James R Goodhand
email: j.goodhand@qmul.ac.uk

Nikolaos Kamperidis
email: n.kamperdis@qmul.ac.uk

Huyenly Nguyen
email: huyenly@doctors.org.uk

Mahmood Wahed
email: m.wahed@qmul.ac.uk

Professor David S Rampton
email: d.rampton@qmul.ac.uk

APPLICATION OF THE WHO FRACTURE RISK ASSESSMENT TOOL (FRAX®) TO PREDICT NEED FOR DEXA SCANNING AND TREATMENT IN PATIENTS WITH INFLAMMATORY BOWEL DISEASE AT RISK OF OSTEOPOROSIS

Keywords: Inflammatory bowel disease < Disease-based, Osteoporosis < Topics, Crohn's disease < Disease-based, Ulcerative colitis < Ulcerative colitis Crohn's disease Osteoporosis

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

There is an argument that in the statistical analysis T-scores should not be compared as they are a so probably acceptable.

We acknowledge this, but have already presented raw BMD data to accompany the T-scores in Table 2.

The main concern with this cohort is how representative they are of IBD patients in general -this is only lightly touched upon in the Discussion and this should be strengthened -the comment that the findings are generalisable is difficult to confirm. There are some suggestions that the cohort is not typical of all IBD patients; most studies find that CD is more frequently associated with reduced BMD than UC, that age and disease duration are risks and often that use of steroid an independent risk factor. That these weren't found may suggest that these patients were not generally representative and were selected for screening because they had more severe disease, especially UC where there is an overrepresentation of patients with pancolitis compared with community based studies. We have re-structured the Discussion, with the sub-headings 'Key Results', 'Limitations of FRAX in IBD', and 'Sources of bias', to allow discussion of these points (and those raised below). We acknowledge the likely (and deliberate) selection bias in our selection of patients perceived to be at risk and thus the limited validity of extrapolating conclusions from this group of patients to IBD 'all-comers'. The term 'generalisablity' has been removed.

There are BSG guidelines concerning screening for reduced BMD and it would be interesting to know how many of these patients met these criteria at the time of screening. In brief, the BSG guidelines 1 read 'Without a reliable scoring system, common sense suggests that patients with the features shown in Boxes 1 and 2, such as age over 70, together with those with very active disease, those with disease responding poorly to treatment, and those with poor nutrition should be considered for DEXA. Those requiring steroids need special consideration'.

Where possible (see supplemental Table 1) we have completed the proportions with each of these factors in our dataset. Despite not having collected data on recent weight loss, physical activity, anticonvulsant use, oestrogen exposure, calcium exposure, visual acuity and neuromuscular disorders, overall only 2.6% (3/116) patients in our cohort did not fulfil at least one of these risk criteria. We now refer to this in the Discussion.

Another confounding not discussed is that many patients were taking a Ca/Vit D supplement and so this may have had an effect on BMD.

We include below the revised baseline demographics [supplemental table 2], the proportions of patients with low bone mineral density [supplemental table 3], and the FRAX scores (pre and post BMD inclusion) and recommended treatments compared with our current practice [supplemental table 4] for all the patients not on calcium and vitamin D.

Having excluded patients taking oral calcium and vitamin D supplements, the clinical FRAX® score alone, when compared to the FRAX score including the BMD result, had a sensitivity of 100% (95%CI: 60%-100%), specificity of 49% (95%CI: 35%-63%), positive predictive value of 24% (95%CI: 11%-42%) and negative predictive value of 100% (95%CI: 83%-100%) in identifying those patients with IBD needing BMD measurement (intermediate risk) or preventive therapy (high risk). Thus calcium and vitamin D, allowing for the reduction in sample size, has no major effect on the usefulness of FRAX.

It is mentioned that a weakness of FRAX scores is that it is dependent on BMI but it is also dependent on steroid usage for more than 3 months and so sustained use for this period would require a re-calculation of the score.

We acknowledge this important point in the Discussion. The FRAX score does not take into account steroid dose and duration. The major finding of this study is the application of the FRAX tool to determine the need for screening with DEXA scanning in IBD patients perceived to be at risk of osteoporosis. In practice, the FRAX tool in fact already recommends DEXA scanning for all patients of medium or smaller build who are on steroids, and who have a secondary cause of osteoporosis (IBD): as a result, the dose and duration of corticosteroid usage in influencing pre-DEXA FRAX score is irrelevant. When a DEXA scan has been done, the deleterious effects of high cumulative doses of corticosteroids (as opposed to use of any dose) will be reflected in a lower BMD and With these provisos the study suggests that a large number of DEXA scans (36%) in this cohort (with probably more severe disease than is typical) were not necessary. This has the potential for considerable reduction in costs. In addition the use of bisphosphonates was particularly poorly directed on the basis of fracture risk reduction; 9/11 who received them did not need them but 1/11 who did were not prescribed them. Individualising bone protection in IBD care has the potential for cost reduction, decreasing the risk of side effects in patients who are unlikely to benefit but maximising benefit to those who are at greatest risk of fracture.

We have incorporated this last point into the Conclusions.

These observations are important and seem to be novel and so I would recommend acceptance of this paper for AP&T despite the reservations about the cohort and it's generalisability to all patients with IBD.

Reviewer: 3

Comments for Transmission to the Authors As indicated above, we have re-structured the Discussion with the sub-headings 'Key Results', 'Limitations of FRAX in IBD', and 'Sources of bias'. We now include paragraphs addressing the duration and dose of corticosteroids, and the importance of vertebral fractures in IBD and include these references.

2. There is a selection bias in that patients recruited for the study were only those deemed (by clinical judgement) to have risk factors for fracture. It neglects those patients who were not selected by the clinicians originally, who may have clinically silent risks and objectively still be at risk of fracture. This should be stated in the discussion. This is now further emphasised in the Discussion.

3. In order to substantiate the predictive ability of FRAX in IBD, it should also accurately predict those (clinically pre-determined) low-risk patients not requiring testing/treatment. Such a cohort of patients at low-risk for fracture, having undergone DXA, had not been identified in this study. Is it possible for the authors to identify (and incorporate into the analysis) IBD patients who were not considered high risk for osteoporosis, but who underwent DXA?

We acknowledge this important point in the Discussion. Unfortunately in our population only 3 patients were identified with no risk factors (see above) according to the current BSG guidelines. We have no data on the applicability of this tool in patients perceived by our clinicians as being at low risk, and unfortunately we cannot justify prospective use of DEXA scanning, for reasons of both ethics and cost (approx 300 euros/DEXA scan, or 30000 euros for 100 low risk patients), to obtain such information.

4. Findings of the Fracture Intervention trial studied the ability of FRAX to predict vertebral fractures in the general population > 40. Briefly, although FRAX (with or without femoral neck BMD) did predict vertebral fractures, the strongest predictor of future vertebral fracture was the presence of vertebral fracture at baseline. These findings may be useful for comment in the current study, given that IBD patients have a high prevalence of vertebral deformities, which are not captured in the FRAX tool (Donaldson et al, JBMR 2009.24:1793-99). This is important. Interestingly, the BMD at the lumbar spine in our cohort of patients was significantly lower than in the femoral spine. We have now included this and these references in the Discussion. We very much hope that the changes we have made to the manuscript make it acceptable for publication.

INTRODUCTION

In the general population, the majority of fragility fractures occur in patients with bone mineral densities (BMD) in the osteopaenic rather than osteoporotic range, suggesting that factors other than BMD need to be taken into account when determining both the need for screening and anti-osteoporosis treatment 1 . In keeping with this, while patients with IBD are at increased risk of osteoporosis [START_REF] Silvennoinen | A controlled study of bone mineral density in patients with inflammatory bowel disease[END_REF][START_REF] Staun | Bone mineral content in patients with Crohn's disease. A longitudinal study in patients with bowel resections[END_REF][START_REF] Jahnsen | Bone mineral density is reduced in patients with Crohn's disease but not in patients with ulcerative colitis: a population based study[END_REF][START_REF] Robinson | Osteoporosis and determinants of bone density in patients with Crohn's disease[END_REF][START_REF] Schoon | Osteopenia and osteoporosis in Crohn's disease: prevalence in a Dutch population-based cohort[END_REF] , low BMD itself confers only a modest increase in the risk of fracture [START_REF] Vestergaard | Fracture risk is increased in Crohn's disease, but not in ulcerative colitis[END_REF][START_REF] Bernstein | The incidence of fracture among patients with inflammatory bowel disease. A population-based cohort study[END_REF][START_REF] Loftus | Long-term fracture risk in patients with Crohn's disease: a population-based study in Olmsted County, Minnesota[END_REF][START_REF] Van Staa | Inflammatory bowel disease and the risk of fracture[END_REF][START_REF] Card | Hip fractures in patients with inflammatory bowel disease and their relationship to corticosteroid use: a population based cohort study[END_REF][START_REF] Ananthakrishnan | Fracture-Associated Hospitalizations in Patients with Inflammatory Bowel Disease[END_REF] such that routine screening with dual x-ray absorptiometry (DEXA)

scanning is not justified according to recent British Society of Gastroenterology and American Gastroenterological Association guidelines [START_REF] Lewis | Guidelines for osteoporosis in inflammatory bowel disease and coeliac disease[END_REF][START_REF]American Gastroenterological Association medical position statement: guidelines on osteoporosis in gastrointestinal diseases[END_REF] .

Based on data from population-based cohorts, the World Health Organisation (WHO) have developed a web-based fracture risk assessment tool (FRAX®), that integrates an individual's risk factors and reports the 10-year probability of hip or other major osteoporotic fracture (clinical spine, forearm, hip or shoulder fracture) [START_REF] Kanis | on behalf of the National Osteoporosis Guideline Group. Case finding for the management of osteoporosis with FRAX® -assessment and intervention thresholds for the UK[END_REF][START_REF] Kanis | behalf of the World Health Organisation Scientific Group. Assessment of osteoporosis at the primary health care level[END_REF][START_REF] Kanis | Case finding for the management of osteoporosis with FRAX--assessment and intervention thresholds for the UK[END_REF] . It divides the level of risk into high, intermediate and low and can be calculated with or without BMD score (Figure 1a). Using guidelines from the National Osteoporosis Guidelines Group (NOGG) the online tool recommends the need for treatment, DEXA scanning or lifestyle advice/reassurance (Figure 1b) [START_REF] Kanis | Approaches to the targeting of treatment for osteoporosis[END_REF] .

In an attempt to prevent unnecessary DEXA scanning and anti-osteoporotic therapy, we aimed to assess the accuracy of pre-BMD FRAX scores in identifying those patients with IBD needing BMD measurement (intermediate risk) and/or preventive therapy (high risk)

with and without DEXA scan.

METHODS

Study design and setting

We conducted a retrospective evaluation of the FRAX® tool in adult patients (age≥20) Using electronic case note review, cross-sectional data including demographics, disease type, duration, activity and extent according to the Montreal classification, therapies, including cumulative dose of corticosteroids in the preceding three years, and previous surgery at the time of scanning were determined. Disease activity at the time of scanning was recorded and defined for Crohn's disease as abdominal pain and/or diarrhoea with a CRP>10mg/ml, and for UC as diarrhoea with rectal bleeding.

Bone mineral density measurement

All patients underwent DEXA scanning using the same Hologic Discovery A scanner (system number 80933), installed at St Bartholomew's Hospital in 2004. We used WHO guidelines to define bone mineral density: a T-score of ≥ -1 denotes normal bone; a T-score between -1 and -2.5 denotes osteopaenia, and a T-score of ≤ -2.5 denotes osteoporosis.

FRAX score

FRAX® scores were calculated with the online tool using the UK algorithm [START_REF] Kanis | on behalf of the National Osteoporosis Guideline Group. Case finding for the management of osteoporosis with FRAX® -assessment and intervention thresholds for the UK[END_REF][START_REF] Who | Fracture Risk Assessment Tool[END_REF] . In brief, this requires the completion of 12 fields: age (years); sex (male or female); height (cm); weight (kg); history of minimal trauma fracture (defined as a fracture in adult life occurring spontaneously, or arising from trauma which, in a healthy individual, would not have resulted in a fracture); history of parental hip fracture; corticosteroid exposure (defined as more than 7.5mg prednisolone [or equivalent] in the preceding three months); a concomitant diagnosis of rheumatoid arthritis; a secondary cause of osteoporosis; daily alcohol intake of more than 3 units and whether they smoked or not at the time of the scan.

All patients were considered to have a secondary cause of osteoporosis (namely, IBD). Data on personal and family history of fracture, rheumatoid arthritis, alcohol and smoking habits were confirmed by telephone interview.

FRAX® scores were calculated and patients' risk stratified using the age-adjusted NOGG recommendations on the basis of clinical factors alone; FRAX scores were then redetermined with BMD (g/cm 2 ) included [START_REF] Who | Fracture Risk Assessment Tool[END_REF] . All analyses were two tailed and p values <0.05 considered significant. Univariate analysis of baseline demographic data comparing those IBD patients with normal BMD and those who were osteoporotic or osteopaenic was carried out using chi-squared test for categorical data and Student's t-test for continuous variables. Paired FRAX® scores within risk groups were compared using paired Student's t test, and sensitivity analyses were undertaken to determine the accuracy of pre-BMD NOGG recommendations with those after the BMD was taken into account for the need for screening with DEXA scanning. Finally, NOGG treatment recommendations were compared with our patients' current treatments based on BMD alone using Chi-squared analysis.

Ethical Consideration

Because we were evaluating the appropriateness of use of results of DEXA scans already completed in our IBD service, we did not require formal ethical approval according to the guidelines of the UK National Research Ethics Service (NRES) [START_REF]Differentiating audit, service evaluation and research[END_REF] .

RESULTS

We identified 178 IBD outpatients who had had DEXA scans between 2005 and 2009. The telephone questionnaire response rate to determine the FRAX® score was 72% giving a study population of 116 patients (Figure 3).

Baseline demographics in relation to bone mineral density

There were no significant differences between those patients with normal or low (t-scores ≤ -1, i.e BMDs in the osteopaenic and osteoporotic range) BMD in terms of gender, ethnicity, disease type, duration, location, extent, behaviour and/or activity (Table 1 3). The mean FRAX® scores in each of the three risk groups fell significantly in all groups for both major and hip fractures following the inclusion of BMD scores (Table 3).

Overall, the clinical FRAX® score alone, when compared to the FRAX score including the BMD result, had a sensitivity of 100% (95%CI: 70%-100%), specificity of 40% (95%CI: 31%-50%), positive predictive value of 16% (95%CI: 9%-27%) and negative predictive value of 100% (95%CI: 90%-100%) in identifying those patients with IBD needing BMD measurement (intermediate risk) or preventive therapy (high risk).

Indications for anti-osteoporosis treatment

On the basis of their BMD alone and as part of our routine practice, eleven patients were started on treatment for osteoporosis with a bisphosphonate following their DEXA scan. In this small group of patients, there was no significant difference between our existing practice and that suggested by NOGG (post BMD inclusion) in the 97 patients at low or intermediate risk for fracture (Table 3). However, according to the NOGG guidance, 9 of these 97 patients were being treated unnecessarily.

Moreover, in the high risk group, while treatment was recommended by NOGG in 11/19 (58%) patients, only 1 patient was being treated with a bisphosphonate (p<0.01). In keeping with the existing literature [START_REF] Lewis | Guidelines for osteoporosis in inflammatory bowel disease and coeliac disease[END_REF] , our patients with IBD, on the basis of their t-scores, had lower BMDs than the general population. In patients with IBD who are perceived by their clinicians as being at increased risk of osteoporosis and/or osteopaenia, the clinical criteria alone of the FRAX® score can be used to predict those patients who need DEXA scans and/or consideration for specific treatment of osteoporosis. In our series of 116 such patients, by using this score over 5 years we could have prevented 36% DEXA scans. Our data also suggest that patients who carry a high clinical risk of fracture are frequently not considered for treatment when this decision is based on t-scores alone. Furthermore, according to NOGG, 8% of our patients were over-treated with bisphosphonates.

Limitations of FRAX® in IBD patients

There are some limitations to these conclusions. First, the FRAX® algorithms are based on general population cohort studies undertaken in people over the age of 40, and have not been validated in IBD populations. The FRAX® tool calculates risk for patients under the age of 40 using data for individuals aged 40. Data for people under the age of 40, particularly in IBD-specific populations, is unlikely to be available for several decades, since in order for enough fractures to occur to determine actual risk, prospective cohort studies in young patients will take 30 -60 years to complete. It is likely that the fracture risk in the 69 (59%) of our subjects who were under 40 was over-estimated, because of age-related loss in BMD.

Second, body mass index, a component of the FRAX® score, does not fluctuate as much in the general population as it does in IBD, where periods of active disease may precipitate substantial weight loss. In order to avoid over-estimating the fracture risk in IBD, it may therefore be best to complete the FRAX® score during periods of remission: in our group about a third of our patients had active disease at the time of scanning.

Third, although the FRAX® tool includes corticosteroids as a dichotomous risk factor, it does not take into account dose or duration of exposure to steroids. In practice, the FRAX tool in fact already recommends DEXA scanning for all patients of medium or smaller build who are on steroids, and who have a secondary cause of osteoporosis (IBD): as a result, the dose and duration of corticosteroid usage is irrelevant as a potential influence on pre-DEXA FRAX score. When a DEXA scan has been done, the deleterious effects of high cumulative doses of corticosteroids (as opposed to use of any dose) will be reflected in a lower BMD and incorporated in the post-DEXA FRAX score.

Potential Sources of Bias

When evaluating the validity of extrapolation of our results to other IBD populations, three potential sources of bias need to be considered.

First, because the data was collected by retrospective electronic case-note review and a telephone interview, the data is subject to discrepancies in interpretation, missing data, and recall bias, in particular for cigarette smoking and alcohol consumption, and it is possible that because of these factors the FRAX ® scores were underestimated.

Second, the patients included were selected deliberately because they were perceived by their treating clinicians to be at risk of osteoporosis and had for that reason had a DEXA scan arranged. In this context, however, only 2.6% (3/116) patients would not have fulfilled the current BSG criteria for DEXA screening (data not shown) [START_REF] Lewis | Guidelines for osteoporosis in inflammatory bowel disease and coeliac disease[END_REF] . We have no data on the applicability of this tool in patients perceived as being at low risk: unfortunately we cannot justify prospective use of DEXA scanning to obtain such information on ethical grounds and for reasons of cost (approximately €300/DEXA scan, or €30000 for 100 low risk patients).

Conversely, it is not known what proportion of patients in a consecutive cohort of unselected IBD patients would be scored as being at intermediate and/or high risk based on a pre-DEXA FRAX test, and thus the service implications and costs in terms of DEXA screening that such an approach would create.

As a result of a deliberate patient selection policy, our cohort is characterised by patients with a tendency towards extensive and severe disease, which probably accounts for the lack of differences seen in the univariate analyses between patients with normal and low BMDs (Table 1) that have been reported in previous prospective studies of IBD patients, for example, in disease duration [START_REF] Pollak | Femoral neck osteopenia in patients with inflammatory bowel disease[END_REF][START_REF] Ardizzone | Altered bone metabolism in inflammatory bowel disease: there is a difference between Crohn's disease and ulcerative colitis[END_REF][START_REF] De Jong | Longitudinal study of bone mineral density in patients with Crohn's disease[END_REF][START_REF] Van Hogezand | Ileum resection is the most predictive factor for osteoporosis in patients with Crohn's disease[END_REF] and steroid exposure [START_REF] Stockbrugger | Discordance between the degree of osteopenia and the prevalence of spontaneous vertebral fractures in Crohn's disease[END_REF][START_REF] Pigot | Low bone mineral density in patients with inflammatory bowel disease[END_REF][START_REF] Hela | Osteoporosis and Crohn's disease[END_REF] . In addition, about half of our patients were taking calcium and vitamin D supplements. Sub-analyses (data not shown) of patients taking and not taking calcium and vitamin D supplements showed that they had little effect on BMD and did not alter the usefulness of the pre-DEXA FRAX score as a screening tool in these patients. Compared with the general population, patients with IBD are more prone to vertebral than hip fractures [START_REF] Vestergaard | Fracture risk is increased in Crohn's disease, but not in ulcerative colitis[END_REF][START_REF] Vestergaard | Fracture risk in patients with celiac Disease, Crohn's disease, and ulcerative colitis: a nationwide follow-up study of 16,416 patients in Denmark[END_REF] . In keeping with this observation, t-scores were significantly lower at the lumbar spine than at the femoral neck in our patients. Although FRAX® (with or without femoral neck BMD) predicts vertebral fractures in the general population, the strongest risk factor for future vertebral fracture in Donaldson's study was the presence of a vertebral fracture at baseline [START_REF] Donaldson | FRAX and risk of vertebral fractures: the fracture intervention trial[END_REF] . Because vertebral fractures can be asymptomatic, they are frequently overlooked by both patients and clinicians: subclinical fractures reportedly occur in up to 20% IBD patients [START_REF] Heijckmann | High prevalence of morphometric vertebral deformities in patients with inflammatory bowel disease[END_REF][START_REF] Siffledeen | Vertebral fractures and role of low bone mineral density in Crohn's disease[END_REF] . It is likely then that in a proportion of our patients, the FRAX® scores may have underestimated the risk of subsequent fracture.

Conclusions

Although patients with IBD are at risk for low bone mineral density this confers only a small increase in the risk of fragility fracture.

  post-DEXA FRAX score.

5 .

 5 The authors did collect an impressive array of data, including cumulative corticosteroid use, calcium and vitamin D supplementation, etc. Could the authors comment on the influence of results if those patients taking vitamin D/calcium supplements were excluded from the analysis? See comments above regarding calcium supplements.

  screened for low BMD with DEXA scanning at Barts and the London NHS Trust from 2005-2009.IBD ParticipantsPatients attending our specialist IBD out-patient clinics with an established diagnosis of IBD whose clinicians considered them to be at risk of osteoporosis and who had therefore had a 2005 and 2009 were eligible for inclusion. In those patients who had duplicate scans to monitor BMD (n=29), only the results of the first scan were included.Because we were interested in anti-osteoporosis prescribing based on BMD results alone we excluded patients (n=15) taking bisphosphonates before 2005, after the current DEXA scanner was installed (see below).

  Statistical analyses were conducted using SPSS (version 16) and Prism (version 4) software.

Figure 4 .

 4 Figure 4. Comparison of the post-DEXA NOGG recommendations and clinical practice at Barts and The London NHS Trust (BLT) in the low, intermediate and high risk groups stratified according to the pre-DEXA FRAX®. The numbers represent the number of patients in each group. *, hence 42 DEXA scans could have been prevented.

Table 1 :

 1 Proportion of patients at risk for osteoporosis according to BSG criteria

	Best wishes
	Yours sincerely,
	David Rampton
	Prof Clinical Gastroenterology
	d.rampton@qmul.ac.uk

Table 2 :

 2 Baseline demographic characteristics of patients that were considered to be at risk of osteoporosis at the time

	Characteristic

of the DEXA scanning. Patients who were on calcium/vitamin D supplementation at the time of DEXA scanning have been excluded. *, initiated after DEXA scanning T-scores of: ≥ -1 were considered as corresponding to normal BMD. T-scores < -1 were considered as corresponding to low BMD Age Mean age (yrs) [SEM] 36.6 [2.09] 41.9 [3.13] 0.19 Mean age at diagnosis (yrs) [SEM] 23.5 [1.65] 23.5 [2.19] 0.99 BMI Body mass index 23.5 [0.70] 22.9 [0.62] 0.53 Disease duration Disease duration (yrs) [SEM] 14.9 [2.17] 15.7 [1.83] 0.

Table 3 :

 3

	DEXA Site		UC (16)	Crohn's (43)	p -value
	Lumbar	Mean BMD [SEM]	0.97 [0.04]	0.97 [0.02]	0.89
	spine
		Mean [SEM] Z score	-0.59 [0.33]	-0.53 [0.21]	0.88
		Mean [SEM] T score	-0.94 [0.32]	-0.94 [0.20]	0.99
		Osteopaenia n (%)	6 (37.5%)	15 (34.5%)
					0.98
		Osteoporotic n (%)	2 (12.5%)	6 (14%)
	Femoral	Mean BMD [SEM]	0.94 [0.04]	0.89 [0.02]	0.32
	neck
		Mean [SEM] Z score	-0.14 [0.23]	-0.40 [0.14]	0.32
		Mean [SEM] T score	-0.41 [0.25]	-0.71 [0.14]	0.29
		Osteopaenia n (%)	5 (31.3%)	20 (46.5%)
					0.44
		Osteoporotic n (%)	0 (0%)	1 (2.3%)

Prevalence of reduced bone mineral density (BMD) measured at the lumbar spine and femoral neck in Crohn's disease and UC. Patients who were on calcium/vitamin D supplementation at the time of the DEXA scanning have been excluded. T-scores of: ≥ -1 were considered normal; -1 and -2.5 osteopaenic; ≤ -2.5 as osteoporosis, were used to categorise BMD except in 13 patients under 20, when Z scores were used. T-scores were significantly lower at the spine than the hip (p<0.01).

F o r P e e r R e v i e w

Barts and The London NHS Trust: The Royal London Hospital, St Bartholomew's Hospital and The London Chest Hospital

Table 4 :

 4 To assess the accuracy of pre-BMD FRAX scores in identifying at risk IBD patients needing BMD measurement (intermediate risk) and/or therapy (high risk).

	Current	bisphosphonate
	treatment based on t -scores
	alone

FRAX scores (pre and post BMD inclusion) and the recommended treatment compared with our current practice. Patients on calcium/vitamin D supplementation at the time of the DEXA scanning have been excluded. Patients have been stratified by risk (low, intermediate or high) according to the pre-BMD FRAX® score. † Paired data were compared using students paired t-test. ‡ Treatment recommendations were compared to current use of bisphosphonates using Chi-squared analysis. S, denotes spine; FN, femoral neck; M, major; H, Hip.

Objectives:

Results: On DEXA scans, 47% [38/81] and 12% [10/81] patients with Crohn's disease were osteopaenic and osteoporotic, respectively; equivalent figures for patients with UC were 34% [12/35] and 14%

[5/35]

. The clinical FRAX® score alone, when compared to the FRAX score including the BMD result, had a sensitivity of 100% (95%CI: 70%-100%), specificity of 40% (95%CI: 31%-50%), positive predictive value of 16% (95%CI: 9%-27%) and negative predictive value of 100% (95%CI: 90%-100%) in identifying those patients needing BMD measurement (intermediate risk) or preventive therapy (high risk).

Conclusions:

In patients with IBD perceived to be at risk of osteoporosis and/or osteopaenia, the clinical FRAX score alone can accurately predict the risk of osteoporotic fracture and thereby reduce the need for DEXA scans and unnecessary anti-osteoporosis treatment.

  Overall, 47% [38/81] patients with Crohn's disease and 34%[12/35] patients with UC were osteopaenic, and 12% [10/81] Crohn's patients and 14% [5/35] UC patients were osteoporotic at either the femoral head or lumbar spine. There was a strong positive correlation between the t-score measured at the femoral neck and at the lumbar spine (R=+0.66 [0.55, 0.75] 95%CI); the mean t-score was significantly lower at the lumbar spine compared with the femoral neck for both Crohn's and UC (Table2).

	Frax®scores with and without BMD measurement: sensitivity analysis
	According to clinical FRAX® scores alone, 36% (42/116) of patients were at low risk for
	fracture, 47% (55/116) at intermediate risk and 16% (19/116) at high risk. Although there
	was a trend for the measured t-scores to decrease with increasing FRAX risk this was not F r o statistically significant (Table
	P
	e
	e r
	R
	e
	v i e
	w
	). Similarly, there
	was no difference in the proportions of patients treated with corticosteroids, 5
	aminosalicylates, immunosuppressants, or monoclonal antibodies against tumour necrosis
	factor (anti-TNFs). No difference was seen in the cumulative mean [SEM] dose of
	prednisolone between groups (normal BMD 1242 [331]mg vs low BMD 1437 [274]mg

p=0.77). The number of patients on calcium/vitamin D preparations was significantly higher in the low BMD group for both UC and CD (Table

1

).

Table 1 :

 1 The web-based WHO FRAX® tool is a readily available, easy to use, free and rapid way of assessing fracture risk based on clinical factors alone that could be utilized in IBD outpatient clinics to predict those patients who need DEXA scans and/or consideration for specific treatment of osteoporosis. If the patient is at low risk then no DEXA scan is required. If the FRAX® score based on clinical risk factors alone is intermediate and/or high then patients should undergo DEXA scanning and once the BMD is known the FRAX® score recalculated to determine the need for specific anti-osteoporosis treatment. Individualizing bone protection using the FRAX® score in IBD care has the potential to reduce the cost related to DEXA scanning; it may also obviate the risk of side effects in patients who are unlikely to benefit from treatment, and ensure appropriate treatment for those who are at greatest risk of fracture. Baseline demographics of patients considered to be at risk for osteoporosis at the time of DEXA scanning.

	Characteristic			Normal BMD	Low BMD*	P -Value
				(n=51)	(n=65)
	Sex	Male Female		19 (37.2%) 31 (60.7%)	31 (47.7%) 34 (52.3%)	0.34
	Age	Mean age (yrs) [SEM]		37.7 [1.7]	40.9 [2.2] 0.29
		Mean age at diagnosis (yrs) [SEM]	26.1 [1.9]	25 [1.8] 0.68
	BMI	Body mass index		24 [0.55]	22.7 [0.62] 0.15
	Disease duration	Disease duration (yrs) [SEM]	12.7 [1.4]	15.1 [1.6] 0.28
	Ethnicity	Caucasian		40 (78.4%)	53 (81.5%)
		South Asian		8 (15.7%)	11 (16.9%)	0.44
		Afro-Caribbean		3 (5.8%)	1 (1.5%)
	Disease Type	Crohn's disease Ulcerative Colitis		33 (62.7%) 18 (35.3%)	48 (73.8%) 17 (26.2%)	0.27
	Crohn's disease (Montreal Classification)	A1: Age ≤16 F A2: 17-40 A3: >40 o L2: Colonic r L1: Ileal		8 (24.2%) 18 (54.5%) 0 (0%) 5 (15.1%) 4 (12.1%)	13 (27.1%) 25 (52.1%) 5 (10.4%) 5 (10.4%) 12 (25%)	0.17 0.39
		L3: Ileocolonic P +L4 : Upper GI		24 (72.7%) 2 (6.1%)	30 (62.5%) 7 (14.6%) 0.33
	Activity	B1: Inflammatory e B2: Stricturing B3: Penetrating e +p: Perianal r Active Crohn's disease	12 (36.4%) 9 (27.3%) 12 (36.4%) 11 (33.3%) 11 (33.3%)	14 (29.2%) 21 (43.8%) 13 (27.1%) 11 (22.9%) 0.43 0.32 13 (27.1%) 0.55
	Medication	Budesonide Prednisolone	R	10 (30.3%) 12 (36.4%)	18 (37.5%) 0.64 21 (43.8%) 0.51
	Surgery UC Classification) (Montreal	5 ASA Immunosuppression Anti-TNF Calcium and vitamin D preparations e v i 21 (63.6%) 20 (60.1%) 3 (9.1%) 15 (45.5%) Previous laparotomy 17 (51.5%) e E1: Proctitis 2 (11.1%) E3: Total 7 (38.9%) w E2: Left 8 (44.4%)	33 (68.8%) 0.63 29 (60.4%) 0.99 2 (4.2%) 0.38 31 (64.6%) 0.001 24 (50%) 0.82 1 (5.9%) 13 (76.5%) 2 (11.8%) 0.11
	Activity	Active UC		6 (33.3%)	9 (52.9%) 0.32
	Medications	Prednisolone		16 (88.9%)	13 (76.5%) 0.4
		5 ASA		17 (94.4%)	13 (76.5%) 0.18
		Immunosuppression		11 (61.1%)	9 (52.9%) 0.74
		Calcium and vitamin D3	10 (55.6%)	12 (70.6%) 0.03
	Surgery	Colectomy		1 (5.6%)	4 (23.5%) 0.18

Table 2 :

 2 Prevalence of reduced bone mineral density (BMD) measured at the lumbar spine and femoral neck in Crohn's disease and UC.Application of the FRAX ® tool in patients with IBD at risk of osteoporosis.12

	T-scores of: ≥ -1 were considered normal; -1

Table 3 :

 3 FRAX scores (pre and post BMD inclusion) and the recommended treatment compared with our current practice. Patients have been stratified by risk (low, intermediate or high) according to the pre-BMD FRAX® score. † Paired data were compared using Students paired t-test, ‡ Treatment recommendations were compared to current use of bisphosphonates using Chi-squared analysis. S, denotes spine; FN, femoral neck; M, major; H, Hip. WHO case finding strategies, adapted from Kanis JA on behalf of the World Health Organization Scientific Group (2008)[START_REF] Kanis | behalf of the World Health Organisation Scientific Group. Assessment of osteoporosis at the primary health care level[END_REF] .FIGURE 1b. NOGG age-related treatment thresholds without BMD inclusion into the FRAX® score, adapted from[START_REF] Kanis | on behalf of the National Osteoporosis Guideline Group. Case finding for the management of osteoporosis with FRAX® -assessment and intervention thresholds for the UK[END_REF] . Study overview using STROBE criteria[START_REF] Vandenbroucke | Strengthening the Reporting of Observational Studies in Epidemiology (STROBE): explanation and elaboration[END_REF] .

	Current treatment based
	on t -scores alone

A B

ACKNOWLEDGEMENTS

All authors approved the final manuscript, and declare no financial conflict of interest. JRG conceived the study, collected and analysed the data and prepared the manuscript. HN, NK, and MW participated in study design, collected and analysed the dataset. DSR participated in study design, interpretation of its results and preparation of the manuscript. We are grateful to Dr James Lindsay and Dr Louise Langmead, for their constructive criticism whilst reviewing the work and for allowing us to include their patients. We would also like to thank the National Association of Crohn's and Colitis and CORE/British Society Paediatric Gastroenterology and Nutrition (BSPGHAN) for funding JRG's research.


Thomas Batard

Michel Berthier

Christophe Saint-Jean

Clifford Fourier Transform for Color Image Processing

come

Introduction

During the last years several attempts have been made to generalize the classical approach of scalar signal processing with the Fourier transform to higher dimensional signals. The reader will find a detailed overview of the related works at the beginning of [START_REF] Brackx | The Two-Dimensional Clifford-Fourier Transform[END_REF]. We only mention in this introduction some of the approaches investigated by several authors. Motivated by the spectral analysis of color images, S. Sangwine and T. Ell have proposed in [START_REF] Sangwine | Hypercomplex Fourier Transforms of Color Images[END_REF] and [START_REF] Ell | Hypercomplex Fourier Transforms of Color Images[END_REF] a generalization based on the use of quaternions: a color Thomas Batard Lab. MIA, La Rochelle University, France, e-mail: tbatar01@univ-lr.fr Michel Berthier Lab. MIA, La Rochelle University, France, e-mail: michel.berthier@univ-lr.fr Christophe Saint-Jean Lab. MIA, La Rochelle University, France, e-mail: christophe.saintjean@univ-lr.fr 1 corresponds to an imaginary quaternion and the imaginary complex i is replaced by the unit quaternion µ coding the grey axis. A quaternionic definition is also given by T. Bülow and G. Sommer in the context of analytic signals, for signals defined on the plane and with values in the algebra H of quaternions [START_REF] Bülow | The hypercomplex signal -a novel approach to the multidimensional analytic signal[END_REF]. Concerning analytic signals, M. Felsberg makes use of the Clifford algebras R 2,0 and R 3,0 to define an appropriate Clifford Fourier transform [START_REF] Felsberg | Low-level image processing with the structure multivector[END_REF]. A generalization in the Clifford algebras context appears also in J. Ebling and G. Scheuermann [START_REF] Ebling | Clifford Fourier Transform on Vector Fields[END_REF]. The authors mainly use their transformation to analyze frequencies of vector fields. Using the same Fourier kernel, B. Mawardi and E. Hitzer obtain an uncertainty principle for R 3,0 multivector functions [START_REF] Mawardi | Clifford Fourier transformation and uncertainty principle for the Clifford algebra cl 3,0[END_REF]. The reader may find in [START_REF] Brackx | The Two-Dimensional Clifford-Fourier Transform[END_REF] definitions of Clifford Fourier transform and Clifford Gabor filters based on the Dirac operator and Clifford analysis. One could ask the reason why to propose a new generalization. An important thing when dealing with Fourier transform is its link with group representations. We then recall in section 2 the usual definition of the Fourier transform of a function defined on an abelian Lie group by means of the characters of the group. The definition we propose in section 3 relies mainly on the generalization of the notion of characters, that is why we study the group morphisms from R 2 to Spin(3) and Spin [START_REF] Ebling | Clifford Fourier Transform on Vector Fields[END_REF]. These morphisms help to understand the behavior of the Fourier transform with respect to well chosen spinors. We treat in section 4 three applications corresponding to specific bivectors of R 4,0 . They consist in filtering frequencies according to color, hue and chrominance part of a given color. In section 5, we show that for particular choices of group morphisms and under well chosen identification with quaternions, the Clifford Fourier transform we propose coincides with the definitions of S. Sangwine [START_REF] Ell | Hypercomplex Fourier Transforms of Color Images[END_REF] and T. Bülow [START_REF] Bülow | Hypercomplex spectral signal representations for the processing and analysis of images[END_REF].

Fourier transform and group actions

Let us recall briefly some basic ideas related to the group approach of the definition of the Fourier transform. Details can be found in the Appendix, see also [START_REF] Smach | Generalized Fourier Descriptors with Applications to Objects Recognition in SVM Context[END_REF] for examples of applications to Fourier descriptors. Let G be a Lie group. The Pontryagin dual of G, denoted G, is the set of equivalence classes of unitary irreducible representations of G. It appears that if G is abelian, every irreducible unitary representation of G is of dimension 1, i.e. is a continuous group morphism from G to S 1 . This is precisely the definition of a character. It is well known that the characters of R m are given by

(x 1 , • • • , x m ) -→ e i(u 1 x 1 +•••+u m x m ) with u 1 ,u 2 , • • • , u m real. This shows that R m = R m . The characters of SO(2) are the group morphisms θ -→ e inθ
for n ∈ Z and the corresponding Pontryagin dual is Z. The characters of Z/nZ are the groups morphisms u -→ e i 2πku n for k ∈ Z/nZ, from which we deduce that Z/nZ is its own Pontryagin dual.

In the general case (provided that G is unimodular), the Fourier transform of a func-

tion f ∈ L 2 (G; C) is defined on G by f (ϕ) = G f (x)ϕ(x -1 )dν(x)
(for ν a well chosen invariant measure on G). Applying this formula to the case From now on, we deal with the abelian group G = (R 2 , +) since this paper is devoted to image processing applications.

G = R m , resp. G =SO(2), resp. G = Z/nZ
Let us make some crucial remarks about the case n = 2. Let f be a real or complex valued function defined on R 2 . Its Fourier transform is given by

f (a, b) = R 2
f (x, y)e -i(ax+by) dx dy

Identifying C with (R 2 , 2 ), we have S 1 =SO(2) and the action of S 1 on C, given by the complex multiplication, corresponds to the action of the group SO(2) on (R 2 , 2 ). Hence, we can define a Fourier transform in L 2 (R 2 , (R 2 , 2 )) using the action of group morphisms from R 2 to SO(2) on (R 2 , 2 ). These ones are real unitary representations of the group R 2 of dimension 2.

The Fourier transform of f ∈ L 2 (R 2 , (R 2 , 2 )) defined above can be written in the Clifford algebra language. Indeed, from the embedding of (R 2 , 2 ) into R 2,0 , f may be viewed as a R 1 2,0 -valued function

f (x, y) = f 1 (x, y)e 1 + f 2 (x, y)e 2
where e 2 1 = e 2 2 = 1 and e 1 e 2 = -e 2 e 1 . From this point of view, the Fourier transform of f is given by

f (a, b) = R 2 [cos((ax + by)/2)1 + sin((ax + by)/2)e 1 e 2 ]( f 1 (x, y)e 1 + f 2 (x, y)e 2 )
[cos(-(ax + by)/2)1 + sin(-(ax + by)/2)e 1 e 2 ] dx dy using the fact that the action of Spin(2) on R 1 2,0 corresponds to the action of SO(2) on (R 2 , 2 ) (see Appendix). We can write this last formula in the following form:

f (a, b) = R 2 ( f 1 (x, y)e 1 + f 2 (x, y)e 2 )⊥ϕ a,b (-x, -y)dx dy
where ϕ a,b is the morphism from R 2 to Spin(2) that sends (x, y) to exp[((ax + by)/2)(e 1 e 2 )] and ⊥ denotes the action v⊥s = s -1 vs of Spin(2) on R 1 2,0 , and more generally the action of Spin(n) on R 1 n,0 .

Note that group morphisms from R 2 to Spin(2) followed by the action on R 1 2,0 correspond to the action of group morphisms from R 2 to SO(2) on (R 2 , 2 ). In other words, they are real unitary representations of R 2 of dimension 2 too.

Remark 1. As in the standard case, where the Fourier transform of a real valued function is defined by embedding R into C, we define here the Fourier transform of a real valued function by embedding R into R 2 .

Starting from these elementary observations, we now proceed to generalize this construction for R n -valued functions defined in R 2 . In other words, we are looking for a generalization of the action of group morphisms to SO(2) on the values of a (R 2 , 2 )-valued function.

3 Clifford Fourier transform in L 2 (R 2 , (R n , Q)) Let f ∈ L 2 (R 2 , (R n , Q))
where Q is a positive definite quadratic form. We propose to associate the Fourier transform of f with the action of the following group morphisms on the values of f , depending on the parity of n.

If n is even, then we consider the morphisms

ϕ : R 2 -→ SO(Q) If n is odd, then we embed (R n , Q) into (R n+1 , Q ⊕ 1)

and consider the morphisms

ϕ : R 2 -→ SO(Q ⊕ 1)
Thus the generalization we propose is based on the computation of real unitary representations of dimension n or n+1 of the abelian group R 2 . The main fact is that we no more consider equivalent classes of representations. This means in particular that the Fourier transform we define depends on the positive definite quadratic form of R n .

Remark 2. Recall that up to a change of the basis, a positive definite quadratic form is given by the identity matrix. Thus, f may always be viewed as a (R p , 2 )-valued function(p denotes n if n is even and n + 1 if n is odd). As a consequence of the change of the basis, SO(Q) become SO(p) and group morphisms from R 2 to SO(Q) become group morphisms from R 2 to SO(p).

As for the case of R 2 -valued functions, we can rewrite the Fourier transform in the Clifford algebra language, using the fact that the action of Spin(p) on R 1 p,0 corresponds to the action of SO(p) on R p . Moreover, it appears to be much more easier to compute group morphisms to Spin(p) rather than group morphisms to the matrix group SO(p).

If n is even, then from the embedding of R n into R n,0 , f may be viewed as a R 1 n,0valued function:

f (x, y) = f 1 (x, y)e 1 + f 2 (x, y)e 2 + • • • + f n (x, y)e n ,
where e 2 i = 1 and e i e j = -e j e i . Denoting by ϕ a group morphism from R 2 to Spin(n), we define the Clifford-Fourier transform of f by

f (ϕ) = R 2 ϕ(x, y) f (x, y)ϕ(-x, -y)dx dy = R 2 f (x, y)⊥ϕ(-x, -y)dx dy.
If n is odd, we first embed R n into R n+1 . Then, from the embedding of R n+1 into R n+1,0 , f may be viewed as a R 1 n+1,0 -valued function:

f (x, y) = f 1 (x, y)e 1 + f 2 (x, y)e 2 + • • • + f n (x, y)e n + 0e n+1 ,
where e 2 i = 1 and e i e j = -e j e i . Denoting by ϕ a group morphism from R 2 to Spin(n + 1), we define the Clifford-Fourier transform of f by

f (ϕ) = R 2 ϕ(x, y) f (x, y)ϕ(-x, -y)dx dy = R 2 f (x, y)⊥ϕ(-x, -y)dx dy. Remark 3. If n is even, the Clifford Fourier transform of f is a R 1 n,0 -valued function. If n is odd, the Clifford Fourier transform of f is a R 1 n+1,0 -valued function.
Remark 4. For Q=1 on R, the Fourier transforms we define correspond to the standard Fourier transforms of R-valued functions.

From now on, we deal with the case n = 3 since this paper is devoted to color image processing. However, we have seen above that we treat the cases n = 3 and n = 4 in the same manner, by computing group morphisms from R 2 to Spin(4).

3.1 The cases n=3,4: Group morphisms from R 2 to Spin [START_REF] Ebling | Clifford Fourier Transform on Vector Fields[END_REF] This part is devoted to the computation of group morphisms from R 2 to Spin(4).

Using the fact that the group Spin( 4) is isomorphic to the group Spin(3) × Spin(3), we first compute group morphisms from R 2 to Spin(3).

One can verify that Spin(3) is the group

Spin(3) = {a1 + be 1 e 2 + ce 2 e 3 + de 3 e 1 , a 2 + b 2 + c 2 + d 2 = 1}
and is isomorphic to the group of unit quaternions.

Proposition 1. The group morphisms from R 2 to Spin(3) are given by

(x, y) -→ e 1 2 (ux+vy)B
where B belongs to S 2 3,0 , the set of unit bivectors in R 3,0 (see Appendix), and u and v are real.

Proof. We have to determine the abelian subalgebras of the Lie algebra spin(3) = R 2 3,0 of the Lie group Spin(3). More precisely, as the exponential map of R 2 is onto, group morphisms from R 2 to Spin(3) are given by Lie algebra morphisms from the abelian Lie algebra R 2 of R 2 to spin(3). Taking two generators (

f 1 , f 2 ) of R 2 , any morphism ϕ from R 2 to spin(3) satisfies: ϕ( f 1 ) × ϕ( f 2 ) = 0.
We deduce that Im(ϕ) is an abelian subalgebra of R This shows that two commuting elements of R 2 3,0 are colinear and that the abelian subalgebras of R 2 3,0 are of dimension 1. If we write ϕ(

f 1 ) = 1 2 uB and ϕ( f 2 ) = 1 2 vB for some u, v ∈ R and B ∈ S 2
3,0 , we see that the morphisms from R 2 to R 2 3,0 are parameterized by two real numbers and one unit bivector, and are given by

ϕ u,v,B : (x, y) → 1 2 (ux + vy)B.
Consequently, the group morphisms from R 2 to Spin(3) are the morphisms ϕ u,v,B with ϕ u,v,B : (x, y) → e where I denotes the pseudoscalar of R 4,0 and the following relations hold:

uu + vv = 1 uv + vu = 0.
The morphism χ: Spin(4)-→ Spin(3)× Spin(3) with

χ(u + Iv) = (u + v, u -v)
is an isomorphism. An alternative description of Spin(4) relies on the following fact: the morphism ψ :

H 1 × H 1 -→ SO(4) defined by (τ, ρ) -→ (v -→ τvρ)
(where v is a vector of R 4 considered as a quaternion) is a universal covering of SO(4) (see [START_REF] Postnikov | Lec ¸ons de Géométrie[END_REF]). This means that Spin( 4) is isomorphic to H 1 × H 1 . We will use this remark later on to compare our transform to Sangwine's and Bülow's ones. 2 However, this writing is not convenient to determine group morphisms to SO(4) since it doesn't provide explicitly the rotations in R 4 that φ u,v,B,w,z,C generates by its action on R 1 4,0 . The solution comes from an "orthogonalization" of the corresponding Lie algebras morphism from R 2 to R 2 4,0 , namely the linear map

φ u,v,B,w,z,C (X,Y ) = T (0,0) φ u,v,B,w,z,C (X,Y )
where T denotes the linear tangent map. By definition:

φ u,v,B,w,z,C (X,Y ) = d dt φ u,v,B,w,z,C (exp(t(X,Y ))) t=0
The exponential map of R 2 being the identity map, we get

φ u,v,B,w,z,C (X,Y ) = d dt φ u,v,B,w,z,C (t(X,Y )) t=0 = d dt e 1 2 t(uX+vY )B + e 1 2 t(wX+zY )C 2 + I e 1 2 t(uX+vY )B -e 1 2 t(wX+zY )C 2 t=0 = (uX + vY )B + (wX + zY )C 4 + I (uX + vY )B -(wX + zY )C 4
The orthogonalization of the morphism φ u,v,B,w,z,C consists in decomposing the bivector φ u,v,B,w,z,C (X,Y ) for each X,Y into commuting bivectors whose squares are real. The corresponding spinor is written as a product of commuting spinors of the form e F i with F 2 i < 0. These ones represent rotations of angle -F 2 i in the oriented planes given by the F i 's. In our case, the bivector φ u,v,B,w,z,C (X,Y ) is decomposed into F 1 + F 2 where

F 1 = 1 8 (X(u + w) +Y (v + z))(B +C + I(B -C)) F 2 = 1 8 (X(u -w) +Y (v -z))(B -C + I(B +C))
(see the Appendix for details). The group morphisms φ u,v,B,w,z,C from R 2 to Spin(4) can then be written as This is the convenient form to describe group morphisms from R 2 to SO(4).

φ u,v,B,w,z,C (x, y) = e [ (
To conclude this part, let us remark that the expression of the morphisms φ u,v,B,w,z,C may be simplified. Indeed, when B and C describe S 2 3,0 ⊂ R 4,0 , the unit bivectors

D = 1 4 (B +C + I(B -C)) ID = 1 4 (B -C + I(B +C))
describe S 2 4,0 , the set of unit bivectors in R 4,0 .

Therefore, the morphisms φ u,v,B,w,z,C are parametrized by four real numbers and one unit bivector D ∈ S 2 4,0 , and may be written

Φ u,v,w,z,D (x, y) = e 1 2 [(x(u+w)+y(v+z))D] e 1 2 [(x(u-w)+y(v-z))ID] .

The cases n=3,4: The Clifford Fourier transform

From the computation of group morphisms from R 2 to Spin(4), we give an explicit formula of the Clifford Fourier transform

f of f ∈ L 2 (R 2 , (R 3 , Q)) or L 2 (R 2 , (R 4 , Q)). Definition 1. Let f ∈ L 2 (R 2 , (R 3 , Q)), resp. L 2 (R 2 , (R 4 , Q)) and denote by f the embedding of f into the Clifford algebra Cl(R 4 , Q ⊕ 1), resp. Cl(R 4 , Q). The Clif- ford Fourier transform of f is given by f (u, v, w, z, D) = R 2 f (x, y)⊥ Φ u,v,w,z,D (-x, -y)dx dy = R 2 e 1 2 [(x(u+w)+y(v+z))D] e 1 2 [(x(u-w)+y(v-z))ID] f (x, y) e -1 2 [(x(u+w)+y(v+z))D] e -1 2 [(x(u-w)+y(v-z))ID] dx dy.
• Decomposing f as f || + f ⊥ with respect to the plane generated by the bivector D,

we get f (u, v, w, z, D) = R 2 f || (x, y)e [-(x(u+w)+y(v+z))D] dx dy + R 2 f ⊥ (x, y)e [-(x(u-w)+y(v-z))ID] dx dy
Indeed, the plane generated by ID represents the orthogonal of the plane generated by D in R 4 .

Proposition 3. The Clifford Fourier transform is left-invertible. Its inverse is the mapˇgiven by

ǧ(a, b) = R 4 ×S 2 4,0 g(u, v, w, z, D)⊥ Φ u,v,w,z,D (a, b) du dv dw dz dν
where ν is a unit measure on S 2 4,0 .

Proof. We have to verify that ˇ

• ( f )(λ , µ) = f (λ , µ) for all (λ , µ) ∈ R 2 . ˇ• ( f )(λ , µ) = = R 4 ×S 2 4,0 R 2 f || (x, y)e [-(x(u+w)+y(v+z))D]
dx dy e [(λ (u+w)+µ(v+z))D] du dv dw dz dν

(1) + R 4 ×S 2 4,0 R 2 f ⊥ (x, y)e [-(x(u-w)+y(v-z))ID] dx dy e [(λ (u-w)+µ(v-z))ID] du dv dw dz dν (2) It is sufficient to prove that (1) = f || (λ , µ). (1) = R 4 ×S 2 4,0 R 2 f || (x, y)e [(λ -x)(u+w)+(µ-y)(v+z)]D dx dy du dv dw dz dν = R 4 ×S 2 4,0 R 2 f || (x, y)e u(λ -x)D e w(λ -x)D e v(µ-y)D e z(µ-y)D dx dy du dv dw dz dν = R 2 R 3 ×S 2 4,0 f || (x, y) R e u(λ -x)
D du e w(λ -x)D e v(µ-y)D e z(µ-y)D dw dv dz dν dx dy

= R 2 R 2 ×S 2 4,0 f || (x, y)δ λ ,x R
e w(λ -x)D dw e v(µ-y)D e z(µ-y)D dv dz dν dx dy

= R 2 R×S 2 4,0 f || (x, y)δ λ ,x δ λ ,x R e v(µ-y)D dv dz dν dx dy = R 2 S 2 4,0 f || (x, y)δ λ ,x δ λ ,x δ µ,y R e z(µ-y)D dz dν dx dy = R 2 S 2 4,0 f || (x, y)δ λ ,x δ λ ,x δ µ,y δ µ,y dν dx dy = R 2 f || (x, y)δ λ ,x δ λ ,x δ µ,y δ µ,y dx dy = f || (λ , µ)
4 Application to color image filtering

Clifford Fourier transform of color images

For the applications we have in mind to color image filtering, we define a partial Clifford Fourier transform, i.e. we deal with a subset of the set of unitary group representations of R 2 of dimension 4. The subset we consider will depend of the colors we aim at filtering. More precisely, we restrict the definition 1 to the set of group morphisms Φ u,v,0,0,D where the bivector D is fixed.

Definition 2. Clifford Fourier transform with respect to a bivector.

Let f ∈ L 2 (R 2 , (R 3 , Q)), resp. L 2 (R 2 , (R 4 , Q))
and denote by f the embedding of f into the Clifford algebra Cl(R 4 , Q ⊕ 1), resp. Cl(R 4 , Q). The Clifford Fourier transform of f with respect to the bivector D is defined by

f D (u, v) = R 2 f (x, y)⊥ Φ u,v,0,0,D (-x, -y)dxdy = R 2 e 1 2 (
I Q,D (u, v) = f D (u, v) = R 2
f (x, y)⊥ Φ u,v,0,0,D (-x, -y)dxdy.

• Thus, given a color image, we define a set of associated Clifford Fourier transforms parametrized by the set of positive definite quadratic forms on R 3 and unit bivectors in R 4,0 .

As the Clifford Fourier transform in L 2 (R 3 , Q) and L 2 (R 4 , Q), we can show that the Clifford Fourier transform of a color image is invertible.

Proposition 4. Let f ∈ L 2 (R 2 , (R 3 , Q))
and D be a unit bivector in Cl(R 4 , Q ⊕ 1).

Then, the Clifford Fourier transform of f with respect to D is invertible. Its inverse is the mapˇdefined by

ǧ(x, y) = R 2 g(u, v)⊥ Φ u,v,0,0,D (x, y)dudv.
Proof. Decomposing f with respect to the plane generated by

D as f = f + f ⊥ we have f D (u, v) = R 2 ( f (x, y) + f ⊥ (x, y))⊥ Φ u,v,0,0,D (-x, -y)dxdy
This can be written

f D (u, v) = f D (u, v) + f D ⊥ (u, v)
where

f D (u, v) = R 2 f (x, y)⊥ Φ u,v,0,0,D (-x, -y)dxdy = R 2 f (x, y)e -(ux+vy)D dxdy and f D ⊥ (u, v) = R 2 f ⊥ (x, y)⊥ Φ u,v,0,0,D (-x, -y)dxdy = R 2 f ⊥ (x, y)e -(ux+vy)ID dxdy.
Let us remark that each one of the two integrals may be identified with the Fourier transform of a function from R 2 to C. Then, we deduce that there exists an inversion formula (left and right) for the Clifford Fourier transform f D given by

f (x, y) = R 2 f D (u, v)⊥ Φ u,v,0,0,D (x, y)dudv
Indeed, the right term equals

R 2 ( f D (u, v) + f D ⊥ (u, v))⊥ Φ u,v,0,0,D (x, y)dudv = R 2 f D (u, v)e (ux+vy)D dudv + R 2 f D ⊥ (u, v)e (ux+vy)ID dudv (3)
Each one of these integrals may be identified with the inversion formula of the Fourier transform of a function from R 2 to C, hence

(3) = f (x, y) + f ⊥ (x, y) = f (x, y).
The following proposition is useful for applications and in particular for applications to the frequencies filtering developed in the next section. It gives an integral representation of any 3D-valued signal defined on the plane by 3D-valued cosinu-soidal signals. This representation is obtained from the Clifford Fourier transform with respect to some bivector. In this proposition we show that the representation is invariant with respect to the choice of the bivector. In the discrete case, we obtain a decomposition of the signal as a sum of cosinusoidal signals.

Proposition 5. Using the previous notations, if B and D are elements of S 2 4,0 , we have f B (u, v)⊥ Φ u,v,0,0,B (x, y)

+ f B (-u, -v)⊥ Φ -u,-v,0,0,B (x, y) = f D (u, v)⊥ Φ u,v,0,0,D (x, y) + f D (-u, -v)⊥ Φ -u,-v,0,0,D (x, y)
Moreover, the e 4 component of this expression is null.

Proof. Simple computations show that

f B (u, v)⊥ Φ u,v,0,0,B (x, y) + f B (-u, -v)⊥ Φ -u,-v,0,0,B (x, y) (4) = R 2 e -xu+yv 2 (B+IB) e λ u+µv 2 (B+IB) f (λ , µ) e -λ u+µv 2 (B+IB) e xu+yv 2 (B+IB) dλ dµ + R 2 e xu+yv 2 (B+IB) e -λ u+µv 2 (B+IB) f (λ , µ) e λ u+µv 2 (B+IB) e -xu+yv 2 (B+IB) dλ dµ = R 2 2cos(u(x -λ ) + v(y -µ)) f (λ , µ)dλ dµ + R 2 2cos(u(x -λ ) + v(y -µ)) f ⊥ (λ , µ)dλ dµ. Hence (4) = R 2 2cos(u(x -λ ) + v(y -µ)) f (λ , µ)dλ dµ.
This proposition justifies the fact that these filters are symmetric with respect to the transformation (u, v) → (-u, -v).

Color image filtering

We now present applications to color image filtering. The use of the Fourier transform is motivated by the well known fact that non trivial filters in the spatial domain may be implemented efficiently with masks in the Fourier domain. Although it seems natural to believe that the results on grey level images may be generalized, there are not so many works dedicated to the specific case of color images. Let us mention the reference [START_REF] Sangwine | Colour-dependent linear vector image filtering[END_REF] where an attempt is made through the use of an ad hoc quaternionic transform. The mathematical construction we propose appears to be well founded since it explains the fundamental role of bivectors and scalar products in terms of group actions. As explained before, the possibility to choose the bivector D and the quadratic form Q is an asset allowing a wider range of applications.

Indeed, Sangwine et al. proposal can be written in our formalism by considering appropriate D and Q.

The applications proposed in this paper are based on the following fact:

( f D ) || = ( f || ) D and ( f D ) ⊥ = ( f ⊥ ) D .
In other words, the part of the Clifford Fourier transform of f that is parallel to D corresponds to the standard Fourier transform of the part of f that is parallel to D.

The same principle holds for the orthogonal part.

We use low pass, high pass and directional filters on the D-parallel part, resp. D-orthogonal part, leaving the D-parallel part, resp. D-orthogonal unmodified. The choice of the bivector D and the quadratic form Q (that determines the D-orthogonal part) will depend on the colors we aim at filtering. Then, we show the action of such filters using the inversion formula of the Clifford Fourier transform.

There is an other way to decompose a color α = (r, g, b), that is with respect to its luminance and chrominance parts, respectively denoted by l α and v α . Embedding the color space RGB into the Clifford algebra R 4,0 by

i α = r e 1 + g e 2 + b e 3 + 0 e 4 ,
the former corresponds to the projection of i α on the axis generated by the unit vector (e 1 + e 2 + e 3 )/ √ 3, the latter its projection on the orthogonal plane in e 1 e 2 e 3 , called the chrominance plane, represented by the unit bivector (e 1 e 2e 1 e 3 + e 2 e 3 )/3. In what follows we make use of the following fact too: every hue can be represented as an equivalence class of bivectors of R 4,0 . More precisely, we have the following result. Then, there is a bijection between T / and the set (e 1 + e 2 + e 3 ) v for v a unit vector in the chrominance plane. This latter being in bijection with the set of different hues, we conclude that there exists a bijection between T / and the set of hues. (e 2 +e 3 )e 1 (Fig. 3(b)). The first one consists in removing high frequencies of the red components of the image, whereas the second one consists in removing high frequencies of the red hue part of the image. In Fig. 3(a), we can see that both green and cyan stripes are not modified. As in the previous case, this comes from the fact that both green color and cyan color 255 e 2 + 255 e 3 belong to ID. The result is different in Fig. 3(b). The unit bivector 1 √ 2 (e 2 + e 3 )e 1 = 1 √ 2 (e 1 + e 2 + e 3 ) ∧ e 1 represents the red hue, involving that the cyan stripes are blurred. Indeed, unit bivectors representing cyan and red hues are opposite, therefore they generate the same plane. Green stripes are no more invariant to the low pass filter since the green axis e 2 is not orthogonal to the bivector In Fig. 4, the color α has been chosen to match with the color of the background green leaves. As the low pass filter (Fig. 4(a)) removes green high frequencies, the center of flowers containing yellow high frequencies turns red. In Fig. 4(b), background pixels corresponding to green low frequencies appear almost grey.

To conclude this part, we propose to compare the results of two low pass filters on the D-orthogonal part with respect to the same bivector D = e 1 e 4 but changing the quadratic form. As a consequence, the bivector ID differs in the two cases.

For the first one (Fig. 5(a)), we take Q 1 , whereas for the second one (Fig. 5(b)), we construct the quadratic form Q 2 such that Q 2 is given by I 4 in the basis (e 1 , 1 √ 2 (e 1 + e 2 ), i α i α , e 4 ). In other words, we orthogonalize the red, the yellow and the color of leaves which are the main colors in the image.

In Fig. 5(a), the unit bivector ID is e 2 e 3 . Hence, the low pass filter removes green and blue high frequencies but preserves red high frequencies. This explains why the image turns red. In Fig. 5(b), the unit bivector ID is (e 1 +e 2 ) √ 2 i α i α , then it contains the colors of the background and inside the flowers. Therefore, the low pass filter removes all the high frequencies in the image except the ones of the red petals.

For some specific applications, a fine tuning of the quadratic form Q should give better results. To conclude this paper, we show how to recover the hypercomplex Fourier transform of S. Sangwine and the quaternionic Fourier transform of T. Bülow in the Clifford algebras context, using the appropriate morphism from R 2 to Spin(4). First of all, let us recall the definitions of these Fourier transforms.

5.1

The hypercomplex Fourier transform of Sangwine et al.

In [START_REF] Ell | Hypercomplex Fourier Transforms of Color Images[END_REF], the authors define the discrete hypercomplex Fourier transform. It can be extended to R 2 as follows. Let f : R 2 → H, then its hypercomplex Fourier transform is given by

F(u, v) = R 2 e -µ(xu+yv) f (x, y)dx dy where µ ∈ H 0 ∩ H 1 .
There is a freedom in the choice of µ in the hypercomplex Fourier transform as we have a freedom in the choice of the bivector D in the Clifford Fourier transform for color images. In fact, they have the same role, i.e. they decompose the four-dimensional space R 4 into two orthogonal two-dimensional subspaces and decompose the Fourier transform into two standard Fourier transforms. This is shown in the following proposition.

Proposition 7. Let µ = µ 1 i+ µ 2 j + µ 3 k be a unit quaternion. Let f ∈ L 2 (R 2 , (R 4 , Q))
where Q is the quadratic form represented by I 4 in the basis (e 1 , e 2 , e 3 , e 4 ), and let C be the unit bivector e 4 ∧ (µ 1 e 1 + µ 2 e 2 + µ 3 e 3 ). Then, f C given by Proof. We have to determine the four-dimensional rotation that is generated by the action of the unit quaternion e µφ on H given by q -→ e µφ q.

f C (u, v) = R 2 f (x, y)⊥ Φ u,v,0,0,C (-x, -y)dxdy = R 2 e 1 2 (xu+yv)IC e 1 2 (xu+yv)C f (x, y)e -1 2 (xu+yv)C e -1 2 (
It is explained in [START_REF] Ell | Hypercomplex Fourier Transforms of Color Images[END_REF] that this rotation may be decomposed as the sum of two twodimensional rotations of angle -φ in the planes generated by (1, µ) and its orthogonal (with respect to the euclidean quadratic form). Therefore, we can identify this rotation with the action of the spinor

e -φ 2 (C+IC)
on the four-dimensional space R 1 4,0 . As a consequence, the action of group morphisms (x, y) -→ e µ(xu+yv) from R 2 to H 1 on H corresponds to the action of group morphisms (x, y) -→ e -1 2 (xu+yv)(C+IC) from R 2 to Spin(4) on R 1 4,0 . Remark 5. To the best of our knowledge, the authors restrict for their applications to µ taken as the grey axis, i.e.

µ = 1 √ 3 (i + j + k).
In other words, the Fourier transform they propose is decomposed as a standard Fourier transform of the luminance part and a standard Fourier transform of the chrominance part.

The quaternionic Fourier transform of B ülow

The quaternionic Fourier transform [START_REF] Bülow | Hypercomplex spectral signal representations for the processing and analysis of images[END_REF] of a function f : R 2 → R is the quaternion valued function F ( f ) defined by

F ( f )(y 1 , y 2 ) = R 2 exp(-2πiy 1 x 1 ) f (x 1 , x 2 )exp(-2π jy 2 x 2 ) dx 1 dx 2 .
2 The product law needs not to be respected since we just use an isomorphism of vector spaces.

The link between this Fourier transform and the one proposed here is given by the next result.

Proposition 8. Let f ∈ L 2 (R 2 ; Re 4 ) where (e 1 , e 2 , e 3 , e 4 ) is the basis of R 4 that generates R 4,0 . The Clifford Fourier transform of f defined by

f C (2πy 1 , 0, 0, 2πy 2 ) = R 2 f (x 1 , x 2 )⊥ Φ 2πy 1 ,0,0,2πy 2 ,C (-x 1 , -x 2 ) dx 1 dx 2
where C is the bivector -1 4 (e 1 + e 2 )(e 3e 4 )

corresponds to the quaternionic Fourier transform of f seen as a H-valued function under the following identification3 e 1 ↔ i e 2 ↔ j e 3 ↔ k e 4 ↔ 1.

Proof. We have to determine one of the two elements of Spin(3) × Spin(3) that generate the following rotation in H :

f (x 1 , x 2 ) → exp(-2πiy 1 x 1 ) f (x 1 , x 2 )exp(-2π jy 2 x 2 ).
Simple computations show that the rotation

f (x 1 , x 2 ) → exp(-2πiy 1 x 1 ) f (x 1 , x 2 )
can be written in R 1 4,0 as f (x 1 , x 2 ) → e -πx 1 y 1 (e 4 e 1 +e 2 e 3 ) f (x 1 , x 2 )e πx 1 y 1 (e 4 e 1 +e 2 e 3 ) .

In the same way,

f (x 1 , x 2 ) → f (x 1 , x 2 )exp(-2π jy 2 x 2 ) corresponds to f (x 1 , x 2
) → e -πx 2 y 2 (e 4 e 2 +e 1 e 3 ) f (x 1 , x 2 ) e πx 2 y 2 (e 4 e 2 +e 1 e 3 )

By associativity, this shows that exp(-2πiy

1 x 1 ) f (x 1 , x 2 ) exp(-2π jy 2 x 2 ) = e -τ e -ρ f (x 1 , x 2 )e ρ e τ

By definition,

χ(e ρ e τ ) = χ(e πx 1 y 1 e 4 e 1 )χ(e πx 1 y 1 e 2 e 3 )χ(e πx 2 y 2 e 4 e 2 )χ(e πx 2 y 2 e 1 e 3 ).

From simple computations, we get χ(e ρ e τ ) = (e 2πx 1 y 1 e 2 e 3 , e 2πx 2 y 2 e 1 e 3 )

and conclude therefore that (x 1 , x 2 ) → (e 2πx 1 y 1 e 2 e 3 , e 2πx 2 y 2 e 1 e 3 )

is the morphism φ 2πy 1 ,0,e 2 e 3 ,0,2πy 2 ,e 1 e 3 .

From Section 3, this latter may be rewritten Φ 2πy 1 ,0,0,2πy 2 , 1 4 (e 1 +e 2 )(e 3 -e 4 )) . Indeed, we have

Conclusion

We proposed in this paper a definition of Clifford Fourier transform that is motivated by group actions considerations. We defined a Clifford Fourier transform that is associated with the action of all the group morphisms Φ u,v,w,z,D from R 2 to Spin(4), parameterized by four real numbers and one unit bivector. This transform has the property of being left-invertible. For the particular case of a color image, we associate the Clifford Fourier transform with the action of group morphisms Φ u,v,0,0,D , specified by only two real numbers (the frequencies) and where the bivector D is fixed. This transform is parameterized by a quadratic form on R 4 and a unit bivector in the corresponding Clifford algebra. Some previous Fourier transforms based on quaternions are proved to be particular settings of ours. We have treated in this context an application to color image filtering. Future works will be devoted to find applications of the general transform, that should easily deal with relations between colors in the image. Applications to multispectral images such as color/infrared images will be also investigated.

< ϕ(g)(a), ϕ(g)(b) >=< a, b > ∀a, b ∈ V ∀g ∈ G.

•

We now restrict to locally compact unimodular groups. On such groups, we can construct a measure that is invariant with respect to both left and right translations. It a called a Haar measure. From a Haar measure is defined a Haar integral of the group.

Proposition 9. Let G be a locally compact unimodular group, and ν denotes a Haar measure. Then, for f ∈ L 2 (G; C) and h ∈ G, we have

G f (g) dν(g) = G f (gh) dν(g) = G f (
f ∈ L 2 (G; C) is the map f defined on G by f (ϕ) = G f (g)ϕ(g -1
) dν(g).

•

Theorem 1. Inversion formula of the Fourier transform. f (ϕ) is a Hilbert-Schmidt operator over the space of the representation ϕ. There is a measure over G denoted by ν such that f ∈ L 2 ( G; C) and f → f is an isometry. Moreover, the following inverse formula holds:

f (g) = G Trace( f (ϕ)ϕ(g)) d ν(ϕ).

•

Let us now have a closer look on Lie groups. We refer to [START_REF] Helgason | Differential geometry, Lie groups and symmetric spaces[END_REF] for an introduction to differential geometry.

X,Y ) → [X,Y ] from g × g to g is R-bilinear. Moreover it satisfies [X, X] = 0 ∀ X ∈ g for all v in V (1 denotes the unit of Cl(V, Q)) such that for each R-algebra A and each R-linear map f : V -→ A with ( f (v)) 2 = Q(v).1
for all v in V (1 denotes the unit of A), then there exists a unique morphism

g : Cl(V, Q) -→ A of R-algebras such that f = g • i Q .
The solution is unique up to isomorphisms and is given as the (non commutative) quotient

T (V )/(v ⊗ v -Q(v).1)
of the tensor algebra of V by the ideal generated by v ⊗ v -Q(v).1, where v belongs to V (see [START_REF] Postnikov | Lec ¸ons de Géométrie[END_REF] for a proof).

It is well known that there exists a unique anti-automorphism t on Cl(V, Q) such that

t(i Q (v)) = i Q (v)
for all v in V . It is called reversion and usually denoted by x -→ x † , x in Cl(V, Q).

In the same way there exists a unique automorphism α on Cl(V, Q) such that

α(i Q (v)) = -i Q (v)
for all v in V . In this paper we write v for i Q (v) (according to the fact that i Q embeds V in Cl(V, Q)).

As a vector space Cl(V, Q) is of dimension 2 n on R and a basis is given by the set {e i 1 e i is called a k-vector. A 0-vector is a scalar and e 1 e 2 • • • e n is called the pseudoscalar. We denote x k the component of degree k of an element x of Cl(V, Q).

The inner product of x r of degree r and y s of degree s is defined by The outer product of x r of degree r and y s of degree s is defined by

x r ∧ y s = x r y s r+s

These products extend by linearity on Cl(V, Q). Clearly, if a and b are vectors of V , then the inner product of a and b coincides with the scalar product defined by Q.

When it is defined (for example when x is a versor and Q is positive) we denote

x = √ xx †
and say that x is a unit if xx † = ±1.

In this paper, we deal in particular with the Clifford algebra of the Euclidean R n denoted by R n,0 . R k n,0 is the subspace of elements of degree k and R * n,0 is the group of elements that admit an inverse in R n,0 . We denote by S 2 n,0 the set of elements of R 2 n,0 of norm 1.

Let a be a vector in R n,0 and B be the k-vector a 1 ∧ a 2 ∧ • • • ∧ a k , then the orthogonal projection of a on the k-plane generated by the a i 's is the vector and is isomorphic to the group H 1 of unit quaternions. It is also a classical result that Spin(4) is isomorphic to Spin(3)×Spin(3) (see [START_REF] Lounesto | Clifford algebras and Spinors[END_REF] for more information on spinors in R 3 and R 4 ).

The Lie algebra of Spin(n) is R 2 n,0 with Lie bracket

A × B = AB -BA.
As the exponential map from its Lie algebra to Spin(n) is onto (see [START_REF] Helgason | Differential geometry, Lie groups and symmetric spaces[END_REF] for a proof), every spinor can be written as

S = ∞ ∑ i=0 1 i! A i
for some bivector A.

From Hestenes and Sobczyk [START_REF] Hestenes | Clifford algebra to geometric calculus[END_REF], we know that every A in R 2 n,0 can be written as

A = A 1 + A 2 + • • • + A m
where m ≤ n/2 and A j = A j a j b j , j ∈ {1, . . . , m}, for all σ in the permutation group S(m). Actually, as A 2 k is negative we have e A i = cos( A i ) + sin( A i )

A i A i
The corresponding rotation R i : x -→ e -A i xe A i acts in the oriented plane defined by A i as a plane rotation of angle 2 A i . The vectors orthogonal to A i are invariant under R i . It then appears that any element R of SO(n) is a composition of commuting simple rotations, in the sense that they have only one invariant plane. The vectors left invariant by R are those of the orthogonal subspace to A. If m = n/2 this latter is trivial. The previous decomposition is not unique if A k = A j for some j and k with j = k. In this case infinitely many planes are left invariant by R.

2 3 ,

 3 0 whose dimension is inferior or equal to 2. If a = a 1 e 1 e 2 + a 2 e 3 e 1 + a 3 e 2 e 3 and b = b 1 e 1 e 2 + b 2 e 3 e 1 + b 3 e 2 e 3 satisfy a × b = 0, then the structure relations of R 2 3,0 , i. e. e 1 e 2 × e 3 e 1 = e 2 e 3 , e 3 e 1 × e 2 e 3 = e 1 e 2 , e 2 e 3 × e 1 e 2 = e 3 e 1 , imply (a 1 b 2a 2 b 1 )e 2 e 3 -(a 1 b 3a 3 b 1 )e 3 e 1 + (a 2 b 3a 3 b 2 )e 1 e 2 = 0.

1 2 (

 2 ux+vy)B . Let us recall what group is Spin(4). Every τ in Spin(4) is of the form τ = u + Iv = (a1 + be 1 e 2 + ce 2 e 3 + de 3 e 1 ) + I(a 1 + b e 1 e 2 + c e 2 e 3 + d e 3 e 1 )

Proposition 2 . 1 8 1 8 1 2 (ux+vy)B , e 1 2

 21111 The group morphisms from R 2 to Spin(4) are the morphisms φ u,v,B,w,z,C that send (x, y) to e [x(u+w)+y(v+z)][B+C+I(B-C)] e [x(u-w)+y(v-z)][B-C+I(B+C)] with u, v, w, z real and B, C two elements of S 2 3,0 . Proof. The group law of Spin(3) × Spin(3) being (a, b), (c, d) → (ac, bd), the group morphisms from R 2 to Spin(3) × Spin(3) are the morphisms ϕ u,v,B,w,z,C defined by ϕ u,v,B,w,z,C : (x, y) → e (wx+zy)C with u, v, w, z real and B, C two elements of S 2 3,0 .By χ -1 , the group morphisms from R 2 to Spin(4) are the φ u,v,B,w,z,C that send (x, y)

4 +I 1 8 1 8

 411 ux+vy)B+(wx+zy)C [(x(u+w)+y(v+z))(B+C+I(B-C))] e [(x(u-w)+y(v-z))(B-C+I(B+C))]

xu+yv)ID e 1 2Definition 3 .

 13 (xu+yv)D f (x, y)e -1 2 (xu+yv)D e -1 2 (xu+yv)ID dxdy. • It follows the definition of the Clifford Fourier transform of a color image. Clifford Fourier transform of a color image. Let I be a color image. We associate to I a function f ∈ L 2 (R 2 , (R 3 , Q)) defined by f (x, y) = r(x, y)e 1 + g(x, y)e 2 + b(x, y)e 3 + 0e 4 where r, g and b correspond to the red, green and blue levels. The Clifford Fourier transform of I with respect to Q and D is the Cl(R 4 , Q ⊕ 1)valued function I Q,D defined by

Proposition 6 .

 6 Let T be the set of bivectors T = {(e 1 + e 2 + e 3 ) ∧ i α , α ∈ RGB} with the following equivalence relation: B C ⇐⇒ B = λC for λ > 0Then, there is a bijection between T / and the set of hues.Proof. We have (e 1 + e 2 + e 3 ) ∧ i α = (e 1 + e 2 + e 3 )v α

Fig. 1

 1 Fig. 1 Original images

Fig. 2 4 Fig. 2 (Fig. 3

 2423 Fig. 2 The Clifford-Fourier transform H Q 1 ,e 1 e 4

1 √ 2 (e 2 +

 122 Fig. 4 The Clifford-Fourier transform F Q 1 , vα ∧e 4 vα ∧e 4

4 Fig. 5

 45 Fig. 5 Low pass filters in the ID part

2 e 1 ↔

 21 xu+yv)IC dxdy corresponds to the hypercomplex Fourier transform of f seen as a H-valued function under the identification i e 2 ↔ j e 3 ↔ k e 4 ↔ 1.

where τ = πx 2 y 2 (e 4 e 2 + e 1 e 3 )

 223 and ρ = πx 1 y 1 (e 4 e 1 + e 2 e 3 ).

1 4 e 2 e 3 + 4 = 1 4 e 1 (e 3 -e 4 ) + e 2 (e 3 -e 4 ) = 1 4 (e 1 +

 341342341 e 1 e 3 + I(e 2 e 3e 1 e 3 ) = 1 4 e 2 e 3 + e 1 e 3e 1 e 4e 2 e e 2 )(e 3e 4 )

  x r • y s = x r y s |r-s| if r and s are positive and byx r • y s = 0 otherwise.

  P B (a) = (a • B)B -1 The vector a -(a • B)B -1 = (a ∧ B)B -1is called the rejection of a on B.C. The spinor group Spin(n)a i ∈ R 1 n,0 , a i = 1 or equivalently Spin(n) = {x ∈ R n,0 , α(x) = x, xx † = 1, xvx -1 ∈ R 1 n,0 ∀v ∈ R 1 n,0 }.It is well known that Spin(n) is a connected compact Lie group that universally covers SO(n) (n ≥ 3). One can verify that Spin(3) is the group {a1 + be 1 e 2 + ce 2 e 3 + de 3 e 1 , a 2 + b 2 + c 2 + d 2 = 1}

with {a 1

 1 , . . . , a m , b 1 , . . . , b m } a set of orthonormal vectors. ThusA j A k = A k A j = A k ∧ A j whenever j = k and A 2 k = -A k 2 < 0This means that the planes encoded by A k and A j are orthogonal and implies thate A 1 +A 2 +•••+A m = e A σ (1) e A σ (2) . . . e A σ (m)

  leads to the usual definition of the Fourier transform, resp. Fourier coefficients, resp. discrete Fourier transform.

Traditionally, the Fourier transform in L 2 (R m , (R n , 2 )) is defined by n standard Fourier transforms in L 2 (R m , R) on each one of the components, embedding R into C. Using group representations theory, we are able to define Fourier transforms that treat jointly the different components.

  hg) dν(g). Locally compact abelian groups and compact groups are unimodular.Definition 8. Fourier transform on locally compact unimodular groups. Let G be a locally compact unimodular group with Haar measure ν. The Fourier transform of

	•
	Remark 6.

  Definition 9. Lie group and Lie algebra. A real C ∞ Lie group is a topological group endowed with a structure of real C ∞manifold. The Lie algebra of G is (isomorphic to) the tangent space of G at the neutral element e: T e G. It is usually denoted by g. It can be made into an algebra over R by considering the Lie bracket [ , ] that satifies: (

  [START_REF] Bülow | Hypercomplex spectral signal representations for the processing and analysis of images[END_REF]

  • • • e i k , i 1 < i 2 < . . . < i k , k ∈ {1, . . . , n}}and the unit 1. An element of degree k∑ i 1 <•••<i k α i 1 ...i k e i 1 e i 2 • • • e i k

Available at http://mia.univ-larochelle.fr/ → Production → Démos

The product law needs not to be respected since we just use an isomorphism of vector spaces.

Thomas Batard, Michel Berthier and Christophe Saint-Jean

Acknowledgments

This work is partially supported by the "Communauté d'agglomération de La Rochelle".

Appendix A. Lie groups representations and Fourier transforms

From the groups theory approach, the basic structure we need to define Fourier transforms is locally compact unimodular groups. Let us start by the definition of the dual of a topological group G, that is the set of the equivalence classes of its unitary irreducible representations, denoted by G . We refer to [START_REF] Vilenkin | Special functions and the theory of group representations[END_REF] for details.

Definition 4. Group representation.

Let G be a topological group and V be a topological vector space over R or C.

In general, V is an Hilbert space. If V is finite-dimensional, then the representation is said to be finite, and the dimension of V is called the degree of the representation.

Then, the representation ϕ is said to be irreducible if W and {0} are the only subspaces of V that are invariant by ϕ.

) and (ϕ 2 ,V 2 ) be two linear representations of the same group G. We say that they are equivalent if there exists an isomorphism γ :

• From now on, V is a C-vector space equipped with a hermitian form < , >. where f : R → G satisfies

f is called a one-parameter subgroup.

•

To compute group morphisms from R 2 to Spin(3) and Spin(4), we use the following result on Lie groups morphisms.

Proposition 10. Let G and H be two C ∞ Lie groups, and exp G , exp H be the corresponding exponential maps. Let φ : G → H be a Lie group morphism. The linear tangent map of φ at g, denoted by T g φ , is the linear map from T g G to T φ (g) H given by

Then, if we note e the neutral element of G we have φ (exp G (X)) = exp H (T e φ (X)).

(5)

•

The map T e φ is a Lie algebra morphism, i.e. it satisfies

From ( 5), we deduce that if the group G is connected and the exponential map of G is onto, then the Lie group morphisms from G to H are determined by Lie algebras morphisms from g to h.

B. Clifford algebras

Let V be a vector space of finite dimension n over R equipped with a quadratic form Q. Formally speaking, the Clifford algebra Cl(V, Q) is the solution of the following universal problem. We search a couple