Jo Caers

Dirk Hose

Ine Kuipers

Tomas Jan Bos

Els Van Valckenborgh

Eline Menu

Elke De Bruyne

Hartmut Goldschmidt

Ben Van Camp

Bernard Klein

♦ Dirk Hose

Els Van Valckenborg

Karin Vanderkerken
email: karin.vanderkerken@vub.ac.be

Hae-Jae Cha

Thymosin beta4 has tumor suppressive effects and its decreased expression results in

est destinée au dépôt

Introduction

Βeta-thymosins are a family of small peptides that were originally proposed to be thymic hormones [START_REF] Goldstein | Thymosins: chemistry and biological properties in health and disease[END_REF]. They were identified as actin monomer binding proteins, controlling the availability of actin for polymerization. They therefore may have a crucial role in regulating cellular functions involving actin polymerization/depolymerization cycles.

Currently, 15 -thymosins have been identified and characterized as highly conservative 5-kDa peptides containing 40 to 44 amino acid residues. In most mammalian tissues, thymosin- 4 (Tβ4), the most abundant thymosin peptide, T10 and T15, have been studied as important members of the -thymosin family [START_REF] Hannappel | beta-Thymosins[END_REF]. Several studies reported that these genes are overexpressed in solid tumors, which could be correlated to the angiogenic and metastatic potential of the studied tumors [START_REF] Chen | Roles of thymosins in cancers and other organ systems[END_REF].

Multiple Myeloma (MM) is a hematological malignancy characterized by the accumulation of monoclonal plasma cells (PC) in the bone marrow (BM). MM cell biology can be dissected into the interactions of MM cells with their surrounding stroma (matrix proteins, cytokines and BM cells) and in the acquisition of essential changes in cell behavior such as self-sufficiency in growth signals, evasion of apoptosis and acquisition of invasive and spreading capacities [START_REF] Caers | Unraveling the biology of multiple myeloma disease: cancer stem cells, acquired intracellular changes and interactions with the surrounding micro-environment[END_REF]. Earlier reports indicated that T4 was downregulated in RNA from primary human MM cells and cell lines [START_REF] Gondo | Differential expression of the human thymosin-beta 4 gene in lymphocytes, macrophages, and granulocytes[END_REF]. This observation is in contrast to the results obtained in most solid tumors where an upregulation is often seen in malignant cells compared to their normal counterparts. Cha et al showed that overexpression of T4 resulted in an increased metastatic capacity of lung cancer cells and increased angiogenic response [START_REF] Cha | Role of thymosin beta4 in tumor metastasis and angiogenesis[END_REF]. Since migration, invasion and associated angiogenesis are key features in MM biology, we were interested in studying T4 expression in a large panel of MM patients and its functionality in the murine 5TMM model.

Material and methods

Gene expression analysis on human myeloma cells.

T4 expression was analyzed in purified PCs from BM samples obtained from 14 healthy donors, 11 patients with monoclonal gammopathy of unknown significance (MGUS) and 298 previously untreated multiple MM patients at the University Hospitals of Heidelberg or Montpellier [START_REF] Hose | Induction of angiogenesis by normal and malignant plasma cells[END_REF]. Of these, 209 MM-patients were treated by high dose therapy and autologous stem cell transplantation (ASCT). Biotinylated complementary RNA (cRNA) was amplified according to the Affymetrix labeling protocol (Affymetrix, Santa Clara, CA, USA). cRNA from a first group of patients (7 normal donors, 7 MGUS-and 65 MM-patients) was hybridized to the human U133 A and B. This group is termed hereafter the HM1-group. A second independent validation group of patients (7 normal donors, 16 MGUS-and 233 MM-patients) was termed the HM2 group. For these patients the U133 2.0 GeneChip was used. These micro-array data were previously used for several analyses, but thymosin beta 4 expression was never analyzed before [START_REF] Sprynski | The role of IGF-1 as a major growth factor for myeloma cell lines and the prognostic relevance of the expression of its receptor[END_REF][START_REF] Hose | Inhibition of aurora kinases for tailored risk-adapted treatment of multiple myeloma[END_REF]. HM2 data were corrected for batch effect due to the usage of different labelling kits according to Johnson et al [START_REF] Johnson | Adjusting batch effects in microarray expression data using empirical Bayes methods[END_REF]. Expression data were gcrma-normalised and analyzed by the bioconductor packages for R. For patient characteristic see supplementary table 1.

The 5T2MM and 5T33MM murine models of myeloma.

The 5TMM models originated in elderly C57Bl/KaLwRij mice [START_REF] Radl | Multiple myeloma and related disorders. Lessons from an animal model[END_REF]. The 5T33MMvivo (5T33MMvv) cells grow in vitro stroma-dependently with a limited survival while the 5T33MMvitro (5T33MMvt) cell line is a clonally identical variant that originated from an in vitro culture of 5T33MMvv cells, growing BM stroma-independently in RPMI-1640 supplemented with 10% bovine serum 1% natriumpyruvate, 100 U/ml penicillin, 100 µg/ml streptomycin and 2 mM L-glutamine (all from Biowhittaker, Verviers, Belgium) [START_REF] Manning | A model of multiple myeloma: culture of 5T33 murine myeloma cells and evaluation of tumorigenicity in the C57BL/KaLwRij mouse[END_REF].

Quantification of intracellular protein levels of Tβ4 and F-Actin -GActin

Enzyme-Linked Immunosorbent Assays (ELISA) for measuring Tβ4 concentrations were performed according to the manufacturer's instructions (Immundiagnostik, Bensheim, Germany). Cells (10 7 ) were lyzed in a phosphate buffer containing 0.14 M NaCl, 2.6 M KCl, 8 mM Na2HPO4, 1.4 M KH2PO4 and 1% Triton X100 and sonicated with an ultrasound finger. Protein levels and ratios between F-Actin and G-Actin were determined using the G-actin / F-actin in vivo assay kit (Cytoskeleton Inc, Denver, USA).

Quantitative real-time PCR

Quantitative real-time PCR (qRT-PCR) was performed using the ABI Prism 7700 Sequence Detection System. For the detection of both human and mouse Tβ4 mRNA and the endogenous reference gene GUS, Assays on Demand (Applied Biosystems) were used. To verify the results obtained with the microarrays studies, Tβ4 expression was measured in 3 cell lines and in 3 patient samples and their correlations statistically verified using a Spearman correlation test.

Generation of 5T33MMvt cells overexpressing T4

A lentiviral transferplasmid encoding mouse Tβ4 (mTβ4) was constructed. The mTβ4 gene was obtained from H.J. Cha (NIDCR, NIH, Bethesda, USA [START_REF] Cha | Role of thymosin beta4 in tumor metastasis and angiogenesis[END_REF]) and inserted into the transferplasmid pHR'tripCMV-IRES-tNGFR-SIN [START_REF] Breckpot | Lentivirally transduced dendritic cells as a tool for cancer immunotherapy[END_REF]. mTβ4-encoding lentiviral vector particles were produced in 293T cells, collected, ultracentrifugated and their viral titer determined [START_REF] Breckpot | Identification of new antigenic peptide presented by HLA-Cw7 and encoded by several MAGE genes using dendritic cells transduced with lentiviruses[END_REF]. After transduction, 5T33MMvt cells were surface stained using a house made biotinylated anti-tNGFR antibody and purified by FACS sorting into a 6-well plate (Becton Dickinson, FACSVantage). Next, they were analyzed for Tβ4 expression by RT-PCR. The 5T33MMvt cells overexpressing Tβ4 are termed in the following 5T33MMvt Tβ4+ .

In vitro and in vivo effects of T4-overexpression

In vitro proliferation was assessed by measuring DNA synthesis using a 3 H-thymidine incorporation assay, as described earlier [START_REF] Caers | Antitumour and antiangiogenic effects of Aplidin in the 5TMM syngeneic models of multiple myeloma[END_REF]. Apoptosis sensitivity of the MM cells was analyzed by staining with FITC labeled-annexin V and propidium iodide according to the manufacturers' instructions (BD Biosciences, Erembodegem, Belgium). In vitro migration studies were performed using Transwell chambers and 10% fatal calf serum as chemoattractant and were quantified through flow cytometry. To determine the effect of Tβ4 overexpression on survival, groups of 10 C57BLKaLwRij mice were intravenously injected with either 5x10 5 5T33MMvt Tβ4+ or wild type 5T33MMvt cells. Animals were sacrificed when they showed signs of morbidity, namely hind limb paralysis. Kaplan-Meier analysis was used to determine a difference in the survival.

Results and Discussion

Different studies indicated a pivotal role of T4 in the metastatic process of solid tumors [START_REF] Nummela | Thymosin beta4 is a determinant of the transformed phenotype and invasiveness of S-adenosylmethionine decarboxylase-transfected fibroblasts[END_REF][START_REF] Goldstein | Thymosin beta4: a new molecular target for antitumor strategies[END_REF]. An adenoviral-based overexpression of Tβ4 was applied in a colon cancer and melanoma model showing increased growth, motility and invasive capacities in vitro and a larger tumor load in vivo [START_REF] Wang | Overexpression of the thymosin beta-4 gene is associated with increased invasion of SW480 colon carcinoma cells and the distant metastasis of human colorectal carcinoma[END_REF][START_REF] Wang | Overexpression of the thymosin beta-4 gene is associated with malignant progression of SW480 colon cancer cells[END_REF]. Since proliferation, migration and invasion are part of the hallmarks in the biology of MM, we were interested in investigating an involvement of T4 in this disease. We first investigated the Tβ4-expression pattern in 298 primary MM-cell samples and 14 normal plasma cell samples from healthy donors. Given the differential T4 expression in MM patients, we subsequently investigated a possible prognostic value and influence on event free (EFS) and overall survival (OS) in our patient population. As T4 was expressed in all MM-cells, we examined the survival of 209 patients by comparing patients with T4 expression above (T4 high ) and below (T4 low ) the median (Figure 1.b). Patients with T4 high vs. T4 low show a significantly longer median EFS (n=209, 37.6 months vs. 26,2, P < .05), but only a trend regarding OS (P =.1). Concerning EFS, multivariate analyses on T4 expression with either ISS or β2-microglobulin indicated an independent (P =.04) prognostic value of T4 expression regarding ISS (P<.001), but not (P =.07) regarding β2-microglobulin (P=.003) levels. In multi-variate analyses for OS, ISS and β2-microglobulin appear as significant (P =<.001) variables, whereas T4 expression fails to reach independence (P =.09 and P =.1, respectively).

A supervized analysis of expression data comparing the T4 high vs. T4 low group identified over 300 significantly differentially expressed genes. These genes are listed in supplementary table 2. Analysis of their biological function allowed separation in main functional categories and this distribution is illustrated in figure 1.c. Signal transduction, protein metabolism and nuclear functions were the largest categories, but 19 genes were implicated in cytoskeletal organization and 32 genes in lymphoid differentiation and immunoglobulin processing. In general these gene clusters indicate a biological difference between MM cells of the two patient groups .

The data obtained in MM-patients were also seen in the 5T33MM murine MM model by qRT-PCR demonstrating a decreased mRNA expression in 5TMM cells compared to normal BM cells (P<.001, Figure 1.d). Competitive ELISA confirmed these results on protein level (results not shown). To study functional effects of T4, the gene was overexpressed using a lentiviral expression vector. The 5T33MMvt cell line was stably transduced and after subcloning, a 99% pure clone with strong t-NGFR expression was obtained. qRT-PCR confirmed the overexpression of Tβ4 compared to control cells (Figure 2a).

To assess the functional involvement of differential Tβ4 expression we used the 5T33MMvt and 5T33MMvt Tβ4+ cell lines. In a 3 H thymidine assay, 5T33MMvt Tβ4+ cells showed a significant decrease in DNA synthesis compared to control cells (P<.05). 5T33MMvt Tβ4+ cells showed an significantly increased sensitivity to vinca-alkaloids (vinblastin) and bortezomib (Figure 2.b P <.001 for both bortezomib and vinblastin).

Likewise, bortezomib induced apoptosis was higher in 5T33MMvt Tβ4+ vs. 5T33MMvt cells (P<.05, Figure 2.c). In addition to affect survival pathways, Tβ4 overexpression reduced migratory capacities of 5T3MM cells; the percentages of cells that migrated in basal conditions and in 10%FCI was significantly lower in 5T33MMvt Tβ4+ compared to control cells (P<.05, Figure 2.d). The relative increase after stimulation (compared to basal conditions) was however similar in both populations. We further examined the effects of Tβ4 expression on tumor development and survival of diseased mice by injecting mice intravenously with 5T33MMvt Tβ4+ or control cells. In this study, the mean survival of mice injected with control cells was significantly shorter 65,9 days (SD 6.6 days), compared to 88,9 days (SD 9,3 days) for mice injected with 5T33MMvt Tβ4+ cells (P< .05, Figure 2.f). These in vivo results confirm data obtained using the in vitro proliferation and apoptosis assays.

In solid tumors, T4 expression is frequently upregulated in malignant and metastatic cells. In these cancers, higher T4-expression resulted in increased metastatic and invasive capacities of tumor cells, while proliferation remained unaffected [START_REF] Cha | Role of thymosin beta4 in tumor metastasis and angiogenesis[END_REF]. In hematological disorders, malignant plasma cell disorders, such as plasma cell leukemia and MM were the rare disorders that showed a decreased T4-expression. [START_REF] Gondo | Differential expression of the human thymosin-beta 4 gene in lymphocytes, macrophages, and granulocytes[END_REF][START_REF] Shimamura | Expression of the thymosin beta 4 gene during differentiation of hematopoietic cells[END_REF]. In contrast to solid tumors, publications on the function of T4 in hematological conditions are scanty but indicate some inhibitory activity. T4 was initially isolated and purified from a thymic protein preparation, called thymosin fraction-5. Addition of this protein fraction to different leukemic cell lines resulted into decreased growth responses [START_REF] Spangelo | Presence of a peptide component of thymosin fraction-5 manifesting discrete cytostatic properties in HL-60 human promyelocytic leukemia cells[END_REF].

Similar inhibitory effects were recently described for T4 on hematopoietic stem cells [START_REF] Bonnet | Thymosin beta4, inhibitor for normal hematopoietic progenitor cells[END_REF], bone marrow derived mast cells [START_REF] Leeanansaksiri | Thymosin beta 4 and its N-terminal tetrapeptide, AcSDKP, inhibit proliferation, and induce dysplastic, non-apoptotic nuclei and degranulation of mast cells[END_REF] and human promyelocytic leukemia cells [START_REF] Huang | Thymosin beta4 and AcSDKP inhibit the proliferation of HL-60 cells and induce their differentiation and apoptosis[END_REF], in agreement with the results presented here. Whereas a mechanistic explanation of this discrepancy is beyond the scope this paper, further investigations are clearly merited.

Since Tβ4 has been shown to bind G-actin in a 1:1 manner and thus affects the polymerization of G-Actin into F-Actin, we analyzed in a semi-quantitative way, intracellular G-actin and F-Actin. This quantification showed a lowered G-Actin-F-Actin ratio after Tβ4 overexpression (Figure 2.e). F-Actin is of particular importance in cytoskeleton changes involved in cellular migration and in microtubuli organization controlling the mitotic spindle [START_REF] Menu | The Factin content of multiple myeloma cells as a measure of their migration[END_REF][START_REF] Xu | Actin and microtubules: working together to control spindle polarity[END_REF]. In line with these results, vinca-alkaloids (e.g. vinblastine used here) with micro-tubulin [polymerization] inhibitory activity, had more affect on the proliferation capacities of 5T33MMvt Tβ4+ cells than on control cells (fig 2.b). Since immunohistochemical studies also showed a nuclear staining of Tβ4 in 5TMM cells (results not shown), involvement of other pathways might also be implicated. Supervized gene analysis comparing T4 high vs. T4 low found different groups of genes differently expressed, included genes involved in cytoskeleton organization, nuclear homeostasis, lymphocyte differentiation and protein metabolism, which might indicate that the role of Tβ4 is more complicated as initially supposed.

In conclusion, our results propose a tumor suppressive function of Tβ4-expression in MM with impact on survival. Tβ4 was downregulated in MM cells of patients compared to the normal BM plasma cells and studies with the murine 5T33MM model show a decreased in vitro and in vivo tumor growth for cells overexpressing the Tβ4 gene.

Authorship

JC and DH were the principal investigators and took primary responsibility for the paper. JC, DH, IK, TJB, EDB and EM participated in the laboratory work for this study. BVC, EVV, BK and KV coordinated the research. HG and BK were responsible for patient recruitment and patient data. JC, DH, TJB, BK and KV wrote the paper. The authors reported no potential conflicts of interest.

Running heads J Caers et al. Thymosin beta 4 as tumor suppressor in MM

Supplementary Table S1. Clinical data for patients undergoing high-dose chemotherapy and autologous stem cell transplantation. Clinical patient data for age, serum-β2-microglobulin, and plasma cell infiltration in the Heidelberg/Montpellier-group 1 (HM1) and -2 (HM2). Median value and range are given. NA, not available; ISS, International Staging System. S2. Supervised gene analysis resulted in identification of differently expressed genes in myeloma cells from T4 high and T4 low patients from the validation group. After identification of the biological function of the gene, these were grouped into different classes. survival of T4 high and T4 low patients. Patients with T4 low MM had statistically significantly decreased event free survivals compared to patients with T4 high MM (P<.001), while also their overall survival tended to be shorter. Figure 1.c illustrates the differently expressed genes between T4 high and T4 low patient groups. After identification of the gene, these were grouped into similar biological function. A complete list of the genes can be found in supplementary table 2.

Supplementary Table

Characteristic

T4

  is significantly lower expressed in MM cells of the HM1 group (P<.05) and HM2 group (P<.001) compared to normal plasma cells. This holds true for a significantly lower T4 expression in its pre-malignant stage (MGUS), its early (Durie Salmon stage I) or late stage (Durie Salmon II and III) in both HM1 and HM2 groups (P<.001) (Figure 1.a). No relevant correlation could be found between TB4-expression and percentage of plasma cell infiltration in the bone marrow smear. Gene expression assessed by DNAmicroarray correlates well with qRT-PCR performed on MM patient samples (coefficient of correlation r = 0.993, P< .001). These data are in agreement with results from Gondo et al. showing an decreased T4-expression in a small number of MM samples by northern blot analysis (5).

Figure 1 .

 1 Figure 1.a shows the micro-array data obtained for the Tβ4 expression in CD138+ sorted

Figure 1 .

 1 d A similar gene expression pattern was observed in the murine 5TMM models where Tβ4 mRNA expression in 5T33MM and 5T2MM invaded BM was lowered compared to normal BM cells (Figure 1.d).

Figure 2 .a

 2 Figure 2.a Schematic representation of the modified lentiviral transfer plasmid and

Figure 2 .

 2 Figure 2.b. 3-H thymidine uptake revealed a decreased DNA synthesis rate in 5T33MMvt Tβ4+ cells compared to wild type cells. Incubation with the anti-MM agent

Figure 2 .

 2 f C57Bl/KaLwRij mice were injected with 5T33MMvt wild type and 5T33MMvt Tβ4+ cells. Kaplan-Meier analysis showed a significantly different survival between these 2 groups with a mean survival of mice injected with 5T33MMvt wild type of 65,9 days (SD 6.6 days), compared to 88,9 days (SD 9,3 days) for mice injected with 5T33MMvt Tβ4+ cells. (P < .05). Abbreviations: LTR: long terminal repeat; gag: frameshifted gag gene; RRE: rev-responsive element; CMV: cytomegalovirus promotor trip: central polypurine tract + termination sequence; Ires: internal ribosomal entry site; tNGFR: truncated form of the nerve growth factor receptor

Erik Quartier, and Carine Seynaeve for excellent technical assistance.

Financial support: This work was financially supported by the Fund for Scientific Research-Vlaanderen (FWO-Vlaanderen), the Association for International Cancer Research and by a grant from the European Commission FP6 to MSCNET. E Menu is a post-doctoral fellow of FWO-Vlaanderen.


Tomas Tuma

Paul Hurley

On the incoherence of noiselet and Haar bases

Noiselets are a family of functions completely uncompressible using Haar wavelet analysis. The resultant perfect incoherence to the Haar transform, coupled with the existence of a fast transform has resulted in their interest and use as a sampling basis in compressive sampling. We derive a recursive construction of noiselet matrices and give a short matrix-based proof of the incoherence.

Introduction

The noiselet basis, originally described in [START_REF] Coifman | Noiselets[END_REF], has garnered interest recently because noiselets [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF] are maximally incoherent to the Haar basis and (2) have a fast algorithm for their implementation. Thus, they have been employed in compressive sampling to sample signals that are sparse in the Haar domain [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF]. The work presented here was motivated by the observation that it had not been previously shown in a straightforward way that the discrete Haar transform is maximally incoherent to a discretized version of the noiselet transform. Additionally, the exact form of a noiselet matrix needed to be inferred from the original work. The main contributions are the derivation of a recursive, tensor product-based, construction of noiselet matrices, the unitary matrices that result from the noiselet transform for discrete input, and an intuitive proof showing its incoherence to the corresponding Haar matrix.

Preliminaries

General definitions

Definition 1. Let A be an m×n matrix, and B be a matrix of an arbitrary size. The Kronecker product of A and B is

A ⊗ B =    a 11 B • • • a 1n B . . . . . . . . . a m1 B • • • a mn B    .
The Kronecker product (see e.g. [START_REF] Laub | Matrix Analysis for Scientists and Engineers[END_REF]) is a bilinear and associative operator which is not generally commutative. It can be combined with a standard maxtrix multiplication as follows:

(A ⊗ B)(C ⊗ D) = AC ⊗ BD
whenever the products AC, BD exist. This property is sometimes called the mixed product property.

Definition 2. Let A be a m × n matrix. A(k, * ) de- notes the (row) vector (A(k, 1) A(k, 2) . . . A(k, n))
while, A( * , l) similarly denotes the (column) vector (A(1, l) A(2, l) . . . A(m, l)) T .

Noiselets

Noiselets [START_REF] Coifman | Noiselets[END_REF] are functions that are completely uncompressible under the Haar transform. The family of noiselets is constructed on the interval [0, 1) as follows:

f 1 (x) = χ [0,1) (x), f 2n (x) = (1 -i)f n (2x) + (1 + i)f n (2x -1) f 2n+1 (x) = (1 + i)f n (2x) + (1 -i)f n (2x -1)
Here, χ [0,1) (x) = 1 on the definition interval [0, 1) and 0 otherwise. It is shown in [START_REF] Coifman | Noiselets[END_REF] that {f j } is a basis: Theorem 1. The set {f j |j = 2 N , . . . , 2 N +1 -1} is an orthogonal basis of the vector space V 2 N , which is the space of all possible approximations at the resolution 2 N of functions in L 2 [0, 1).

Haar Transform

Haar wavelet transform can be described by a real square matrix. For our purposes, it is advantageous to recursively build the Haar matrix using the Kronecker product [START_REF] Falkowski | Walsh-like functions and their relations[END_REF]:

H n = 1 √ 2 H n/2 ⊗ (1 1) I n/2 ⊗ (1 -1)
.

The iteration starts with

H 1 = 1 . The normalization constant 1 √ 2 ensures that H T n H n = I.
Haar wavelets are the rows of H n .

Matrix construction of noiselets

First we extend and discretize the noiselet functions. Definition 3. The extensions of noiselets to the interval [0, 2 m -1] sampled at points 0, . . . , 2 m -1 is the series .

of functions f m (k, l) f m (1, l) = 1 l = 0, . . . , 2 m -1 0 otherwise f m (2k, l) = (1 -i)f m (k, 2l) + (1 + i)f m (k, 2l -2 m ) f m (2k + 1, l) = (1 + i)f m (k, 2l) + (1 -i)f m (k, 2l -2 m )
where m denotes the range of extension, k = 1, . . . , 2 m+1 is the function index and l = 0, . . . , 2 m -1 is the sample index.

Starting with a 1

× 1 matrix N 1 , a sequence of noiselet matrices N 1 , N 2 , N 4 , . . . , N 2 m of sizes 1 × 1, 2 × 2, 4 × 4, . . . , 2 m × 2 m
, respectively, is generated. The rows of the N n matrix are noiselets which form an orthonormal basis for the space C n . Definition 4. For n = 1, N 1 = 1 . Then the n × n noiselet matrix N n is built up recursively according to:

N n (k, * ) = 1 2 (1 -i 1 + i) ⊗ N n/2 ( k 2 , * )
when k = 0, 2, 4, . . . , n -2 and

N n (k, * ) = 1 2 (1 + i 1 -i) ⊗ N n/2 ( k -1 2 , * )
when k=1,3,. . . ,n-1.

Lemma 1. Let m > 0.

The noiselet matrices N 1 , N 2 , N 4 , . . . , N 2 m are built up from a series of discretised and extended noiselets f m :

N n (k, l) = f m (n + k, 2 m n l), k, l = 0, . . . , n -1.
Proof. Let m > 0 be fixed. For n = 1

N 1 (0, 0) = f m (1, 0) = 1.
By induction, for a matrix of size n = 2 p , p = 1, . . . , m, its basis vector k = 0, 2, 4, . . . , n -2 and vector indices l = 0, . . . , n 2 -1

N n (k, l) = (1 -i)N n/2 ( k 2 , l) = (1 -i)f m ( n 2 + k 2 , 2 m n 2 l) = f m (n + k, 2 m n l).
For the same n, k and l = n 2 , . . . , n -1,

N n (k, l) = (1 + i)N n/2 ( k 2 , l - n 2 ) = (1 + i)f m ( n 2 + k 2 , 2 2 m l n -2 m ) = f m (n + k, 2 m n l).
To see this, observe that f m is zero outside of [0, 2 m -1] and therefore, the first half of samples of f m (k, l) are defined exclusively by the expression

(1 ± i)f m (k, 2l)
whereas the second half of the samples are defined exclusively by

(1 ± i)f m (k, 2l -2 m ).
For k odd (k = 1, 3, . . . , n -1) the proof is similar.

Specially, the noiselet matrix N n for n = 2 m can be found as the "tail" of the function series f m . Indeed, the expression in Theorem 1 becomes N (k, l) = f m (n + k, l) for n = 2 m .

Incoherence of noiselets and Haar

In what follows, we adhere to the terminology of basis coherence which is common in the field of compressive sampling. See for example [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF] for details on these definitions and related literature.

Mutual coherence of two bases is defined as the maximum scalar product of any pair of their basis vectors:

Definition 5. Mutual coherence between two orthonormal bases Ψ, Φ is µ(Ψ, Φ) = max k,j | ψ k , φ j |.
The minimal coherence is usually termed maximal or perfect incoherence, which means that µ(Ψ, Φ) = O(1). In other words, the matrix of scalar products ΨΦ * is "flat". As Candès and Romberg suggest [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF], we will show the perfect incoherence of Haar and noiselets in the following setting. Given an orthonormal n × n Haar matrix H, we compute the matrix of scalar products for a corresponding noiselet matrix N normalized such that N * N = nI. By doing so, the product will be flat with all values having the magnitude of 1.

For clarity of the main proof, it saves some technical work to define a "twisted" noiselet basis.

Definition 6. The twisted noiselet matrix N1 = 1 . Then the n × n twisted noiselet matrix Nn is built up recursively by

Nn (k, * ) = 1 2 Nn/2 ( k 2 , * ) ⊗ (1 -i 1 + i) when k = 0, 2, 4, . . . , n -2 and Nn (k, * ) = 1 2 Nn/2 ( k -1 2 , * ) ⊗ (1 + i 1 -i) when k = 1, 3, . . . , n -1.
The difference between this and the definition of the noiselet matrix N (Definition 4) is that the order of operands in the Kronecker product is changed. In fact, each one is just a permutation of the other.

Lemma 2. For n = 2 m , the bases N n , Nn consist of the same set of basis vectors.

Proof. Indeed, we can write Nn = P n N n where P is the permutation matrix:

P (k, * ) = (1 0) ⊗ P n/2 ( k 2 , * ) k = 0, 2, 4, . . . , n -2 (0 1) ⊗ P n/2 ( k-1 2 , * ) k = 1, 3, . . . , n -1
starting with P = [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF].

The claim holds for n = 1. For n = 2, 4, 8, . . . , 2 m ,

P n N n (k, l) = P n (k, * )N n (l, * ) T
as it can easily be shown that N n is symmetric. Using the recurrent equations for P n and N n and applying the mixed product rule, we get, for k = 0, 2, 4, . . . , n -2,

P n N n (k, l) = 1 2 (1 -i)P n/2 ( k 2 , * )N n/2 ( * , l 2 )
when l = 0, 2, 4, . . . , n -2 and

P n N n (k, l) = 1 2 (1 + i)P n/2 ( k -1 2 , * )N n/2 ( * , l 2 )
when l = 1, 3, . . . , n -1. By induction,

P n N n (k, * ) = 1 2 Nn/2 ( k 2 , * ) ⊗ (1 -i 1 + i)
for even k indices. This situation for odd k is similar.

Now the main result can be shown.

H 1 N * 1 = 1 • 1 = 1 For n = 2 m , m > 1,

  Imaginary part of 64x64 noiselet matrix

Figure 1 :

 1 Figure 1: Noiselet matrix: graphical view. In figures (a) and (b), the black and white colors denote values of -0.25 and 0.25 respectively. In figures (c) and (d), the black, gray and white colors denote values of -0.125, 0 and 0.125 respectively. .

  Theorem 2. Let n = 2 m where m is a non-negative integer. Let N n be the noiselet matrix of size n × n and let H n be the Haar matrix of size n × n. Then H n and N n are maximally incoherent. Proof. Without loss of generality, assume the bases are normalized such that H T n H n = I and N * n N n = nI. For the case of n = 1,

  the incoherence is shown by induction. Suppose we know maximal incoherence holds for n 2 and we want to show it for n. In the induction step, we use the iterative construction of the Haar matrix by means of Kronecker product. By computing the product * n , j = 0, 2, 4, . . . , n -2 and transform it by H n , getting Note the altered normalization factor of noiselets. Now the mixed product property can be applied to get (i, j2 )| = 1 for i = 1, . . . , n 2 . The Kronecker multiplication is only by entries with magnitude 2, thus the resulting magnitudes are1 2 * 2 = 1. The proof is equivalent for j = 1, 3, . . . , n -1.

	H n	N * n = H(N * n P * n ) = (H n N * n )P T n
	we will still be able to conclude on magnitude of the el-
	ements of (H n N * n ), since permutation matrices do not
	change magnitudes. The product H n N * n can be computed per-column; we take the j-th column of N H n N * n ( * , j) = 1 √ 2 H n/2 ⊗ (1 1) I n/2 ⊗ (1 -1) • 1 √ 2 N * n/2 ( * , j 2 ) ⊗ (1 -i 1 + i) 1 2     H n/2 N * n/2 ( * , j  1 + i 2 ) ⊗ (1 1) 1 -i I n/2 N * n/2 ( * , j 1 + i    = 2 ) ⊗ (1 -1) 1 -i 1 2 H n/2 N * n/2 ( * , j 2 ) * 2 I n/2 N * n/2 ( * , j 2 ) * 2i .
	By induction, it follows that |H n/2 |I n/2 N * n/2	N * n/2 (i, j 2 )| = 1 and

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