3D compressed sensing ultrasound imaging

Céline Quinsac, Adrian Basarab and Denis Kouamé Université de Toulouse

IRIT, UMR CNRS 5505 Toulouse, France quinsac@irit.fr

Jean-Marc Grégoire Université de Tours

UMR Inserm U930/ CNRS équipe 5 Tours, France

Abstract—This paper proposes a compressed sensing method adapted to 3D ultrasound (US) imaging. Three undersampling patterns suited for 3D US imaging, together with a nonlinear conjugate gradient reconstruction algorithm of the US image k-spaces, are investigated in vivo radio-frequency 3D US volumes. Reconstructions from 50% of the samples of the original 3D volume show little information loss in terms of normalized root mean squared errors.

Keywords-ultrasound imaging; compressed sensing; reconstruction; 3D; k-space; sparsity

I. INTRODUCTION Ultrasound (US) imaging remains the medical imaging

modality of choice in numerous applications (e.g. pregnancy monitoring, cardiac imaging, blood flow estimation) due to its safety, relative low cost and rapidity. However, in cases like three dimensional scanning for example, the amount of data can be a limiting factor for real-time imaging or simply data storage.

Compressive sampling (CS) or compressed sensing is a novel theory aiming to reduce the amount of data collected during the acquisition that emerged in 2006 [1]. The principle of CS is to measure only a few significant coefficients of a compressible signal and then to reconstruct it through optimization. It differs from compression as the signal is directly acquired (or sampled) in its compressed form. CS has been successfully applied to a number of domains, including medical imaging, in particular in magnetic resonance imaging and tomography [2-4].

The purpose of this paper is to study the applicability of the CS theory for 3D US imaging and to propose a derived method to reconstruct radio-frequency (RF) (and therefore B mode) US volumes. First, the principle of CS is reminded, together with two associated key concepts: sparsity and incoherence. Second, the assumptions of CS are examined for the US case and a CS method suited to echography signals is proposed. Three distinct sampling schemes satisfying CS requirements and adapted to US imaging are compared. Finally, some promising results on 3D US volume reconstruction are presented.

II. THEORY OF CS

A. Principle The principle of CS is to sample and recover sparse signals

from fewer samples than requested by the Shannon sampling

theorem. Signals sparse in a given domain contain only a few significant coefficients in this basis. By compression, i.e. setting the non-significant coefficients to zero, it is possible to recover almost exactly the original signal. For images, well known cases include compression by discrete cosine transform (JPEG format), wavelets (JPEG 2000), etc. Ideally, when acquiring compressible data, only the most significant coefficients would be measured and from those few samples, the signal would be recovered. How can we acquire those few samples without any prior knowledge on the signal?

CS gives an answer to this problem by sampling the signal in a completely incoherent way, the most intuitive sampling strategy being random linear combinations of the signal. Each measurement contains information about the whole signal but the number of measurements is low compared to the Nyquist sampling criteria [1].

The reconstruction of the signal from those few measurements, which is an inverse problem, is then performed through an optimization process. It has been shown that, if the number of measurements is sufficient, amongst all the signals that match those samples, the one with the fewest coefficients in the sparse domain will be the original signal [5].

Mathematically, the idea of CS translates as follows. Consider the signal to recover, for example, an image m in RN, sparse in a basis Ψ of RN×N so that s=Ψm has S non-zero entries (with S << N). Assume m is sampled using a sampling basis A in RK×N, incoherent with the sparsifying basis Ψ, resulting in measurements y=Am, with y in RK. The original image can be reconstructed resolving the constrained optimization problem:

yAmtsm =Ψ ..min1

(1)

where 1

denotes a l1 norm. Equation (1) states that, among all the solutions that verify the measurements, the original image m is the one with minimum l1 norm in the sparsifying basis, i.e. the sparsest in Ψ [5].

The success of that reconstruction is guaranteed with overwhelming probability if:

• K ≥ C·μ²(A,Ψ)·S·log(N), where C is a constant and μ(A,Ψ) is the coherence between the sampling and sparsifying bases [6]:

jknjkAnA Ψ=Ψ

≤≤,max),(

,1μ (2)

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10.1109/ULTSYM.2010.0088

978-1-4577-0381-2/10/$25.00 ©2010 IEEE

where ⋅⋅, is the inner product (correlation).

• the signal m is sparse in a basis Ψ,

• the sampling basis A and sparsifying basis Ψ are incoherent.

The notions of sparsity and incoherence are described below.

B. Sparsity and incoherence The concept of sparsity, necessary for the application of

CS, is mathematically described as follows. m is sparse in the Ψ domain if the support of s=Ψm is small compared to the support of m (or as an approximation if most of the information is contained in a small number of coefficients). This condition is linked to the l1 minimization of s. The l1 norm is used in place of the l0 norm for practical optimization reasons [7]. When designing a CS protocol, the identification of the sparsifying basis is key for the success of the reconstruction [5].

In addition to the sparsifying basis, the sampling basis A should also be chosen with care. For the CS reconstruction to be successful, the sparse basis Ψ and the sampling basis A must be incoherent, that is to say they should not contain correlated elements (2). This property ensures that sparse signals in Ψ will not be sparse in A. The maximum incoherence is obtained with random sampling matrices associated with any fixed sparsifying basis. However, complete randomness is not always achievable due to the constraints in instrumentation so alternative sampling patterns are often considered [8].

III. CHALLENGES OF COMPRESSED SENSING IN 3D ULTRASOUND IMAGING

A. Proposed reconstruction method The idea proposed in this paper for 3D US imaging using

CS is to sample the consecutive RF images in an incoherent way in order to reconstruct their k-space. As US imaging acquisition is performed in the spatial domain, we propose to sample RF signals at random locations. As explained previously, CS is optimal when the acquired samples are linear combinations of the signal to reconstruct. For this reason, the signal reconstruction presented in this paper is not performed in the spatial domain, but in a different basis. We choose here the Fourier domain, or k-space, as the reconstruction basis of US images: spatial random samples are indeed linear combinations of inverse Fourier coefficients.

Here, we propose the following optimization routine to recover the original k-space:

12minarg MyAM

MΨ+− λ (3)

where M is the k-space of the consecutive RF images of the volume (M=Fm), A is the sampling scheme (A=Φ F-1) here, where Φ corresponds to the RF random sample locations and F-1 stands for the inverse Fourier transform), y are the RF US image measurements, Ψ is the sparsifying transform and λ is a coefficient weighting for sparsity in the basis Ψ.

Figure 1. Sampling masks Φ1 (a), Φ2 (b), Φ3 (c), adapted to a spatial sampling of the 3D US volumes. The white pixels correspond to the samples used for CS. The proportion of samples here is 50% of the original volume.

Figure 2. Mean square errors calculated for simulated US images with compressed k-spaces, using wavelet, finite differences and DCT, for different

levels of compression.

The first term of (3) represents the fidelity of the measurements and the second term guarantees the signal sparsity in the basis Ψ.

The main requirements for the success of CS reconstructions are the sparsity of the image in a given basis and the incoherence between the sampling basis A and the sparsifying basis Ψ. For US signals, these key points are discussed below.

B. Proposed sampling patterns Three different sampling schemes Φ1, Φ2 and Φ3 are proposed and evaluated (Fig. 1). Φ1 is a uniform random sampling pattern in the three directions. Φ2, more appropriate for US acquisition, is an alternative pattern proposed here as a strategy to reduce the number of US pulse emissions. It consists of, on each consecutive slice of the volume, uniformly random RF lines and, on each line, uniformly random points. The latter differs from the first as not all the columns (RF signals) are being sampled. The same number of samples is considered in both cases. Whereas Φ2 consists in sampling different RF lines on each slice of the azimuthal direction, with Φ3 the set of unsampled RF lines is always the same in each slice. Consequently, with Φ3 whole axial-azimuthal slices of the volume are not sampled (Fig. 1).

(a)

(b) (c)

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Figure 3. Coarse-scale wavelet TPSF for two different sampling patterns Φ1

(left) and Φ2 (right).

TABLE I. MEAN AND STANDARD DEVIATION OF NRMSE BETWEEN THE RECONSTRUCTION 3D RF US VOLUMES AND THE ORIGINAL VOLUME FOR

DIFFERENT SAMPLING PATTERNS.TABLE 1

Φ1 Φ2 Φ3

0.090 ± 4.4E-4 0.097 ± 4.4E-4 0.094 ± 20E-4

C. Sparsity of US image k-space US images present information redundancy due to the

acquisition process which consists in taking RF lines close to each other. For the application of CS following (3), we assume that the US image k-space is sparse in a basis Ψ. To test this hypothesis, three transforms (Daubechies wavelets, finite differences and discrete cosine transform) have been applied on the frequency domain of a RF image simulated using the Field II US simulation program [9]. Similar transforms have been used for this purpose [2] but here the sparsity of the k-space, rather than the spatial image, was tested [10]. Used parameters and the example scatterer map of a vessel were: transducer center frequency = 3MHz, sampling frequency = 20MHz, number of scatterers = 10,000, number of RF lines generated = 256, size of the object = 40 x 90 mm. The image was then cropped to a 256 by 256 size matrix.

The k-space of the initial image was decomposed in the three bases. A fraction of the most significant transform coefficients was used to form a compressed k-space. The B mode US images resulting from this RF signal k-space compression were compared for different degrees of compression and different transforms. The mean squared errors shown in Fig. 2 allow a quantification of the differences between the different types of compression.

According to Fig. 2, with only 10% of the coefficients of any transform, the errors in the compressed images were relatively low, showing the sparsity of the US images k-space in these bases. The wavelet transform was chosen for the purpose of this paper because of its ease of implementation but other sparsifying bases will be discussed in future works.

D. Incoherence of US imaging sampling To assess the incoherence between the sampling bases A1,

corresponding to a slice of Φ1 and A2, corresponding to a slice of Φ2 or Φ3, and the sparsifying basis Ψ, the Transform Point Spread Function (TPSF), defined by Lustig et al. [2] was calculated for the two different sampling schemes against the wavelet transform:

iTj eAAejiTPSF 11),( −− ΨΨ= (4)

where vectors ei and ej are equal to 1 at the ith and jth position respectively and 0 elsewhere. The maximum sidelobe-to-peak ratio (MSPR) was then considered to reflect the incoherence between the two bases:

),(),(max

iiTPSFjiTPSFMSPR ji≠= (5)

The TPSF of coarse-scale wavelet coefficients for each sampling pattern are shown in Fig. 3.

The results of this comparison show that the Φ2 sampling pattern introduces some 1D coherence in the TPSF and increases the MSPR. However, the interferences caused by the removal of some columns in the sampling scheme are still noise-like. Similarly, the sampling pattern Φ3 would introduce interferences in the third direction (azimuthal) as well (due to the removal of whole slices). We hypothesize that the Φ2 and Φ3 sampling schemes remain sufficiently incoherent to reconstruct an US image k-space from these patterns and therefore a US 3D volume. The results presented in the next section show that this assumption is valid. In addition, these alternative sampling schemes present a significant advantage from the instrumentation point of view.

IV. RESULTS ON A 3D US IMAGE The CS strategy described in (3) was used to reconstruct an

in vivo US volume of mouse embryos, acquired on anaesthetized mice. A single element high resolution scanner SHERPA, developed and commercialized by Atys Medical (Lyon, France) where RF data was available was used (central frequency 22MHz, frame rate 10 images per second, scanning width 16mm, sampling frequency 80M samples/second, emission frequency 20MHz, exploration depth 7.8mm). The volume was then cropped to a 1283 size volume for illustration purposes.

The CS reconstruction of the volume shown in Fig. 4 was performed offline using the sampling masks described in section III.B and 50% of the samples of the originally acquired volume, a db6 wavelet as the sparsifying transform and the optimization routine (3) with λ= 0.4. The optimization routine used was a nonlinear conjugate gradient descent algorithm, see e.g. [11].

The first observation to make is that for all the sampling masks, the CS method (3) provided good reconstructions of the whole volume from only 50% of the samples. The slice that was best reconstructed in each case was always the axial-lateral, where the 2D masks where applied (and then repeated along the azimuthal direction). However this setting could easily be changed for other applications where another slice is more crucial.

Coherent interferences

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Figure 4. 3D CS reconstructions following (3) on an in vivo US volume of mouse embryos (a) using 50% of the samples and the sampling masks Φ1, Φ2 and Φ3. The B-mode volume of the RF random measurements are shown in

(b), (d) and (f) and of CS reconstructed US volume in (c), (e) and (g).

When the coherence increased, i.e. from Φ1 to Φ2 to Φ3, the reconstructions were degraded, as expected. This is particularly visible on the axial-azimuthal plans of Fig. 4. However, considering that absolutely no samples were kept for the axial-azimuthal slice visible on Fig. 4 the result is still quite good. This setting could be used in a situation where the speed of imaging would prevail on the quality of the reconstruction. In addition, despite being less sharp, the image still exhibits the tissue structure and might be sufficient in many applications.

The reconstruction normalized root mean squared errors (NRMSE) calculated on the volumes are given in TABLE I. As expected, the NRMSE on the images reconstructed using the

Φ2 and Φ3 sampling patterns were greater than those using Φ1. This difference is partly explained by the slight coherence of Φ2 and Φ3 with Ψ, as shown on Fig. 3.

V. CONCLUSION In this paper, we showed that the theory of CS is applicable

to echography and can reduce data amount at the price of a reconstruction using the l1 norm. The RF signals can be sampled at random times to provide linear combinations of the desired final image k-space. Through an l1 minimization, the original k-space can be reconstructed in a sparsifying domain such as wavelets and the RF US image subsequently recovered with little loss of information.

Three sampling strategies associated to a k-space reconstruction were proposed in this paper. Future work will include the identification of optimal conditions as well as an investigation of several optimization routines. Additional knowledge about the US images will be inserted in the reconstruction process (statistics of the signal, attenuation). The aim is to reach the fastest and most reliable reconstruction from as little samples as possible. Various applications will also be considered (multidimensional Doppler and tissue characterization).

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