes (3, 5 and 7 voxels) were compared since this parameter might have an impact on the sensitivity of the method to spatial mis-registration, noise and inappropriate choice of the respective PSF FWHM values. Also, results obtained using the mean or the median of the ratios were compared. The mean was expected to be more sensitive to artefacts and noise in contrast to the median, which would tend to discard extreme values. Also, the use of a sliding cube should also make the approach more robust to spatial mis-registration or inaccurate PSF FWHM values used in the process. On the one hand, the ratio map allows a large amount of uncorrelated details such as anatomical structures without significant uptake in the functional image to be discarded. On the other hand, if no structural information is associated with a significant uptake in the corresponding functional image, the low values of the wavelet coefficients may lead to MRM evaluation errors with denominator values close to zero. In order to avoid extreme values that may be generated by the MRM and preserve the activity of the functional image in such configurations, we introduced a fixed threshold T ( ) on the anatomical equation 7 wavelet coefficients. Its value was empirically chosen as 0.1 as it gave satisfactory results in most considered cases, however future studies should investigate the automatic estimation of an optimal value for each case. Therefore an investigation regarding the cube optimal size was carried out in this study.

Validation and comparison study

Analysis

Mean and associated standard deviation were computed on ROI placed on the organs or objects of interest in order to quantify the partial volume effect correction, and compared to the ground-truth when available (synthetic images and simulated datasets). In such cases (for example the spheres), the ROI were defined on the ground-truth, covering the entire structure voxel-by-boxel, and were reported to both uncorrected and corrected images. For clinical datasets, as ground-truth was not available, the improvement between uncorrected and corrected image was reported for ROI placed on both images.

Synthetic images

The proposed algorithm was first validated using synthetic images. All the images considered in this section were generated in 3D and analyzed either on a 2D basis using the original MMA method, or in 3D using the proposed approach. Finally, the GTM methodology was also applied, assuming a perfect knowledge of the ground-truth for the definition of the necessary ROIs in order to eliminate any potential errors that can be associated with a segmentation step. Two different synthetic images were employed, the first one H ( ) to ref figure 1-(

ROI

. The noise intensity (SD value) used was determined through different ROI analysis in the lung and liver from various whole [START_REF] Hatt | A fuzzy locally adaptive Bayesian segmentation approach for volume determination in PET[END_REF] body clinical datasets.

Different combinations of the 3D synthetic images in were considered to compare the performance of the different approaches figure 1 considered. First, functional and anatomical images with complete structural and intensity correlation ( and figure 1-(A)-( 2) figure

) were used to specifically study the accuracy of the correction for spillover effects due to the various contrasts. A second 1-(A)-( 3) combination was analyzed ( and ) in order to examine the value recovery of small objects. These two figure 1

-(B)-(2) figure 1-(B)-(3)
configurations were designed to validate the performance of the local approach for cases where the global approach already leads to satisfactory results, i.e. with a perfect match (structure and intensity) between both modalities. A third combination was used to investigate partial correlation between the two modalities ( ) for which the 2D global MMA is expected to perform with figure 1-(C) (2) and ( 3) reduced qualitative and quantitative accuracy. In each case, the new local 3D MMA approach was compared to the previous 2D global model and to the GTM method.

Finally, tests similar to those already used for the 2D MMA were carried out to evaluate the impact of noise and possible [START_REF] Boussion | A multiresolution image based approach for correction of partial volume effects in emission tomography[END_REF] mis-registration errors, or an inappropriate choice of FWHM parameters. For this purpose, variable noise intensities (SD from 10 to 50 ) % were added to the functional images of . The robustness of the approach against mis-registration errors was evaluated by figure 1

-(C)-(2)
applying different rigid transformations (translation up to 4 voxels (about 4mm), rotation up to 5 ) or inappropriate scaling [START_REF] Dutilleux | An implementation of the algorithme a trous to compute the wavelet transform[END_REF] to the °± % functional image of . The impact of an inappropriate choice of FWHM parameters for the resampling was investigated by figure 1

-(C)-(2)
generating different L images from the H ( ) with FWHM PSF from 4 to 8 mm. These L images were then corrected for ref figure 1-(C)-(1) PVE considering a unique FWHM value of 6 mm.

Simulated images

In the second part of this study, simulated images generated using a segmented brain phantom based on measured T1 MRI images [START_REF] Zubal | Computerized three-dimensional segmented human anatomy[END_REF] were analyzed. The images are a T1-weighted MRI and an associated F-FDG PET. The functional image was generated using the 18 following procedure . Clinically measured plasma time activity curves (TACs) were first used to generate a set of TACs for each [START_REF] Tsoumpas | Study of direct and indirect parametric estimation methods of linear models in dynamic positron emission tomography[END_REF] anatomical region of the brain phantom according to 28 different clinical dynamic frames (1 30sec, 1 15sec, 1 5sec, 4 10sec, 4 30sec, 4

× × × × ×
60sec, 4 120sec, 9 300sec). Pathological parameters were introduced in the parietal and the anterior frontal lobes. The dynamic images

× × ×
were then forward-projected using the maximum ring difference, mash and span as for the patient study, forming projection data of the ECAT HR scanner (spatial resolution of 4.8mm and 5.6mm FWHM transaxially and axially respectively). These projection data were GTM for comparison. The impact of the 3D analysis was observed on this dataset in which voxels are anisotropic.

The second dataset is a whole-body F-FDG-PET/CT image of a lung cancer patient (GE Discovery STE 4-slice PET/CT), acquired PVE corrected PET images as a result of prominent anatomical structures (such as bones) in the CT images which are not present in the FDG PET images. Since no ground-truth is available in this case, anatomical images were semi-automatically segmented to generate ROIs for the tumor, the lungs, soft tissues and bones in order to be used in the GTM method.

RESULTS

Synthetic images

The quantitative results regarding the correction using the three methods can be found in for the three different synthetic figure 1 (D-F) datasets.

contains results of the spillover effects correction for the spheres of constant size and varying intensities ( Figure 1-(D) fig. 1-(A) ). It displays the percentage of recovered intensity (a value of 100 is a perfect recovery of the true activity) in the spheres, demonstrating similar levels of recovery for the three methods, within 2 for all of the different intensities considered. shows quantification

% Figure 1-(E)
results for the constant intensity and variable size spheres ( ). The recovered intensities in the spheres demonstrate that both figure 1-(B) approaches perform with similar accuracy. As the functional image is resampled with the voxel size of the anatomical image as a first step for both methodologies, the impact of tissue-fraction effect is reduced by introducing the higher resolution details of the anatomic imaging when available and correlated, achieving accurate intensity recovery even for the smallest spheres where PVE have the most significant impact.

show the results for the uncorrelated case in which the spheres are different in the functional and Figures 1-(C)-( 4) and - [START_REF] Teo | Partial-volume correction in PET: validation of an iterative postreconstruction method with phantom and patient data[END_REF] anatomical images in terms of both intensity and structure, revealing differences between the 2D global and new 3D local approaches as the new approach does not incorporate uncorrelated anatomical details in the functional image during the correction, whereas the 2D global MMA creates local artifacts related to existing mismatches. In addition, the local approach handles more accurately the differences in absolute signal intensity between the anatomical and functional images. A quantitative comparison of recovered intensities in the spheres is shown in and demonstrates much higher accuracy for the new approach: without correction, the mean error was figure 1-( The GTM method resulted in a mean error of 3.2 . The standard deviation of each measurement associated with the use of the 2D global % MAA approach was much larger than when applying the new model. This is explained by the artifacts that are introduced due to mismatch in structural information between anatomical and functional datasets that the 2D global approach is not able to address. Such a case highlights the limitation of the 2D global MMA and the way the new 3D approach successfully overcomes such issues, with similar accuracy to GTM using the ground-truth ROIs. The measurements in sphere 7 illustrate the fact that no PVE correction can be performed due to the lack of corresponding structure in the anatomical image. It is important however to emphasize that in such a case, the functional activity in the corrected image remains unchanged and no additional artifact is introduced.

The local approach appeared more robust with respect to the increase of noise in the functional image in ): mean figure 1

(C)-(2)
recovered intensity across all structures (excluding sphere 7) and the different noise levels was 99.5 1.1 for the 3D local whereas the ± % 2D global results exhibited much higher standard deviation with a mean recovery intensity of 97.8 27.8 as illustrated in .

± % figure 2-(A)
The impact of an inappropriate choice of FWHM value and of a spatial misalignment between the anatomical and functional images was investigated for the 3D local MMA only as such an analysis has already been performed by Boussion for the 2D global et al. [START_REF] Aston | Positron emission tomography partial volume correction: estimation and algorithms[END_REF] approach only. This analysis demonstrated overall satisfactory robustness with however recovery errors reaching 25 to 50 for certain % spheres although the spatial misalignments and rotations considered were smaller (up to 3mm and 3 respectively). For the present study of °the 3D local MMA robustness, a maximum mean error of 1.9

23.1 was reached with the investigated misalignment, rotations and % ± % inadequate scaling parameters ( ). Regarding the inadequate choice of FHWM parameters, the mean error was 9.4 5.5 and figure 2-(B) ± %

2.8 3 for 2mm and 1mm around the actual exact value ( ).

± % ± ± figure 2-(C)

Simulated images

illustrate the correction obtained using the mean and the median of the ratios respectively (use to establish the Figures 3-(B) and(C) factor ), on the simulated F-FDG brain PET and associated T1-weighted MRI images of . The impact of the cube size (3 3

α 18 fig. 3-(A) × ×
3, 5 5 5 and 7 7 7 voxels in ( 1), ( 2) and ( 3) respectively) is also illustrated. Irrespectively of the cube size, the mean of the ratio led to

× × × ×
major artifacts due to noise and the possible approximations in the FWHM parameters. On the contrary, the use of the median led to better visual result. Only small visual differences (mostly a slight increasing blurring effect) were observed with the three different cube sizes (

). This blurring effect can be explained by the inclusion of additional voxels for the computation of the figure 3-(

C)-(1) to 3-(C)-(3)
median value and a less efficient local modeling. The quantitative analysis as shown in demonstrated higher activity recovery figure 3-(D) when using the smallest cubic window size (3 3 3). All other results were therefore generated with this setting.

× ×

shows the results obtained on the simulated F-FDG brain PET images. The global 2D MMA led to the incorporation of Figure 4-(A) 18 all the MRI details into the corrected PET images ( ), creating artifacts such as uptake corresponding to the skull, whereas figure 4-(A)-(3) the image corrected with the new approach ( ) was free of such artifacts. Further evaluation using a frontal region profile is figure 4-(A)-(4) presented in , demonstrating higher contrast with the local approach. Both grey and white matters are better delineated. The figure 4-( (1 5)) demonstrated higher accuracy of the correction with the figure 4-(C)new approach. In some regions such as amygdala, cerebellum or thalamus, the 2D global MMA and the GTM approach accurately corrected the intensities. However, in other regions such as the frontal or the hippocampus region for instance, it led to an over-or under-estimation of the uptake respectively. Mean error for all analyzed regions was 31.9 8.5 , 21.3 6.8 , 16.7 5.7 and 8.9 2.7

± % ± % % ± % ±
for the non corrected PET, 2D global MMA, GTM and 3D local MMA respectively.

%

Clinical images

As a last evaluation step the new approach was applied and compared to the global MMA and GTM on two clinical cases: a brain and a whole-body acquisition ( and respectively). Despite the good correlation between the T1 MRI ( ) and FDG PET ( fig. 5 6 fig. 5-(A)-( 1) fig.

) images regarding the grey and white matter, non-correlated structures also exist, such as the skull (see red arrows) and the 5-(A)- [START_REF] Rousset | Correction for partial volume effects in PET: principle and validation[END_REF] gadolinium enhancement (see white circle) in the MRI image. These MRI features, which do not match any FDG uptake, were introduced in the corrected image by the 2D global process ( ), whereas the 3D local approach ( ) suppressed these fig. 5-(A)-(3) fig. 5-(A)-( 4) uncorrelated details, leading to a more reliable and visually satisfying PVE correction. In addition, the new approach led to higher contrast between regions as shown in the profiles of . Furthermore, using the SPM software to segment the grey and white matter from figure 5-(B) the MRI images, we obtained quantitative results for the comparison of the two voxel-wise correction methodologies as well as corrected values using the GTM approach in these ROIs. The 2D global MMA led to higher standard deviation values due to the incorporation of additional uncorrelated details, with an average mean intensity variation with respect to the initial PET image in the grey and white matter regions of 1.8 21.1 and 0.2 31.6 respectively. In contrast, higher mean intensity variations and lower standard deviation were

± % ± %
obtained with the new approach: 11.4 6.5 in the grey matter and 2.6 8.3 in the white matter. The new approach therefore led % ± % -% ± %

to less noisy and higher uptake enhancements than the 2D global MMA. By comparison, the use of GTM led to 10 and 19 mean

+ % -%
activity changes for the grey and white matter ROIs respectively.

The whole-body F-FDG PET/CT image ( ) of a patient with lung cancer was analyzed in order to assess the potential of our new approach regarding oncology applications. contains the correction results using both approaches, Figure 6 demonstrating the incorporation of artifacts such as the spine with the global approach (

) whereas the new model allowed fig. 6-(

). Quantitative measurements in this bone region demonstrated an uptake increase of 28.8 25.4 with the 2D fig. 6-(4) ± %

global whereas the 3D local led to a much lower variation (1.9 11.3 ). contains the results of quantitative analysis using ROIs ± % Table 2 placed in the tumor and the lung, revealing an increase in lesion-to-lung ratio of 33.5 with the global approach, 45.8 with the GTM % % method and 54.1 with the 3D local approach. Furthermore, a variation in the whole lung activity concentration of 13.3 with the global % % approach was measured, whereas the new model leads to a smaller variation of only 6.7 thanks to the fact that it disregards anatomical % structures in the lungs without matching FDG uptake. A similar variation of 5.4 of the lung uptake was obtained with the GTM % approach.

DISCUSSION

Multi-modality PET/CT imaging is rapidly becoming the gold standard for diagnostic studies especially in oncology with F-FDG 18 PET/CT. PET/CT systems are now widely used in clinical practice thanks to the automatic fusion of functional and anatomical information they provide. Accurate and efficient PVC in this context might demonstrate significant clinical impact for instance for the detection of small lesions in whole-body PET acquisitions, or for the assessment of therapy response during or after treatment. In addition, new technology developments now allow the simultaneous acquisition of PET and MRI images for clinical brain studies and it is expected [START_REF] Schlemmer | Simultaneous MR/PET imaging of the human brain: feasibility study[END_REF] this will be also extended to whole-body imaging. These developments facilitate the use of anatomical information either during the reconstruction (attenuation correction , or incorporation of a-priori information , ) or as a post-processing step for the

[23 24 ] [ 25 26
] improvement of qualitative and quantitative accuracy of functional images (denoising or partial volume correction , ).

[27 ]

[

The recently introduced MMA methodology for partial volume correction is based on the mutual multi-resolution analysis of a [START_REF] Boussion | A multiresolution image based approach for correction of partial volume effects in emission tomography[END_REF] functional image and the corresponding anatomical one. In contrast to the standard PVE correction approaches using anatomical information , the MMA is voxel-wise and hence does not use ROIs obtained from a segmentation of the anatomical images. The [START_REF] Rousset | Correction for partial volume effects in PET: principle and validation[END_REF] algorithm was validated on synthetic and simulated datasets with accuracy similar to the reference GTM approach with the advantage of not requiring an atlas or a segmentation step as well as leading to PVE corrected images which are subsequently available for further analysis. There was however certain limitations associated with its 2D implementation and the use of a strictly linear and global model.

This model results in the incorporation of every anatomical detail into the functional image and can therefore lead to artifacts in the corrected images where no correlation exists between anatomical and functional details. For example, the methodology has been shown to work well with a combination of FDG PET and T1 MRI brain images , , but its performance with respect to receptor brain PET [9 28 ] imaging has not been demonstrated.

In this study, we have developed and evaluated a new model for MMA, based on a locally adaptive 3D analysis in order to address the limitations of the 2D global MMA previous implementation. An accurate co-registration is considered as a pre-requisite to both MMA and our new proposed approach, although we also demonstrated satisfactory robustness of the method to mis-registration errors. The implementation is based on the extension of the 2D analysis to the 3D and the introduction of a local model. The impact of each of the two modifications has not been evaluated individually. On the one hand, the 3D analysis provides a better representation of the PSF convolution which is a 3D phenomenon, allowing an accurate wavelet decomposition of the image compared to the 2D approach. The local analysis applied to this 3D decomposition allows computing a more accurate model which is subsequently used for the correction, compared to the simple global approach.

In addition, in order to discard uncorrelated details, the initial linear model coefficient was replaced by a new one based on the median value (rather than the mean) of the voxel-by-voxel ratios in a local cube centered on each voxel. Consequently, the 3D anatomical details in the wavelet domain are discarded in the regions where there is no matching functional uptake. The validation of the new model was performed on synthetic and simulated datasets. The results demonstrated similar quantitative performance with the 2D global approach and the GTM in the case of correlated datasets (

). It is worth noting that the GTM was applied using the available figures 1-(A), 1-(B) ground-truth for the definition of the ROIs, leading this way to the best possible results with this approach. In addition, the performance was significantly improved for non-correlated structural and functional image combinations ( , ). The new model was figures 1-(C) figure 4 also evaluated using clinical whole-body and brain datasets. For both these clinical datasets we obtained a significantly improved image quality without artefacts, as well as higher contrast improvements, compared to the 2D global MMA correction (

). However in figures 5 -6 the absence of ground-truth, the absolute accuracy of the recovered activity values and spatial spread of the structures of interest (such as small tumors) could not be directly assessed on these datasets.

Although this new approach allows improved qualitative and quantitative voxel-wise partial volume correction using anatomical information, without assuming homogeneous uptake in regions of interest as the GTM approach, and is applicable to both brain and oncology imaging, one should consider potential pitfalls associated with any postprocessing PVE correction approach based on anatomical information. As the correction is performed based on the anatomical image details, if no or not enough information is available for a specific structure, the PVE correction will be either not possible or incomplete. The new proposed approach is certainly able to handle mismatches between anatomical and functional information. However, in the case of tumour imaging, if the lesion is necrotic in the functional image but there is no corresponding necrosis in the anatomical structure, the correction will be incomplete as only the external outline of the lesion will be corrected. Absence of contrast in the anatomical dataset corresponding to a specific structure would lead to a lack of significant wavelet coefficients in the wavelet decomposition of such a structure, therefore no correction can be deduced and applied to the corresponding functional uptake. However, when a relationship exists between the anatomical signal and the functional one, such an issue may be overcome by using alternative acquisition protocols in order to generate such contrast. For instance, a contrast enhanced anatomical imaging or different sequences available in MRI imaging will be certainly interesting in enhancing the potential of PVC approaches such as our own for newly developed PET/MRI devices. Finally, one can identify the areas of the image where no significant PVE correction was applied as a result of lack of anatomical details by analyzing the MRM. This map indeed contains the correction that was applied to each voxel and very low or zero values correspond to little or no correction. It is also important to emphasize that the tissue fraction effect can be corrected only where the frontier between tissues can be extracted from anatomical images. This effect can in addition only be reduced, not entirely corrected, as it also exists in the anatomical datasets, although with a lower magnitude than in the functional images.

Another potential limitation of voxel-wise partial volume correction approaches based on the use of anatomical images may come from the dependency of the algorithm on the noise level in both the anatomical and functional images. However, our approach demonstrated high robustness versus noise. In addition, the local aspect of the correlation model greatly reduces the sensitivity of the correction to potential artefacts/distortions present in the anatomical images.

Mis-registration is also a limiting factor for multimodality PVE correction approaches such as the one reported in this work. However, current techniques allow fully automated 3D elastic image registration and can accurately align (with errors within 1 to 2 voxels)

[29 ȓ31 ]
whole-body PET and CT images acquired on standalone as well as on combined PET-CT scanners. Furthermore, although both the 2D and 3D approaches are sensitive to a spatial registration error between the anatomical and functional images, the different tests carried out on synthetic images with translation movements up to 4 voxels and rotation movements up to 5 , as well as inappropriate scaling of the °anatomical image showed limited impact with a maximum error of 1.9 23.1 thanks to the use of the sliding cube and the median of % ± % the ratios. However misregistration should be limited to a minimum in order to ensure the most accurate correction, since large translations and rotations might lead to errors up to 25 .

%

Finally, as with any post-processing PVE correction algorithm the new model requires the exact knowledge of both modalities spatial ' resolution (FHWM) and voxel sizes in order to determine the parameters of the wavelet decomposition scheme. However, our new methodology also demonstrated satisfactory robustness versus errors up to 2mm in the choice of the FWHM parameters. After images co-registration, the process is automatic and takes from about a minute to several minutes (depending on the size of the images) on a desktop computer.

j 1 +

 1 discrete convolution shown in , the spatial resolution (or PSF FWHM) of the residual image I depends on both equation 1 the spatial resolution of I and the size of its voxels. This is because the image I is sampled according to the iteration index j (i.e. the level j j of scale in the wavelet transform) and the convolution leads to a non linear dependency (factor 2 ) on the resulting PSF FWHM of the j residual I . The relationship between voxel sizes and the residual FWHM at different scales was determined by applying the trous

  local analysis to account for local differences between H and L. The corrected emission image Lq (same level of resolution as Hq) is therefore obtained using by adding the original uncorrected value of L of the anatomical wavelet coefficients weighted by a local factor : α (6) where stands for the Median of the Ratio Map (MRM). provides the corrected value for each voxel (x, y, z) by adding to α Equation 6

  1) generate the functional images L ( ), and the second one H ( ) used for the correction of PVE via both the 2D figure 1-(2) anat figure 1-(3) global and 3D local methodologies. A Gaussian noise (standard deviation (SD) 2 of the mean in the uniform part of the phantom) was % added in the H images prior to their use for the correction. These 128 128 128 (1 1 1 mm voxels) images contain a cylindrical anat a fixed intensity of 100 and spheres of different sizes and intensities. The first ( ) contains five figure 1-(A)-(1) 2cm diameter spheres with decreasing intensities (200, 120, 90, 70 and 50). The second one ( ) contains four spheres of figure 1-(B)-(1) decreasing diameter (6, 4, 2 and 1 cm) with intensity of 200. The last two images ( ) display spheres common figure 1-(C)-(1) and 1-(C)-(3)to both modalities with however no absolute intensity correlation, as well as two additional structures: one which is present only in the anatomical data H with no corresponding uptake in the functional image and respectively a hot spot in the functional data L without any anat corresponding anatomical structure in H . anat The L images ( ) (same voxel size and dimensions as for H ) were generated by convolving H Gaussian PSF and adding Gaussian noise (SD 10 of the mean intensity). In this first dataset, both H and L images had % anat a voxel size of 1mm . The Gaussian noise approximation is realistic enough for the reconstructed PET images when considering a specific 3

+

  attenuated using the values of the different tissue types contained in the Zubal phantom (muscle, bone, fat, skin), and uncorrected for normalization by applying the inverse normalization factors. These factors and the scatter additive term were both taken from the human study. The randoms contribution was simulated based on the system s detection efficiency factors pattern scaled to the total random events ' of each frame in the acquired human study. Finally, Poisson noise was added to the sinograms and the images were finally reconstructed with filtered backprojection including scatter, attenuation and normalization corrections. The image sizes for both PET and MRI were 128 128 64 (2.35 2.35 2.42 mm voxels). The final static PET image was eventually obtained by summing the last six temporal frames. The were 4.8 mm in plane and 5.6 mm axially for the PET images and 1 mm in all three dimensions for the anatomical datasets. The performance of the method was assessed by comparison with the known ground-truth of the simulation. This dataset was also used to determine the optimal sliding cube size among 3 3 3, 5 5 5 or 7 7 7 voxels (see results section).The approaches were also compared on two clinical images. The first one consists of a clinical T1 MRI (GE 1.5T) and associated FDG brain PET (Philips GEMINI dual slice PET/CT) scans. The MRI images were chosen instead of the CT for the correction to benefit from the improved contrast in the different brain structures. The MRI image contains a hyper-intensity signal in the left occipital lobe and the posterior cingulum due to the gadolinium injection. The PET reconstructed images (using RAMLA 3D (2 iterations, relaxation parameter of 0.05 and a 5mm FWHM 3D Gaussian post-filtering) and CT based attenuation correction) are 128 128 64 (1.41 1.41 2mm voxels) is 512 512 160 (0.47 0.47 1mm voxels). The PET and MRI images were spatially co-registered using mutual information transformations using MIPAV software (Center for Information Technology (CIT National Institutes of Health (NIH)). The FWHM considered was 1mm and 5mm in all three dimensions for the MRI and PET datasets respectively (considering the spatial resolution for the Philips GEMINI PET system of 5.2mm and 5.4mm FWHM transaxially and axially respectively in combination with the 5mm FWHM Gaussian post-filtering applied to the reconstructed image). Qualitative evaluation was carried out using profiles through the frontal and the tempo-occipital regions. Quantitative accuracy was evaluated by white and grey matter quantification (mean intensity and standard deviation) before and after PVE correction using an automatic delineation on the MRI image via the voxel-based morphometry segmentation tool of the SPM software . The same delineation results were used for the ROI based correction using[START_REF] Ashburner | Voxel-based morphometry--the methods[END_REF]

18 55

 18 minutes after injection of 355 MBq (CT: 80mA, 140kVp, PET: 3mins per axial field of view). PET images were reconstructed (voxel size of 4.68 4.68 3.27mm and a matrix size of 128 128 47 voxels) using OSEM (2 iterations, 28 subsets) and CT based attenuation× × 3 × ×correction. The resolution of the original CT image was 0.97 0.97 0.97mm voxels (matrix dimension 421 321 100). The FWHM of the× × 3 × ×PET image was considered as 6.1mm and 6.7mm in plane and axially respectively (matching the spatial resolution of GE Discovery STE), and 1 mm in all three directions for the CT images. Manually drawn spherical regions were placed on the lesion (ROI ) and in the lung lesion (ROI ). An additional ROI in the spinal region (ROI ) was used in order to evaluate the potential of introducing artefacts in the normal bone

  B) spikes on both sides of the profile (see red arrows in ), corresponding to the artifact uptake from the bone figures 4-(A)-(3) and 4-(B) incorporation using the global MMA approach, are absent from the corrected image using the new model. Comparison of ROI values placed in different regions of the brain against the true image (

18 fig. 6 -

 186 fig. 6-(1) and 6-(2)

Figure 1

 1 Figure 1 Three synthetic datasets for which the background region has a fixed value of 100. (A) a correlated case with spheres of constant size and decreasing intensities (from 200 to 50), (B) a correlated case with spheres of constant intensity (200) and decreasing sizes, and (C) an uncorrelated case. For each one of the three synthetic datasets: (1) a high-resolution anatomical image H used to generate the low-resolution ref functional image L (2), (3) the high-resolution anatomical image H used for the PVE correction, (4) the PVE corrected images using the 2D anat global approach and (5) images after PVE correction using the 3D local approach. Percentage of intensity recovery (mean SD) for the three ± datasets corrected for PVE using the GTM (mean only), 2D global or 3D local approach: (D) a correlated case with spheres of constant size and decreasing intensities, (E) a correlated case with spheres of constant intensity and decreasing sizes, and (F) an uncorrelated case. Spheres are numbered on figure 1-(1).

Figure 2 (

 2 Figure 2 (A) Percentage of mean recovered intensity relative to true values (mean SD) across all structures of (excluding sphere 7) ±

Figure 3 Figure 4 (

 34 Figure 3 Optimization and validation for the local modeling parameters on (A) a simulated FDG PET (A)-(2)/MRI (A)-(3) brain dataset (ground truth on figure (A)-(1)) using (B) the mean and (C) the median of the ratio maps methodologies for respectively a (1) 3 3 3, (2) 5 5 5 and (3) 7 7 × × × × × 7 3D cube. (D) intensity recovery in the different brain structures considered in the simulation following the PVE correction.× %

Figure 5 (

 5 Figure 5 (A) a clinical brain T1-weighted MRI/ F-FDG PET with injection of gadolinium contrast: (1) T1-weighted MRI, (2) non-corrected F-FDG 18 18 PET image, (3) 2D global, and (3) 3D local MMA based PVE corrected images. (B) Profile results across the frontal and occipital cortex regions on the uncorrected and PVE corrected PET brain images corresponding to the datasets shown in (A).

Figure 6 a

 6 Figure 6 a clinical whole body CT/ F-FDG PET study: (1) CT, (2) original non corrected PET, (3) PVE corrected images using the 2D global MMA 18 algorithm, and (4) PVE corrected images using the 3D local MMA approach.

  F)

							-
	11.4	% ±	5.2 whereas the 2D global and 3D local corrections resulted in a mean error of 15.1 % -	% ±	6.1 and 2.8 %	% ±	2.4 respectively. %

  Boussion N , Cheze-Le Rest C , Bizais Y , Visvikis D . Quantitative assessment by means of realistic simulated images and patient data of a new method for partial volume correction effects in brain PET . Journal of Nucl Med . 192P -2006 ; 29 . Shekhar R , Walimbe V , Raja S , Zagrodsky V , Kanvinde M , Wu G , Bybel B . Automated 3-dimensional elastic registration of whole-body PET and CT from separate or combined scanners . J Nucl Med . 46 : 1488 -1496 2005 ; 30 . Sorzano CO , Thevenaz P , Unser M . Elastic registration of biological images using vector-spline regularization . IEEE Trans Biomed Eng . 52 : 652 -663 2005 ; 31 . Frost & Sullivan . Emerging Technology Developments in Fusion Technology for Diagnostic Imaging . Technical Insights . 2005 ; 32 . Le Pogam A , Alessio AM , Kandel B , Kirov AS , Barrett O , Fernandez P , Cheze Le Rest C , Visvikis D . Comparison of PET reconstruction resolution recovery and post-reconstruction deconvolution for PET partial volume correction . J Nucl Med . 51 : S2 -577 -2010 ; 33 . Hoetjes NJ , van Velden FH , Hoekstra OS , Hoestra CJ , Krak NC , Lammertsma AA , Boellaard R . Partial volume correction strategies for quantitative FDG PET in oncology . Eur J Nucl Med Mol Imaging . 37 : 1679 -1687 2010 ;

	28 .
	10 14

Table 1

 1 FWHM values of the residual scales for an initial FWHM of 1mm and initial voxel sizes of 0.6, 1 and 1.4 mm. The equation formula provided by the software is presented.

	Voxel size of the initial image (mm)	0.60	1.00	1.40	x
	FWHM residual scale 1 (mm)	1.56	2.61	3.65	2.61 x *
	FWHM residual scale 2 (mm)	3.38	5.64	7.90	5.64 x *
	FWHM residual scale 3 (mm)	6.89	11.48	16.08	11.48 x

*

Table 2

 2 ROI quantification (mean SD of the uptake value (kBq/ml) in different ROIs) for the original whole body PET image and the corrected one using the GTM, the 2D global and the 3D local MMA ± approach. Note that there is no SD in the case of the GTM approach since it is a ROI based PVE correction approach.

	Activity (kBq/ml)	ROI lung	ROI lesion	ROI bone
	ORIGINAL PET	1.5 0.1 ±	8.9 1.4 ±	5.2 0.5 ±
	GTM	1.2	10.9	5.5
	2D GLOBAL	1.3 0.3 ±	10.3 1.6 ±	6.7 1.7 ±
	3D LOCAL	1.4 0.1

± 12.8 1.9 ± 5.3 0.6 ±

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

[START_REF] Yocky | Artifacts in wavelet image merging[END_REF]

 9 14We assumed a constant value of the PSF FWHM in the entire image, which is a simplification considering the potential variation of the PSF FWHM throughout the field of view, especially in the case of whole body imaging. This aspect could be improved by modeling the exact PSF FWHM in each direction according to the spatial position of the analyzed voxel. Finally, an automatic estimation of the threshold T value regarding the wavelets coefficients might improve the results on a case by case basis. Future studies will also investigate the performance of post-reconstruction processing approaches such as the one developed in this study with the incorporation of the PSF and other information into the reconstruction iterative algorithm. Within this context a couple of existing studies have shown a priori similar performance between the post-processing deconvolution and PSF incorporated reconstruction based PVE correction approaches [[ , , with the generic advantage of post-reconstruction approaches being reconstruction algorithm independent. 32 33 ]]

CONCLUSION

We developed an improved voxel-wise methodology to correct for partial volume effects in emission tomography. This new model overcomes limitations encountered in the originally proposed MMA and allows for a more universal approach that can potentially handle any combination of co-registered anatomical and functional images. Our new methodology extends the 2D MMA to a 3D local analysis, in which local details are conditionally taken into account in the correction process. PVE correction was successfully applied to images with either high correlation, for which the 2D MMA correction was already adequate, or more challenging cases for which correlation between anatomical and functional datasets was not complete and for which global 2D MMA failed by introducing artefacts and led to inaccurate quantification. The local 3D process was successfully tested and validated on synthetic, simulated and clinical datasets, with similar performance to the reference GTM method without requiring a segmentation step, producing PVE corrected images, and avoiding artifacts generated by the 2D global approach. In addition it proves to be overall more robust with good accuracy and robustness without particular requirements regarding the structural and intensity correlations between the anatomical and functional images.


Bernard Cornet

Abhishek Ranjan

A remark on arbitrage free prices in multi-period economy

Keywords: Financial exchange economy, arbitrage-free prices, equivalent financial structure JEL Classification: C02, D53

We study the convexity property of the set of arbitrage-free prices for a multi-period financial exchange economy. We provide sufficient conditions for the set of arbitragefree prices to be a convex cone, which includes 2-date model. Further we show that a financial exchange economy with the set of arbitrage-free prices neither convex nor cone can be equivalent to a financial exchange economy with the convex cone set of arbitrage-free prices.

Introduction

This paper is to show that the 2-date model is not sufficient to capture the time evolution of realistic models, as there are many properties which holds in 2 -date model but not in multi-period model. In this paper, we discuss the convexity of the set of arbitrage-free prices as many literature concerning the financial asset uses this concept. Some of the recent literature in 2-date economy shows the existence of equilibrium using the fact that the set of arbitrage-free prices is a convex cone. Unfortunately, this is not true in multiperiod economy making the existence of equilibrium difficult in a financial structure with 3 or more dates. We propose an alternative approach to work in a financial exchange economy with more than 2-dates. We call two financial structure to be equivalent if they have the same consumption equilibrium i.e. (x i , p) i∈I the list of consumption and commodity prices are same in the equilibria of the two exchange economy. Since, we know that utility of every individual i depends only on the consumption part and financial structure only helps in transferring the wealth across time and state, and hence we search for an equivalent financial structure with the set of arbitrage-free prices as a convex cone, and find an equilibrium in the equivalent financial structure. In Section 2, we describe the financial exchange model in T period and study convexity of arbitrage-free prices. In this section we also define the financial structure and the notion of arbitrage-free prices for the given financial structure. In Section 3, we study the properties for the set of arbitrage-free prices to be a convex cone, and also discuss the case when the set of arbitrage-free prices are neither convex nor a cone through examples and propositions. In this section, we also define equivalent financial structure, and give certain conditions under which the financial structures are equivalent. We also show the main result of the paper that, two financial structure can be equivalent even if one has convex cone set of arbitrage-free prices whereas other does not. We illustrate certain examples of the financial structures satisfying these results. We have left few proofs for the Appendix 4.

The model 2.1 Time and uncertainty in a multi-period model

We consider a multi-period financial exchange economy with (T + 1) dates, t ∈ T := {0, • • • , T }. The stochastic structure of the model is described by a finite-tree D of length T and we shall essentially use the same notation as in Debreu (4) and Martine and Quinzii (6) (we refer to Debreu (4) and Martine and Quinzii [START_REF] Magill | Theory of incomplete markets[END_REF] for an equivalent presentation with the information partition). The set D t denotes the set of states (also called date-events) that may occur at date t and the family (D t ) t∈T defines a partition of the set D. At each date t = T , there is a priori uncertainty about which state will prevail in the next date. There is a unique non-stochastic event occuring at date t = 0, which is denoted 0, so D 0 = {0}. Every state ξ = 0 has a unique immediate predecessor denoted ξ -. For each ξ ∈ D, we define ξ + = { ξ ∈ D : ξ = ξ-} as the set of the immediate successors of ξ, and we notice that the set ξ + is nonempty if and only if ξ ∈ D \ D T . Moreover, we define the set of the successors (not necessarily immediate) of ξ as

D + (ξ) = {ξ ∈ D : ∃(ξ 1 , ξ 2 , • • • , ξ k ), ξ k = ξ -, ξ k-1 = ξ - k , • • • , ξ 1 = ξ - 2 , ξ = ξ - 1 }.
We also use the notation

ξ > ξ (resp. ξ ≥ ξ) if ξ ∈ D + (ξ) (resp. ξ ∈ D + (ξ) ∪ {ξ}).

The financial structure

We consider a financial asset structure F with J = {1, • • • , J} assets and every asset j ∈ J is characterized by the couple (ξ j , V j ) ∈ D × R D , where ξ j is the emission node of asset j and V j ∈ R D is its payoff. We will adopt the convention that V j ξ , the payoff of asset j at node ξ, is defined for every ξ and V j ξ = 0 if ξ ∈ D + (ξ j ). The financial asset structure F can be summarized as F = (J , (ξ j , V j ) j∈J ).

We will also denote F = (J , (W j (•)) j∈J ) where the total payoff mapping W j (•) : R → R D is defined by W j (q j ) = V j -q j 1 ξ j1 , and we define W (•) := [W 1 (•), . . . , W J (•)], thus a mapping from R J to R D .

Arbitrage-free notion

We say that (F, q) is arbitrage-free if and only if there exist no portfolio z ∈ R J such that W F (q)z > 0, or equivalently:

W F (q)R J ∩ R D + = {0}.
The set of arbitrage-free prices is denoted by Q F . From the characterization theorem of no-arbitrage as in Angeloni and Cornet (1), we recall that q is arbitrage-free for the financial structure F if there exist λ ∈ R D++ satisfying W T F (q)λ = 0, that is for all j ∈ J , λ • W j F (q) = λ • [V j F -q j 1 ξ j ] = 0, and we denote Q F or Q as the set of arbitrage-free prices q of F.

For every λ ∈ R D++ and j ∈ J , we define

q j F (λ) := (1/(λ(ξ j ))V j F (p)
• λ and q F (λ) = (q 1 F (λ), . . . , q J F (λ)), and we notice that for every λ ∈ R D++ ,

W T F (q F (λ))λ = 0, i.e, λ • W j F (q F (λ)) = 0 for all j.
3 Convexity results and the equivalent financial structure

In this section, we study the important properties of convexity and conity for the set of arbitrage-free prices in the financial exchange economy. The set of arbitrage-free prices may neither be convex nor a cone in a multi-period financial exchange economy unlike 2-date economy where the set of arbitrage-free prices is always a convex cone. We have shown several examples where the set of arbitrage-free prices is neither convex nor a cone. Later, we define the concept of equivalent financial structure to study the financial exchange economy with the set of arbitrage-free prices neither convex nor a cone and show that this financial exchange economy may be equivalent to a financial exchange economy with the convex cone set of arbitrage-free prices.

A convexity result

We show certain results for the set of arbitrage-free prices to be a convex cone. We recall that V j ξ is defined as the payoff of the j th asset at node ξ and ξ j is the node at which asset j is issued.

Proposition 1. The set of arbitrage-free prices Q F is a convex cone if the financial struc- ture F satisfies V j ξ V k ξ 2 = 0 whenever ξ j = ξ k3 .
Now we recall that an asset is called short lived asset if it has non-zero payoff only at the immediate successors of the node at which it is issued i.e the financial asset j is said to be short lived if V j ξ = 0 for all ξ / ∈ (ξ j ) + .

Corollary 1. If the financial structure F has only short lived assets, then the set of arbitrage-free prices Q F is a convex cone.

We notice that all assets in a financial structure F with 2 dates will be short lived asset.

Hence we can deduce corollary 1 to the following corollary:

Corollary 2. The set of arbitrage-free prices Q F for a financial structure F with 2 dates is always a convex cone.

Now we give the proof of proposition 1.

Proof. We first prove that

Q F is cone, that is, for all q ∈ Q F and α ∈ (0, ∞), αq ∈ Q F . Since q ∈ Q F , there exists λ ∈ R D ++ satisfying λ(ξ j )q j = ξ∈D + (ξ j ) λ(ξ)V j ξ , ∀j ∈ J .
Now we define λ ∈ R D ++ as follows :

λ (ξ) =        1 if ξ = 0, α λ(ξ) λ(ξ j ) λ (ξ j ) if there exist j satisfying V j ξ = 0, α λ(ξ) λ(ξ -) λ (ξ -) otherwise.
We notice that λ (ξ j )αq j =

ξ∈D + (ξ j ) λ (ξ)V j ξ , ∀j ∈ J . Hence, αq ∈ Q F , implies Q F is a cone. We now prove that Q F is convex. Since Q F is a cone, It suffices to prove that q + r ∈ Q F , whenever q, r in Q F . Indeed, let q ∈ Q F , r ∈ Q F , then there exist µ ∈ R D ++ , ν ∈ R D ++ such that q j = ξ∈D + (ξ j ) µ(ξ) µ(ξ j ) V j ξ and r j = ξ∈D + (ξ j ) ν(ξ) ν(ξ j ) V j ξ , ∀j ∈ J . Therefore, q j + r j = ξ∈D + (ξ j ) ( µ(ξ) µ(ξ j ) + ν(ξ) ν(ξ j ) )V j ξ .
Now we define λ ∈ R D ++ as follows :

λ(ξ) =        1 if ξ = 0, ( µ(ξ) µ(ξ j ) + ν(ξ) ν(ξ j ) )λ(ξ j ) if there exist j satisfying V j ξ = 0, ( µ(ξ) µ(ξ -) + ν(ξ) ν(ξ -) )λ(ξ -) otherwise.
Clearly

q j + r j = ξ∈D + (ξ j ) λ(ξ) λ(ξ j ) V j ξ . Hence q + r ∈ Q F , implies Q F is convex.
Now when we Consider a financial structure F with only one asset issued at time t = 0 with payoff V 0 and price q 0 , and the rest of the assets are issued at future times. Let's denote the reduced financial structure(the financial structure starting tomorrow) by F and the total payoff matrix for the reduced financial structure is W (q), then the total payoff matrix for F is given by

W (q 0 , q) := -q 0 0 V 0 W (q)
. Proposition 2. If the set of arbitrage-free prices Q F for the reduced financial structure F is a convex cone, and either V 0 ≥ 0 or V 0 ≤ 0, then the set of arbitrage-free prices Q F for the financial structure F is also a convex cone.

Proof. Suppose q(λ) ∈ Q F , then W T (q(λ))λ = 0, where λ = (λ 0 , λ ) 0.

W T (q(λ))λ = 0 implies W T (q(λ ))λ = 0 and V T 0 λ -q 0 λ 0 = 0. Therefore Q F = {(q 0 , q(λ )) | W T (q(λ ))λ = 0, q 0 = V T 0 λ λ 0 and λ 0}.

Clearly ,when V 0 ≥ 0(resp. V 0 ≤ 0), then V T 0 λ ≥ 0(resp. V T 0 λ ≤ 0), hence q 0 ≥ 0 (resp. q 0 ≤ 0). Therefore

Q F = R + × Q F (resp. R -× Q F ) is a convex cone.

Examples of non-convexity

We will show that the set of arbitrage-free prices for a multi-period financial structure with long lived assets may be neither convex nor a cone. Here we illustrate an example in support of this statement : Consider a financial structure F with 3 dates and 2 assets. First asset is issued at time t = 0 and has payoff 1 and -1 at t = 1 and t = 2, respectively. Second asset is issued at t = 1 and has payoff of 1 at t = 2. We notice that the set of arbitrage-free prices

Q F is [(-∞, 0) × (1, ∞)] [(0, ∞) × (0, 1)]. And clearly Q F is neither convex nor a cone.
We show a more general result as a proposition : Proposition 3. Consider the financial structure F with 3 dates, 2 assets, the first one issued at t = 0 and the second one at t = 1 with payoff matrix

W (q) :=     -q 1 0 a -q 2 b 1     .
The set Q of arbitrage-free prices of F is a convex cone if and only if ab ≥ 0.

The set of arbitrage-free prices in terms of a and b can be defined as

Q(a, b) =              R --× R ++ if a ≤ 0 and b ≤ 0 R --× (0, -a b ) ∪ R ++ × (-a b , ∞) ∪ {(0, -a b )} if a < 0 and b > 0 R --× (-a b , ∞, ) ∪ R ++ × (0, -a b ) ∪ {0, -a b } if a > 0 and b < 0 R ++ × R ++ if a ≥ 0 and b ≥ 0
Here we show the graph of the set of arbitrage-free prices(detailed proof is in Appendix).

q 1 q 2 (0, -a b ) a > 0, b < 0 a < 0, b > 0 (0, -a b ) q 1 q 2 q 1 q 2 Figure 1 a ≤ 0, b ≤ 0 a ≥ 0, b ≥ 0 q 1 q 2 q 1 q 2
Proposition 4. Consider the financial structure F with 3 dates, 3 assets, first and second assets issued at t = 0 and third one at t = 1 with payoff matrix

W (q) :=     -q 1 -q 2 0 a c -q 3 b d 1     with ad -bc = 0.
Then the set of arbitrage-free price of F is neither convex nor a cone. Proof. From the characterization theorem of arbitrage-free prices, q ∈ Q if and only if there exists λ = (1, λ 1 , λ 2 ) 0 such that q 1 = aλ 1 + bλ 2 , q 2 = cλ 1 + dλ 2 , and λ 1 q 3 = λ 2 .

From above equations (using ad -bc = 0), we get Q = {(q 1 , q 2 , q 3 ) | q 3 (dq 1 -bq 2 ) = aq 2 -cq 1 , (dq 1 -bq 2 )(aq 2 -cq 1 ) > 0, and (aq 2 -cq 1 )(ad-bc) > 0}.

Suppose that Q is a cone and q = (q 1 , q 2 , q 3 ) ∈ Q, then q 3 (dq 1 -bq 2 ) = aq 2 -cq 1 . Since Q is a cone, αq = (αq 1 , αq 2 , αq 3 ) ∈ Q for all α > 0. Taking α = 1 2 , we get 1 2 = 1, acontradiction. Hence, Q is not a cone. Now we take q = (q 1 , q 2 , q 3 ) ∈ Q and by simple calculations, we get

q 1 (dq 3 + c) = q 2 (bq 3 + a),
Since ad -bc = 0, implies either b = 0 or d = 0, and without loss of generality we can assume b = 0. We let q 1 = 1, q 2 = q 2 -d b and q 3 = bq 3 + a and we notice that the set Q = {(q 2 , q 3 ) | q 2 = bc-ad bq 3 , b = 0, and bc -ad = 0} is not convex. Hence, Q is not convex.

Equivalent financial structure

In this part, we compare the financial equilibria associated to different financial structures.

Here the non-financial primitives of the economy are summarized by the exchange economy E = (X i , P i , e i ) i∈I ) which remains fixed, and only the financial part is changing. Economies with the same non-financial primitives E and different financial parts will be denoted (E, F), (E, F ). Definition 1. Let F, F be two financial structures, we say that the total payoff matrices

W F (p, •) and W F (p, •) are equivalent at a given commodity price p ∈ R L , denoted W F (•) ∼ p W F (•) if ∀λ ∈ R S ++ , W F (q F (λ)) = W F (q F (λ)).
The intuition behind this definition is the following. Financial structures allow agents to transfer wealth across nodes of the date-event tree and thereby give them the possibility to enlarge their budget set. The main consequence of this definition is given in the Appendix and states that, regardless of the standard exchange economy E, consumption equilibria are the same when agents carry out their financial activities through two different structures F and F . Now, we define the financial structures F and F by their total payoff matrices :

W (q) :=     -q 1 -q 2 0 a c -q 3 b d 1     , W (q ) :=     -q 1 0 1 -q 2 0 1     .
Proposition 5. The financial structures with payoff matrices W (•) and W (•) are equivalent if ad -bc = 0. Moreover, Q is neither convex nor a cone and Q is a convex cone.

The proof is given in Appendix. We need Proposition 6 for the proof, which provides a range condition that guarantees two financial structures to be equivalent.

Proposition 6. (a) Assume that for every p ∈ (R H ) S , W F (•) ∼ p W F (•), then F p F .
(b) Assume that T = 2, then the following three assertions are equivalent

(i) W F (•) ∼ p W F (•), (ii) W F (0) = W F (0), (iii) V F = V F , where V F is uniquely defined by W F (q) = -q V F .
The proof is given in the Appendix. Now we consider two financial structures F and F with total payoff matrices W (•) and W (•), respectively. Both F and F has some common assets and payoff of common assets can be represented by A(•), and remaining part is represented by b(•) and b (•), respectively. Then W (•) and W (•) can be written as

W (•) := A(•) B(•) , W (•) := A(•) B (•) .
Claim 3.1. The financial structures F and F defined by their total payoff matrices W (•) and W (•)(as defined above) are equivalent if the financial structures defined by the total payoff matrices B(•) and B (•) are equivalent.

Proof. We need to show that for every λ 0, W (q F (λ)) = W (q F (λ)) if B(q B (λ)) = B (q B (λ)). Now if we assume B(q B (λ)) = B (q B (λ)), then for every z B , there exists z B such that B(q B (λ))z B = B (q B (λ))z B (and vice versa). We need to show that for all z = z A + z B there exists z = z A + z B such that W (q F (λ))z = W (q F (λ))z (and vice versa). Now we take W (q F (λ))z = A(q(λ))z A + B(q(λ))z B = A(q (λ))z A + B(q B (λ))z B = W (q F (λ))z (and vice versa).

Examples of equivalent financial structures

The first example considers financial structure in which all assets are emitted at the same date. Moreover it is easy to see that there is no loss of generality to assume that this date is t = 0. This covers the case of 2 dates for which all assets are emitted at t = 0 but this covers more examples as the above one is described in 3-dates.

Example 1 (All assets emitted at t = 0). Consider one of the following date-event tree

D : t = 0 t = 1 t = 2
or D :

t = 0 t = 1 (a)
The following three financial structures are equivalent

W 1 (•) =   • • 1 0 0 1   , W 2 (•) =   • • 1 0 1 1   , W 3 (•) =   • • -1 0 -1 1   .
(b) The following two financial structures are equivalent

W 1 (•) =   • • 1 0 0 1   , and W 4 (•) =   • • 1 0 -1 1   .
(c) The following two financial structures are not equivalent

W 1 (•) =   • • 1 0 0 1   , and W 5 (•) =   • • 0 0 1 1   .
This result is a consequence of Proposition 5. The intuition behind is that, for example F and F are equivalent since their payoff matrices V F and V F have the same span, that is

V F = V F .
In the case of multi-period financial structures the intuition is different as shown by the following example.

Example 2. Consider the following date-event tree

D : t = 0 t = 1 t = 2 (a )
The following three financial structures are equivalent

W 1 (•) =   • 0 1 • 0 1   , W 2 (•) =   • 0 1 • 1 1   , and W 3 (•) =   • 0 -1 • -1 1   . (b )
The following two financial structures are not equivalent

W 1 (•) =   • 0 1 • 0 1   and W 4 (•) =   • 0 1 • -1 1   .
(c ) The following two financial structures are equivalent

W 1 (•) =   • 0 1 • 0 1   and W 5 (•) =   • 0 0 • 1 1   .
4 Appendix

Equivalent financial structures

Given commodity and asset prices (p, q) ∈ R L × J , the budget set of consumer i is4

B i (p, q, E, F) = {(x i , z i ) ∈ X i × Z i : p(x i -e i ) ≤ W F (p, q)z i },
where W F (p, q) denotes the total payoff matrix asociated to F. We now recall the standard equilibrium notion in this model.

Definition 2. An equilibrium of the financial exchange economy (E, F) is a list p, x, q, z ∈ R L × (R L ) I × R J × (R J ) I such that (i) for every i, (x i , zi ) maximizes the preference P i in the budget set B i (p, q), in the sense that (x i , zi ) ∈ B i (p, q) and B i (p, q) ∩ P i (x) × Z i = ∅, (ii) [Market Clearing] i∈I xi = i∈I e i and i∈I zi = 0.

A consumption equilibrium of (E, F) is a list (p, x) ∈ R L × (R L ) I such that there exist (q, z) ∈ R J × (R J ) I and p, x, q, z is an equilibrium of (E, F).

We introduce an equivalence relation on the set of all financial structures defined on the same set of agents I and the same set of nodes D. Definition 3. The two financial structures F and F are said to be equivalent, denoted F ∼ F , if for every standard exchange economy E, the financial exchange economies (E, F) and (E, F ) have the same consumption equilibria. Also we define the preorder F F that mean that, for every standard exchange economy E, every consumption equilibrium of the financial exchange economy (E, F) is a consumption equilibrium of (E, F ). So clearly F ∼ F if and only if F F and F F.

implies q 2 = λ 3 λ 2 > 0 and q 1 = bλ 3 , hence

Q(b) =        R --× R ++ if b < 0 {0} × R ++ if b = 0 R ++ × R ++ if b > 0.
Thus, Q is a convex cone.

Proof of Proposition 5

Proof. We choose λ = (1, λ 1 , λ 2 ) 0 and we get

W (q(λ)) :=     -(aλ 1 + bλ 2 ) -(cλ 1 + dλ 2 ) 0 a c -λ 2 λ 1 b d 1     , W (q (λ)) :=     -λ 1 0 1 -λ 2 λ 1 0 1     .
Now, we claim that rank W (q(λ)) ≥ 2 and rank W (q (λ)) = 2. Since ad -bc = 0, clearly rank W (q(λ)) ≥ 2 and since λ 0, λ 1 > 0 and the two columns of W (q (λ)) are independent thus rank W (q (λ)) = 2. And we Complete the proof by showing that every column of W (q(λ)) can be written as a linear combination of columns of W (q (λ)). Hence W (q(λ)) = W (q (λ)), and therefore they are equivalent(by proposition 6). Indeed,

    -(aλ 1 + bλ 2 ) a b     = (a + b λ 2 λ 1 )     -λ 1 1 0     +     0 -λ 2 λ 1 1     ,     -(cλ 1 + dλ 2 ) c d     = (c + d λ 2 λ 1 )     -λ 1 1 0     + d     0 -λ 2 λ 1 1     , and
    0 -λ 2 λ 1 1     =     0 -λ 2 λ 1 1     .

Proof of Proposition 6

We prepare the proof of Proposition 6 with a lemma.

Lemma 1. Under the Non-Satiation Assumption (NS), let x, p, q ∈ i∈I X i × R L × R J , then the two following conditions are equivalent :

6

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.35

Documents de Travail du Centre d'Economie de la Sorbonne -2012.35

Note that W j (•) satisfies ∃!ξ j ∈ D, W j ξ = 0 if ξ < ξ j , W j ξ j = -q j , and W j ξ = constant if ξ > ξ j .3 Documents de Travail du Centre d'Economie de la Sorbonne -2012.35

Recall that V j ξ is the payoff of asset j at node ξ

ξ j is the emission node of asset j

Documents de Travail du Centre d'Economie de la Sorbonne -2012.35

For every p = (p(s)) s∈ S , x = (x(s)) s∈ S in R L , we denote by px the vector p(s) • x(s) s∈ S . 10 Documents de Travail du Centre d'Economie de la Sorbonne -2012.35

Proof of Propositions

Proof of Proposition 3

Proof. It is a consequence of following characterization for ab > 0, and we will prove later for ab = 0.

From the Characterization theorem of no arbitrage(1), q ∈ Q(a, b) if and only if there exists

Substituting (4.2) in (4.1), we get

From (4.2), we get q 2 = λ 3 λ 2 > 0, and from (4.3), we have

b and b > 0 or q 2 < -a b and b < 0 Now we consider different subcases for ab > 0. i) a < 0 and b < 0: When a < 0 and b < 0, then we have -a b < 0, and from the above equations q 1 ≥ 0 will imply q 2 ≤ -a b < 0(contradiction), hence only possible solution is q 1 < 0 and q 2 > 0. ii) a < 0 and b > 0: From the above equations, under the condition b > 0 and -a b > 0, we have q 1 < 0 implies 0 < q 2 < -a b , q 1 = 0 implies q 2 = -a b , and q 1 > 0 implies q 2 > -a b . iii) a > 0 and b < 0: From the above equations, under the condition b < 0 and -a b > 0, we have q 1 < 0 implies q 2 > -a b , q 1 = 0 implies q 2 = -a b , and q 1 > 0 implies 0 < q 2 < -a b . iv) a > 0 and b > 0: When a > 0 and b > 0, then we have -a b < 0, and from the above equations q 1 ≤ 0 will imply q 2 ≤ -a b < 0(contradiction), hence only possible solution is q 1 > 0 and q 2 > 0. Now we need to show that Q is convex if ab = 0, i.e. a = 0 or b = 0.

• b = 0 : Q is convex from proposition 1.

• a = 0 : q ∈ Q is arbitrage-free therefore there exists λ = (1, λ 2 , λ 3 ) 0 such that q 1 = λ 3 b and λ 2 q 2 = λ 3 , (i) there is z := (z i ) i∈I ∈ (R J ) I such that x, z, p, q is an equilibrium of (E, F);

(ii) x, p, q satisfies (a') for every i, xi ∈ Bi (F, p, q) and P i (x) Bi (F, p, q) = ∅;

Proof. The implication [(i) ⇒ (ii)] is clear from the definition of a financial equilibrium.

We now show the converse implication [(ii) ⇒ (i)]. Let (x, p, q) satisfy the above conditions (a ) and (b ), then there exists (z i ) i∈I ∈ (R J ) I such that p(x i -e i ) ≤ W (p, q)z i for each consumer i.

Summing up these inequalities over i, and using Condition (b ), one gets 0 ≤ W (p, q) i∈I z i . But under the Non-Satiation Assumption (NS) there exists no arbitrage oppportunity, hence we cannot have W (p, q) i∈I z i > 0. Thus W (p, q) i∈I z i = 0. We now choose a particular consumer i 0 ∈ I, and we define the portfolios (z i ) i∈I by zi 0 = z i 0 -i∈I z i and zi = z i for i = i 0 .

Then, clearly (x, z, p, q) is a financial equilibrium.

Now we give the proof of proposition 6

Proof. (a) Let (x, p) be a consumption equilibrium for the economy (E, F), then there exists (z, q) such that (x, p, z, q) is an equilibrium for the economy (E, F). From LNS, q is arbitrage-free for F, hence there exists λ ∈ R D ++ such that q = q F (λ). We let q = q F (λ) 5 . But, for every p

q ) which implies that Bi (F 1 , p, q) = Bi (F 2 , p, q ). Consequently,

• ∀i, xi ∈ Bi (F, p, q) = Bi (F , p, q ) and ∅ = P i (x) Bi (F, p, q) = P i (x) Bi (F , p, q );

From Lemma 1 , we then deduce that there exists z such that (x, p, z , q ) is an equilibrium for the economy (E, F ). Thus, (x, p) is a consumption equilibrium for the economy (E, F ). 5 Here we only need that ∀i ∈ I, ∀q ∈ QF , ∃q ∈ R J , Bi(F1, p, q) = Bi(F , p, q )

Proof of (ii). Let xi ∈ Bi(F1, p, q1) then there exists z1 ∈ R J

z ∈ W F (p, q (λ)). Now we will prove (i) implies (iii).

We know that W F (•) ∼ p W F (•) if ∀λ ∈ R S ++ , W F (p, q F (λ)) = W F (p, q F (λ)). So, for all z, there exists z such that W F (p, q F (λ))z = W F (p, q F (λ))z , (andviceversa), that is,

Which implies that for all z, there exists z such that V F (p)z = V F (p)z , (and vice versa). Hence, V F = V F .