c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11

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A reduced complexity estimation algorithm for ultrasoundimages de-blurring

Alessandro Palladini ∗, Nicola Testoni, Luca De Marchi, Nicolò SpecialeARCES/DEIS - University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

a r t i c l e i n f o

Article history:

Received 16 October 2008

Accepted 21 February 2009

a b s t r a c t

In this paper we propose a deconvolution technique for ultrasound images based on a

Maximum Likelihood (ML) estimation procedure. In our approach the ultrasonic radio-

frequency (RF) signal is considered as a sequence affected by Intersymbol Interference (ISI)

and AWG noise. In order to reduce the computational cost, the estimation is performed

Keywords:

Ultrasound images

Homomorphic deconvolution

Deconvolution

Viterbi algorithm

with a reduced-state Viterbi algorithm. The channel effect is estimated in two different

ways: either measuring the transducer response with an experimental setting or with blind

homomorphic techniques. We observed an enhancement in image quality with respect to

different metrics. Extensive tests were made to estimate the quantization alphabet that

gives the best performances.

1D model for the acquired signal, treating each scanline sep-

1. Introduction

Ultrasound (US) is a cost effective, mobile, noninvasive, harm-less, and suitably accurate imaging technique. The maindrawback is that signal resolution is limited by the transducerfinite bandwidth and by the non-negligible width of the trans-mitted acoustic beam. Image restoration techniques, such asdeconvolution, can be employed to improve the resolutionof US images and their diagnostic significance. Many algo-rithms based on different techniques for enhancing contrastand improve visual quality of US images have been reportedbut most of them are often too computational demanding tobe applied in a real-time processing context.

Two main approaches are most common when dealingwith US image deconvolution. The first incorporates thePoint Spread Function (PSF) estimation procedure within thedeconvolution algorithm. This approach often leads to thedevelopment of computationally heavy algorithms, usually far

from satisfying the real-time signal processing constraints dis-tinctive of the US biomedical investigation environment. Inthe second approach, PSF and true image estimation are two

∗ Corresponding author. Tel.: +39 051 2093775.E-mail address: apalladini@arces.unibo.it (A. Palladini).

0169-2607/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights resdoi:10.1016/j.cmpb.2009.02.016

© 2009 Elsevier Ireland Ltd. All rights reserved.

disjoint tasks. Within this approach, these procedures can beimplemented by relatively simple algorithms, often suitablefor real-time implementation.

Theoretically speaking, as the PSF is a band-limited func-tion and due to the presence of noise, signal deconvolution isan ill-posed problem. To obtain a stable algorithm, deliveringa unique solution, additional constraints must be imposed.Therefore, designing a method which exhibits the most suit-able compromise among computational complexity, reliabilityand portability for biomedical real-time imaging applica-tions is still an open challenge. Good reviews of the existingapproaches are in [1,2].

In this paper we propose a deconvolution technique forultrasound images based on a Maximum Likelihood (ML)estimation procedure. The overall structure of the algorithmis shown in Fig. 1: we decided to process the radio-frequency (RF) signal before envelope extraction adopting a

arately and independently. Moreover, the system PSF andthe original RF signal are estimated with two disjoint proce-dures.

erved.

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b

Fig. 1 – Overall structure of the proposed deconvolutionp

twvo

dsaufolob

spaSpoWS

2

Epsdid

ssc

z

Trsvw

is projected on the wavelet domain, where the final impulsesmoothness can be easily set by means of the depth of

rocedure.

System PSF is estimated with two different techniques: inhe former we measured it by illuminating a metal wire in aater-filled tank while in the second we applied a modified

ersion of the blind homomorphic techniques presented in [3]n the in vivo US signal.

In a discrete-time context the blurring effect of the trans-ucer can be thought as an Intersymbol Interference (ISI) on RFignal. Therefore, we formulated the deconvolution problems ML sequence estimation problem, processing each scanlinesing the Viterbi algorithm [4], which is an optimum solution

or the ML estimation of discrete-value sequence in presencef ISI and memoryless noise [5]. To efficiently characterize the

arge dynamic range and nonuniform amplitude distributionf the RF signal we adopted a nonuniform quantization alpha-et.

The paper is organized as follows: Section 2 introduces theignal model and the proposed approach for the deconvolutionroblem. The estimation procedure for the system impulsend the ML-based de-blurring procedure are presented inections 3 and 4, respectively. The obtained performance onhantom and real US in vivo images and the comparison tother US 1D signal deconvolution algorithms in literature,LSD [6] and FWD [7], are presented in Section 5. Finally,

ection 6 concludes the paper.

. Signal model and proposed approach

chographic signals result from the interaction between theressure wave generated by the transducer and the tissuetructure. A comprehensive model for the received signal isiscussed in [8]. Moreover, since in modern ultrasonic imag-

ng system the RF signal is sampled and processed in digitalomain, we considered a discrete-time model.

Under the assumptions of weak scattering, narrow ultra-ound beam and linear propagation, the observed sequence ofignal samples z = {z1, . . . , zK}, where K is the sequence length,an be modelled by

= x ∗ h + n. (1)

he sequence of sample of the original signal (real tissue

eflectivity) x = {x1, . . . , xK}, is corrupted by the acquisitionystem impulse response, whose effect is modelled as a con-olution with the minimum phase sequence h = {h1, . . . , hL},here L is the PSF length. An additional sequence of additive

i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11 S5

gaussian white noise samples n = {n1, . . . , nK}, is taken intoaccount to model instrumental noise. The main motivationsupporting the choice of 1D model instead of a 2D model isthe need to develop a real-time algorithm capable to processthe RF signal during its acquisition, without storing it.

Blind deconvolution algorithms can be classified in twomain classes, according to the stage at which the system PSFh is estimated:

(1) A priori identification methods: in this approach the sys-tem PSF is estimated separately from the original signalx and later used in combination with an estimation algo-rithm in order to restore the original signal. This approachis suitable when prior information on the imaging sys-tem is available and leads to computationally simplealgorithms.

(2) Joint identification methods: in this approach the system PSFand the original signal are estimated simultaneously. Thesuccess of these methods is strongly dependent on priorinformations about the original signal and the PSF, typi-cally incorporated in the form of parametric models. Thisapproach often leads towards more complex algorithms,hardly suitable for real-time implementation.

For these motivations we adopted the first approach, usinga modified homomorphic algorithm for PSF identification andML estimation procedure for signal restoration.

3. PSF estimation procedure

Different techniques are available to perform the PSF estima-tion: a common method for measuring h is the insonificationof a metal wire sank in a water-filled tank. It is however pos-sible to estimate h directly from the sampled data: if H(ωk),the discrete Fourier transform of h, is supposed to be mini-mum phase [3,9–11], all the informations regarding the shapeof the imaging system impulse response are contained into itsamplitude spectrum.

The minimum phase version of h can be easily estimatedin the Cepstral domain [9] selecting the low-frequency Lc sam-ples of the cepstrum [12] of z, where Lc is the so-called cut-offparameter. In order to improve the estimation of the acquisi-tion system impulse response, the polynomial reconstructionproperties of the wavelet transform can also be exploited [3],projecting the input signal amplitude spectrum on the waveletdomain. While the first technique produces good filters for theestimation of x but features unwanted amplitude ringings, thesecond one generates much smoother filters, however proneto instability.

Differently, our method combines Cepstral and waveletdomain estimations to ease the trade-off between smooth-ness and stability. First the input signal amplitude spectrum

the Wavelet reconstruction tree; then, the result is broughtin the Cepstral domain where minimum phase and stabil-ity can be enforced through a wise choice of the cut-offparameter.

S6 c o m p u t e r m e t h o d s a n d p r o g r a m s i n

Table 1 – A performance comparison of some significantsetups for different impulsive response estimationalgorithms: cepstrum based (CEP), Wavelet based (WAV),combined Wavelet/cepstrum (MIX).

Method MSE [dB] Distance Length

CEP −18.1 12.6 × 10−3 41

WAV −18.4 3.5 × 10−3 38MIX −18.4 13.2 × 10−3 42

If Cz is the cepstrum of the received ultrasonic signal z, thecepstrum of h is calculated using

Ch(k) =

⎧⎪⎨⎪⎩

Cz(k)Wk k = 0

2�{Cz(k)Wk} k ∈ [1, Lc − 1]

0 k ∈ [Lc, L − 1]

(2)

where Wk are the weights obtained from the Inverse FourierTransform of the equivalent filter corresponding to theWavelet analysis. The time domain minimum phase estimateh is then calculated inverting the cepstrum Ch.

Table 1 compares the performance of these estimationtechniques evaluated using the following metrics:

• the mean square error (MSE) of the deconvolution of a water-tank recorded pulse by means of the estimated one;

• the minimum distance from the unitary circle of the zerosof the filter corresponding to the estimated impulse;

• the number of coefficients which retain the 99.9% of thecorresponding estimated impulse response energy.

where the first one quantifies how good the estimatedimpulse response approximates the shape of the water-tankrecorded pulse, the second one helps in evaluating how stablethe final deconvolution is going to be, and the third one mea-sures how many computational resources should be allocated.

4. ML-based deconvolution

In general, a deconvolution algorithm can be interpreted asestimating x from the noisy signal z. Once the system PSF isknown, a trivial deconvolution procedure consist in estimatingx by applying the inverse filter h−1 to z:

x = h−1 ∗ z = x + h−1 ∗ n. (3)

Unfortunately, the deconvolution noise term h−1 ∗ n is notnegligible and the resulting image quality is often very low.The deconvolution problem must be reformulated possiblyincluding prior information on the noise and on the originalsignal.

Our approach focuses on a ML approach. Given z, h andthe statistics of the noise n, the deconvolution problem can

be formulated as ML sequence estimation problem, where theestimate x of the unknown signal sequence x is chosen bymaximizing the probability density function p(z|x). Assum-ing that sample zk are statistically independent, p(z|x) can be

b i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11

written as

p(z|x) =∏

k

p(zk|x). (4)

Although (4) does not take into account signal correlation, itsuse leads to a tractable mathematical model which fits quitewell a wide class of signals. If n is modelled as additive whitegaussian noise with variance �2

n , Eq. (4) can be rewritten as

p(z|x) =∏

k

1√2��2

n

exp

{−[yk − zk]2

2�2n

}, (5)

where yk is given by

yk =L−1∑i=0

xk−ihi (6)

and L is the system impulse response length. Assuming �2n

to be constant, the maximization of (5) is equivalent to theminimization of the metric

�(x, z) =∑

k

[yk − zk]2 (7)

So, the ML estimation of the original signal x can be writtenas

xML = argminx

�(x, z) (8)

Although formally simple, the obtained Eq. (8) leads to a com-putationally expensive minimization algorithm, ill-suited forreal-time biomedical image processing.

4.1. Viterbi algorithm based estimation

In modern ultrasound systems the measured RF signal isquantized, usually with a 12-bit resolution for RF signal, andwith an 8-bit resolution for the envelope detected signal. In thequantized domain the distortion and noise-free signal sam-ple xk can be modelled as a discrete-valued random process:this approach allows us to reformulate the estimation prob-lem (8) as a discrete-value sequence estimation, which canbe optimally solved with a low computational cost by Viterbialgorithm (VA) [4].

Modeling the signal sample xk with a uniform distribu-tion over a finite alphabet X = {x1, . . . , xM}, the distorted signalsample yk can be thought as the output of a finite-statediscrete-time Markov random process, with finite-state spaceY = {y1, . . . , yJ} of dimension J, as shown in Fig. 2(b). For aFIR filter of length L and a signal alphabet X of size M, theMarkov random process associated to the distorted signal yk

has J = ML−1 states. The set of signal sequences of lengthL − 1, xL−1 = {x0, . . . , xL−2}, are therefore mapped one to one onthe state space Y, and the original sequence x can be obtained

by estimating the state sequence y, from the observation of z.

The ML estimation of a Markov random process statessequence can be computed in a recursive manner through theVA built over the trellis-diagram [4] associated to the process.

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11 S7

Fig. 2 – (a) Model of the acquired US signal zk with intersymbol interference h and white gaussian noise nk and (b)finite-state machine model of the distorted signal yk.

Fp

Tsspaaetbdf(ete

viM

c

sdek

point. Due to the reduced-states trellis diagram, the perfor-mance curve is clearly sub-optimal with respect to the MLone. However, for SNR >19 dB our approach needs only 1 dB

ig. 3 – ML estimation trellis diagram for M = 2, L = 3 (a),roposed reduced complexity estimation trellis diagram (b).

he trellis-diagram is a graphic way to represent a finite-tate process in a redundant form: states, transitions andequences are represented by means of nodes, branches andaths. Such diagram shows all the possible transitions fromnd to each state yj at step k and can recursively generatell the possible state-sequences of a certain length K. Sinceach sequence is in fact associated to a unique path insidehe trellis, VA will then estimate the ML state-sequence yK

y recursively generating all the possible paths of length K,iscarding the concurrent ones by considering their distancesrom the observed sequence zK , computed by means of the Eq.6). Finally, the minimum distance path will be chosen as thestimated sequence. An example of trellis diagram associatedo a finite-state Markov process is shown in Fig. 3(a): at step k,ach state yj is connected with only M states at step k + 1.

The drawback of this procedure is that even for not so largealues of M and L, the VA becomes already impractical, sincets computational and memory demands is proportional to

L−1. Thus, for real-time biomedical image processing appli-ation, computational cost reduction is mandatory.

In the proposed approach, we associate the trellis-diagram

tates to the distortion and noise-free signal symbols xj. Byoing so, we reduce the trellis to an M-state diagram, whereach state at step k in connected with all the states at step+ 1, as shown in Fig. 3(b). After k-steps, VA will generate

all the possible signal symbols sequences xk = {x0, . . . , xk−1},computing the correspondent distorted sample yk with theEq. (6), discarding the concurrent paths inside the trellis bas-ing on their distance from the observed sample zk and finallychoosing the minimum distance path as estimate of x.

This corresponds to limiting the possible survived pathsin the ML trellis at step k to M, discarding the exceeding onesaccording to their distance from the observed partial sequencezk = {z0, . . . zk}. Decision based on partial observation of thewhole sequence zK leads to optimal estimation only if k > L.As the algorithm starts discarding path from k = 2, the finaldecision at step K will be a sub-optimal estimation of thecomplete sequence x[5]. Nevertheless, this approach offers adrastic computational and memory requirements reduction,making the algorithm suitable for real-time application.

To evaluate the performance of the proposed algorithmwith respect to the ML one in terms of symbol estimation error,we processed random binary sequences of length K = 106, cor-rupted by a measured impulse response h and AWG noise withdifferent powers.

Fig. 4 shows the simulation results obtained by averagingbit error rate (BER) values, performing 100 simulations for each

Fig. 4 – Algorithms performance comparison for binarysequence estimation.

s i n b i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11

dard A-law compounding block, to deal with the nonuniformsignal amplitude statistics. Then, they were processed by VA,with the measured and estimated system response and withincreasing quantization levels, starting from acceptable value

Table 2 – Average values of the image enhancementevaluation metrics computed on the in vivo prostaticgland frames dataset.

Algorithm RG-L RG-A CG PSNR SSIN

VA 256 1.77 4.88 3.38 23.04 0.941VA 128 1.77 5.11 3.42 23.05 0.914

S8 c o m p u t e r m e t h o d s a n d p r o g r a m

to achieve the ML algorithm performance, providing howevera drastic reduction of the computational demand.

The whole ultrasound signal deconvolution is obtained byprocessing each scanline with the proposed reduced com-plexity VA. The algorithm can be both applied to RF signalsor envelope detected signals. Since the envelope detectioncauses the loss of the information carried by signal phase, RFsignal processing is still preferable.

4.2. Codebook definition

The VA based estimation procedure performs signal restora-tion on the finite alphabet X = {x1, . . . , xM}. This operation isperformed immediately after the last block of analog front-end (analog to digital converter) of the echographic equipmentand therefore, on the digitalized signal. Such processing canthus be thought as a re-quantization of the digital signal.

In order to obtain a satisfactory signal quality, in terms ofsignal to quantization noise ratio (SQNR), the alphabet X mustbe properly defined, according to the desired dynamic rangeand to the statistical properties of the signal. In particular, thenumber of quantization levels M should be as big as possible,in order to obtain the highest SQNR. On the other hand, Mshould be kept as small as possible, to limit the computationalcost of the VA. The desired dynamic range depends on thesignal analysis procedures to be applied and is determined bydifferent factors.

Due to large tissue attenuation, ultrasonic echo RF signal ischaracterized by a very large desired dynamic range (qualita-tively about 160 dB), which can hardly be obtained by an ADCwithout a time gain compensation (TGC) block [13]. For thesereasons, in modern digital US equipment, a preliminary TGCis applied to the RF signal before its conversion, in order topartially compensate the tissue attenuation and reduce therequired dynamic range to about 100 dB. Then, usually up to1024 levels (12 bits) are employed for signal quantization.

For imaging purposes, the standard grayscale digitalimages are represented with 256 different values (8-bit), whichcorresponds to a dynamic range of about 48 dB. Moreover, dueto human sight discrimination properties the alphabet sizecan be reduced to 32 different values (5-bit), without signifi-cant loss of image quality [14].

US signals are characterized by an extremely nonuniformstatistics. To represent this wide dynamic with a satisfac-tory signal to noise ratio, without any prior information onthe signal, a large number of quantization levels is neces-sary. Thus, a uniform quantization alphabet in not suitable forboth RF signal and envelope processing. By adopting a nonuni-form quantization better performance and a reduction of therequired quantization levels can be achieved. Theoretically,once the statistical distribution of amplitude of the desired sig-nal is known, the optimal quantization codebook can be builtthrough Lloyd–Max rules [15]. Another simpler way to obtain anonuniform codebook is the following: a reversible nonlineartransform (amplitude compression) is applied on the signal,before VA operating with a uniform codebook. The uncom-

pressed estimated sequence x is then obtained by applying adynamic expansion on the estimated compressed sequencexcmp provided by VA, as shown in Fig. 5. To obtain a SQNRnot dependent by the (unknown) statistical distribution of the

Fig. 5 – VA algorithm operating on a nonlinear codebook.

signal, the nonlinear transform applied to the original signalsamples must be [15]

zcmp k = A ln |zk|sgn(zk) (9)

However, such transformation is not invertible, due to thesingularity in the origin. In practice, suboptimal invertiblefunctions like A-law and �-law can be employed. In thepresent work, we adopted a A-law function.

5. Application to ultrasonic images results

To verify the effectiveness of the proposed algorithm as ade-blurring method for biomedical images we tested it onan US signals database which comprises synthetic phantom(CIRS Model 047) acquisitions and in vivo TRUS acquisitionsof prostatic glands (264 frames), both obtained with a com-mercial ultrasound equipment (MYLAB90 Esaote S.p.a.). Thesystem impulse response h was obtained with two differentprocedures: it was measured with an experimental settingconstituted by an acoustic mirror in a water-filled tank and itwas estimated from the echographic signal through the homo-morphic based deconvolution procedure presented in Section3.

Processing both RF signal and signal envelope, de-blurringperformance was evaluated in terms of signal resolutionimprovement and image quality enhancement . To quantifythe resolution improvement we measured the lateral and axialresolution gain at −6 dB (RG-L/A) [16]; the peak signal to noiseratio (PSNR) and the structural similarity index (SSIN) wereused to compute the dissimilarity between the original andprocessed image, in terms of loss of correlation, luminancedistortion and contrast distortion [17]; finally we measured theimage contrast enhancement on phantoms by means of thecontrast gain (CG) [18].

All signals were first processed column-wise with a stan-

VA 64 1.77 5.16 3.46 22.98 0.913VA 32 1.80 5.50 3.62 23.20 0.915FWD 1.01 1.72 1.17 17.54 0.775WLSD 1.57 4.46 1.46 16.39 0.762

i n b i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11 S9

M

ttuoTba

rsAgffoRibrtp

Table 3 – Average values of the image enhancementevaluation metrics computed on the phantom framesdataset.

Algorithm RG-L RG-A CG PSNR SSIN

VA 256 1.83 3.85 9.77 22.06 0.871VA 128 1.86 3.52 10.07 22.01 0.869VA 64 1.95 3.88 11.06 22.10 0.871VA 32 2.14 5.40 12.29 22.00 0.871

Fp

c o m p u t e r m e t h o d s a n d p r o g r a m s

= 32 up to standard value for images M = 256. We verifiedhat the deconvolution applied on RF provide better results inerms of visual image quality and respect to the proposed eval-ation metrics. Moreover, we found that deconvolution basedn estimated system impulse provides better performance.his may be due to the system response aberration causedy tissue velocity propagation inhomogeneities, which can beccounted only by the estimation procedure.

For in vivo and phantom RF signals processing, the averageesults over the whole datasets obtained with the estimatedystem response are shown in Tables 2 and 3 , respectively.s it can be clearly seen, the proposed algorithm provides aood resolution increase in both axial and lateral directionsor both in vivo and phantom acquisitions, with better per-ormance on the in vivo frames. By increasing the numberf quantization levels M, we observed small variation for theG in both axial and lateral directions and for the CG. The

mages SSIN is quite close to the maximum value of 1, with

etter performance on in vivo frames. SSIN and PSNR are cor-elated to image distortion due to deconvolution procedure:hey are both almost constant as M increases. The algorithmrovides good performance also for the minimum acceptable

ig. 6 – Comparison between B-mode image before (a) processinrocessing (c) of an in vivo prostatic gland acquisition.

FWD 0.89 1.20 1.70 18.24 0.873WLSD 1.86 1.32 2.38 18.75 0.856

value M = 32, allowing for an important computational costreduction.

Finally, we compared algorithm performance to other ultra-sound signal deconvolution algorithms in literature: WLSD [6]and FWD [7]. As clearly shown by the results in Tables 2 and 3,the proposed algorithm outperforms both of them.

Figs. 6 and 7 compare to the original B-Mode images(Figs. 6(a) and 7(a)) the visual quality of images processed withthe proposed approach applied to the signal envelope domain(Figs. 6(b) and 7(b)) and to the RF signal (Figs. 6(c) and 7(c)), as it

g, after envelope processing (b) and after RF signal

S10 c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 5 S ( 2 0 0 9 ) S4–S11

ssin

r

Fig. 7 – Comparison between B-mode image before (a) proceprocessing (c) of an in vivo prostatic gland acquisition.

can be seen, resolution and contrast in the processed imagesare much better than in the original ones and processing RFsignal provides better performance.

6. Conclusion

This work proposed a deconvolution technique for US images,based on a reduced-states sequence estimation procedure,suitable for real-time applications. Simulation results shownthat, for high SNR, the proposed algorithm approaches the MLestimation performance in terms of BER, while its computa-tional complexity is sensibly lower.

In particular the obtained results show that the proposedalgorithm outperforms WLSD and FWD in terms of RF signalresolution improvement, measured as axial resolution gainand in terms of image quality, quantified by the peak sig-

nal to noise ratio and the structural similarity index, used tocompute the dissimilarity between the original and processedimage; finally we measured the image contrast enhancementby means of contrast gain.

g, after envelope processing (b) and after RF signal

Further development of this algorithm comprises the pos-sibility of estimating the system impulse response withadaptive techniques and processing signals distorted withnon-minimum phase impulses.

Conflict of interest statement

None declared.

Acknowledgments

The authors gratefully acknowledge Prof. L. Masotti (Univer-sity of Florence, Italy) and his group for providing B-Modeimages. This work has been partially supported by MIURwithin the framework of FIRB founding initiative.

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