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Advanced Acoustic Microimaging Using Sparse Signal Representation for the Evaluation of Microelectronic Packages Guang-Ming Zhang, David Mark Harvey, and Derek R. Braden
Abstract—Acoustic microimaging (AMI) has been widely used to nondestructively evaluate microelectronic packages for the presence of internal defects. To detect defects in small devices such as BGA, flip-chip, and chip-scale packages, high acoustic frequencies are required for the conventional AMI systems. The acoustic frequency used in practice, however, is limited by its penetration through materials. In this paper, a novel acoustic microimaging technique, which utilizes nonlinear signal processing techniques to improve the resolution and robustness of conventional AMI, is proposed and investigated. The technique is based on the concept of sparse signal representations in overcomplete time-frequency dictionaries. Simulation and experimental results show its super resolution and high robustness. Index Terms—Acoustic microimaging (AMI), microelectronic packages, overcomplete dictionaries, sparse signal representations.
I. INTRODUCTION
A
COUSTIC MICROIMAGING (AMI) has been widely used for the analysis of flip-chip and BGA devices. With conventional time-domain AMI (TAMI), a focused ultrasonic transducer alternatively sends pulses into and receives reflected echoes from discontinuities within the sample. Since the echoes are separated in time, based on the depths of the reflecting features in the sample, a gate corresponding to a time window can be used to select a specific depth or interface to view. A mechanical scanner moves the transducer over the sample, producing C-scan (interface scan) images. At each - position, only the peak intensity value and the polarity of the echo within the gate are displayed in the C-scan image. As advanced microelectronic packages are continually being produced smaller and thinner, detection of the internal features and defects in the packages is approaching the resolution limits for TAMI. To increase AMI resolution, one way is to use high acoustic frequencies. However, higher frequencies provide less penetration through materials. A better alternative is to utilize signal processing techniques to improve the resolution [1]–[3]. In [1], frequency-domain AMI (FAMI) which produces individual-frequency fast Fourier transforms (FFT)-filtered images Manuscript received June 8, 2004; revised December 17, 2004. This work was supported by The Engineering and Physical Sciences Research Council (EPSRC) of the U.K. G.-M. Zhang and D. M. Harvey are with the General Engineering Research Institute, Liverpool John Moores University, Liverpool, L3 3AF, U.K. (e-mail:
[email protected]). D. R. Braden is with the Delphi Delco Electronics Systems, Kirkby, Liverpool, L33 7XL, U.K. Digital Object Identifier 10.1109/TADVP.2005.853553
was presented by Semmens and Kessler. Although FAMI reveals some features or defects that are at or below the accepted resolution limits in TAMI, some significant features may be lost or may not be resolved in a single-frequency image, due to spectrum overlapping and frequency shifting. In [2], deconvolution techniques were used to increase the axial resolution in A-scan signals. However, the conventional deconvolution techniques are not effective in most cases for AMI signals because the waveform and frequency of the reflected echo in an A-scan is quite different from the incident waveform due to the frequencydependent attenuation. Recently, a high-resolution AMI technique, which is based on matching pursuit (MP) [4], was introduced in [3]. The experimental results in [3] showed its higher resolution and robustness in comparison to TAMI and FAMI. This paper presents results of further research on time-frequency atomic AMI. A novel nonlinear signal processing technique, taken from the field of sparse signal representation [5], is proposed to improve the resolution and robustness of conventional AMI. This paper is organized into eight sections as follows. The model of AMI signals is briefly described in Section II. The basic principle of sparse signal representation-based AMI (SSRAMI) is presented in Section III. Section IV discusses overcomplete time-frequency dictionaries for AMI. Detailed implementation of SSRAMI is presented in Section V. Simulation results are presented in Section VI, while experimental results are given in Section VII. Finally, we conclude in Section VIII. II. BASIC MODEL A. Modeling of the AMI Signals In reflection mode AMI, the fundamental information is contained in the received radio frequency (RF) signal called the A-scan which displays the echo depth information in the sample at each , coordinate. Fig. 1(a) shows AMI echo components at typical boundaries in a flip-chip package mounted on ceramic , , and are chip top, chip–solder substrate, where bond and solder bond–ceramic substrate echoes, respectively. can be expressed Mathematically, the observed A-scan as follows (1) is reflected echoes displayed in the A-scan correwhere sponding to different interfaces in the device being examined . The reflected and possibly corrupted by additive noise
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dependents on the acoustical The reflection coefficient impedance value of the material involved, defined by the following: (3) is the intrinsic impedance of the material through where is traveling and is that of the which the incident pulse next material which is encountered by the pulse. Since the acoustic impedance of air (0.0004 g/cm s) is far smaller than solid (e.g., Aluminum is 17.0 g/cm s), equals 1 approximately at the solid–air interface. This means that the incident pulse is almost fully reflected at the solid–air interface, and the minus symbol expresses that the polarity of the reflected echo is reversed relative to the incident pulse. Therefore, acoustic microimaging has been widely used for detecting gap-type (air, vacuum) defects such as voids, delaminations, and cracks inside packages. In addition, in most cases, the pulses are band-limited, implying the following description: (4) where is modulation frequency, and scription parameters. Hence, the echo form:
are the detakes the following
(5) The two-sided spectra of are peaked in the vicinity of the frequencies and can be assumed to be enclosed within a band , namely, of width
(6)
B. Reflection Coefficient AMI and Amplitude-Polarity AMI
Fig. 1. AMI echoes at typical boundaries in flip-chip package mounted on (a) ceramic substrate and A-scans obtained using (b) a 230-MHz transducer and (c) a 50 MHz transducer, respectively.
echoes can be modeled by the convolution of the ultrasonic pulse with a function representing the reflectivity properties of can the interrogated materials. Hence, the reflected echo be represented in terms of the reflection coefficient and the at the location of interface , as folincident pulse lows: (2)
For conventional TAMI, the magnitude, phase polarity, and time of the echoes are used to characterize the interfaces. Hence, the size, location and type of defect inside BGA packages and solder joints can be evaluated. When a pulse is transmitted inside a package, ultrasonic waves may be absorbed, scattered, and reflected whenever the acoustic properties of materials change so that the incident pulse is different at different , positions for the interface being investigated [6], [7]. Evidently, from (3) and (5), it is better to characterize the internal features of packages using instead of because is directly related to the acoustical properties of materials. Therefore, it is expected that reflection coefficient AMI enhances the inspection in both accuracy and resolution when compared to amplitude-polarity AMI. In fact, conventional deconvolution techniques improve the image resolution of the C-scan by employing reflection coefficient imaging. However, standard TAMI and FAMI can only perform amplitude AMI. As it will be seen next, the proposed AMI technique can easily perform both amplitude-polarity AMI and reflection coefficient AMI.
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Fig. 2. C-scans of chip–solder bond interface produced from the detection of a flip-chip package soldered on a ceramic substrate (dimension of the flip-chip package is 8:19 8:30 0:86 nm). (a) Using 230-MHz transducer. (b) Using 50-MHz transducer.
2
2
III. SPARSE SIGNAL REPRESENTATION (SSR)-BASED AMI Fig. 1 shows A-scan signals measured from a flip-chip package which is mounted on a ceramic substrate, using Sonoscan’s D9000 system. Because the solder balls are relatively small in comparison with the resolution of a 50-MHz transducer, echoes from solder bond top and bottom shown in Fig. 1(c) are superimposed. The two echoes from the radio frequency (RF) signal acquired by a 230-MHz transducer shown in Fig. 1(b) are clearly seen. The electronic gates were set to image the chip–solder bond interface as shown in Fig. 1(b) and (c). Fig. 2 shows the corresponding C-scan outputs. Obviously, more detail is observed in Fig. 2(a) rather than Fig. 2(b) which is blurred. To avoid this blur, we need to separate the echoes within . the gate and find the appropriate echo component, i.e., Many signal processing techniques have been developed for separating components of signals [2], [5], [8]. A powerful tool is sparse signal representation, which was successfully applied to the problem of blind source separation [9]. In this paper, sparse representations of ultrasonic signals are exploited to separate the ultrasonic echoes in A-scans. From Section II-A, the A-scan signal was modeled as a sum of the reflected ultrasonic echoes. Combining (1) and (2), gives the following when ignoring the noise (7) The goal is to infer both the reflection coefficients and the incident pulses according to the observed signal y, and then produce C-scan images. A three-stage process for SSRAMI is proposed. First, a priori selection of a possibly overcomplete signal dictionary in which the ultrasonic pulses are assumed to be sparsely representable. Second, separating the incident pulses by exploiting their sparse representability. Third, selecting an appropriate echo and producing a C-scan output. A. Selection of an Overcomplete Dictionary According to a priori knowledge of the ultrasonic transmitted pulse, first select a dictionary of generating atoms (or ele, each one a vector in Hilbert space , which ments)
represents activity of an incident pulse . The dictionary can , with generating atoms for be viewed as a matrix of size each column, where is the length of the ultrasonic signal to be processed. In particular, the dictionary can be overcomplete and contain linearly dependent subsets, and in particular need not be a basis. In SSRAMI, ultrasonic signals are decomposed in an overcomplete dictionary because it is more flexible in terms of how the signal is represented. B. Separating the Ultrasonic Pulses In general, the ultrasonic A-scan can be assumed to have a very sparse representation in a proper signal dictionary though not in the time domain. The problem of separating the ultrasonic pulses is, therefore, formulated as the follows: Given the observed A-scan and the overcomplete dicsuch tionary , find the vector of coefficients in (7) and is as sparse as possible. that Estimating involves solving an undetermined set of equations if without sparse requirements on the solution. The additional requirement for sparsity would be to minimize the support of , i.e., minimize the number of places where is nonzero. Hence, we need to solve the problem [10] (8) ), soGiven an overcomplete dictionary (Implying requires enumerating subsets of the dictionary lution of looking for the smallest subset able to represent the signal. Unfortunately, the complexity of such a subset search grows exponentially with . However, under certain conditions, which have been found empirically and theoretically [5], [10], [11], it can be considered equivalent to the constrained optimization problem, given by (9) This problem is commonly known as basis pursuit (BP), introduced in the pioneering work on this subject [5]. It is a convex optimization problem, which can be cast as a linear programming problem and solved by modern interior point methods, even for very large and .
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Fig. 3. The tune-frequency energy distribution of the A-scan in Fig. 1(b) and time-frequency window.
In practical ultrasonic nondestructive detection, ultrasonic signals are commonly contaminated by background noise. in (1) is a white Gaussian process Suppose that the noise with zero mean and variance , the BP decomposition can be adapted to the case of noisy signal by modifying to the following unconstrained form: (10) A computational efficient solution to (P3), as well as the rules for choosing the parameter , will be discussed in Section V. C. Selecting an Appropriate Echo and Producing a C-Scan Output of AMI Fig. 3 displays in phase space the decomposed result of the A-scan shown in Fig. 1(b). The darkness of the time-frequency image increases with the energy value, and each time-frequency atom selected by the BP method is represented by a Heisenberg box. The process of selecting the proper echo and producing the C-scan image is suggested here as follows. First, a time-frequency window as depicted in dotted line in Fig. 3 is determined in terms of the frequency of transducer and the interface to be investigated in the microelectronic package. Suppose the time-frewith time width quency window centers at position and frequency width . Secondly, in the given time-frequency window we search for the time-frequency atoms whose centers are lying in the window, and among them pick up the one with the biggest decomposed coefficient. The time-frequency atom selected and its coefficient is used as the approximation and reflection coefficient . of the expected incident pulse Finally, the reflection coefficient AMI is carried out by directly displaying these coefficients selected at their corresponding position of the C-scan image. Amplitude-polarity AMI can also acbe carried out by reconstructing the reflected echoes cording to (2) using and . The peak intensity value and polarity of the reconstructed echo is displayed at the corresponding position of the C-scan image.
The determination of time-frequency window is a crucial step, on which the image quality depends in great degree. Some a priori information such as the center frequency of the transducer in use and its fundamental bandwidth, and package structure should be incorporated. In a broadband AMI system there is frequency downshifting due to frequency dependent attenuation. Moreover, even at the same interface, there are minor differences in center frequency for the incident pulses at different – positions. Hence, calibration is usually carried out preceding the processing. The average center frequency of window. A is utilized as the center frequency position reasonable selection for the window width and is the width of Heisenberg box of expected atom multiplied by a constant factor. The factor given empirically is to compensate for the changes of center frequency, and reduce the loss of useful echoes reflected in the interface to be investigated. If a small window is used, a clear image would be expected but some features existing in the practical interface may not be displayed. On the contrary, the acoustic image might be a little blurred. An alternative to the above method is to examine the internal features and possibly defects by a succession of C-scans produced by a succession of time-frequency windows as shown in is chosen when become low. SpeFig. 3, where small cific features may yield more information at one time-frequency window rather than another. Therefore, SSRAMI can bring out image details that might not be visible with conventional TAMI and FAMI. D. Other Sparse Signal Representation Approaches Apart from the BP approach, there are several other popular approaches searching for a sparse solution of (7). Mallet and Zhang [4] have suggested a matching pursuit method for decomposing signal over a given dictionary . In MP, this is done by successive approximations of by orthogonal projections on atoms of . Let an initial approximation and residual . At stage k, it identifies the dictionary atom that best correlates with the residual and then adds to the current approximation a scalar multiple of that atom, so that , where is computed by the -inner and , i.e., , , and product of . After iterations, a matching pursuit decomposes the signal into (11) When the dictionary is orthogonal, the method works perfectly. atoms and the algoIf the object is made up of only rithms is run for steps, it recovers the underlying sparse structure exactly. When the dictionary is not orthogonal, MP may failed to super-resolve and not be sparsity-preserving that has been demonstrated by several examples [5] although in most cases MP delivers a sparse representation. For certain dictionaries such as wavelet packet and cosine packet dictionaries, certain special subcollections of the atoms in these dictionaries amount to orthogonal bases. Hence, there are a wide range of orthogonal bases in these dictionaries, in
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fact such orthogonal bases for signals of length . The best orthogonal basis (BOB) approach proposed by Coifman and Wickerhauser [12] adaptively picks a single best basis from these bases. This algorithm in some case delivers near-sparse representation. In particular, when has a sparse representation in an orthogonal basis taken from the dictionary, BOB is expected to work well. However when is composed of a moderate number of highly nonorthogonal components, the method may not deliver sparse representation because finding an orthogonal basis prevents it from finding a highly sparse representation.
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Gabor frames seem to be the most suitable dictionaries for AMI in terms of the following features. A Gabor frame is optimally concentrated in both time and frequency. Experimental results [18] shows real Morlet wavelet which has the same function format with (12) is one of the best-performing wavelets in ultrasonic nondestructive inspections. Atoms in Gabor frames match ultrasonic echoes very well as shown in Section VI. The discretised scale parameter can be selected in an arbitrary way in some implementations (refer to Section V). The smaller the scale , the higher the AMI accuracy (refer to Section VI). However, unlike wavelet packet (WP) dictionaries and cosine packet (CP) dictionaries, there is no fast implementation algorithm.
IV. OVERCOMPLETE DICTIONARIES FOR AMI A dictionary is an indexed collection of elementary wave. The waveforms , called atoms or eleforms ments are discrete-time signals of length The construction/selection of dictionary in the proposed SSRAMI is crucial in order that the atoms in will be well matched to the ultrasonic incident pulse in (2). Ideally, we hope the atoms themselves to be adapted to . Lewicki and Sejnowski [13] proposed an algorithm for learning a dictionary by viewing it as probabilistic model of the observed signal. In addition, one can build for a special application from existing dictionaries by incorporating some prior information [14]. In this paper, we test and compare several existing dictionaries for AMI. Over the last few years, a lot of dictionaries have been proposed and most of them are overcomplete. Popular examples of overcomplete dictionaries include: wavelet packets and cosine packets dictionaries [12], which contain atoms, representing transient harmonic phenomena with a variety of durations and locations; Gabor frames [15]; wavelet frames [16]; frequency dictionaries obtained by sampling the frequencies more finely; and trivial dictionaries, such as Dirac dictionary and Heaviside dictionary [5]. From (1) and (4)–(6), the ultrasonic echoes appear as transients in the observed A-scan, exhibiting time-frequency localization characteristics. As a result, dictionaries consisting of atoms, which have a compact support both in time and frequency, are suitable for our applications. A. Gabor Dictionaries The real Gabor dictionary is defined by [17], where
and (12)
where is the scale of the function, its translation and its frequency modulation; window function is Gaussian function , constant and factor normalize , and is the phase of the real Gabor vectors. In practical applications, signal decomposition is performed in the diswhere is composed of all crete Gabor dictionary with and . is a frame of Duabechies [15] proved that . It follows that satisfies the same , with the same frame frame inequalities as bounds A and B, by a change of variable .
B. Wavelet Packet Dictionaries and Cosine Packet Dictionaries The WP and CP dictionaries were proposed by Coifman and Meyer in [12]. When it comes to one dimension-discrete time signals of length N, each of the dictionaries contains waveforms. A WP dictionary consists of a standard orthogonal wavelet dictionary, the Dirac dictionary, and a collection of oscillating waveforms spreading across a range of frequencies and duration. A CP dictionary is made up of standard orthogonal Fourier dictionary, and a range of Gabor-like elements: sinusoids of various frequencies weighted by windows of various widths and locations. A WP dictionary requires a depth parameter (the depth of the WP table) to specify the degree of finest frequency partition. Similarly, a CP dictionary requires a depth parameter (the depth of CP table) to specify the degree of finest -fold time partition. The resulting dictionaries are the overcomplete with atoms. V. IMPLEMENTATION OF SSRAMI From Section III, there exist several approaches to decompose ultrasonic signals in overcomplete dictionaries, thus resulting in different implementation methods of SSRAMI. A. BP-Based AMI BP chooses the decomposition with minimum norm of coefficients as described in (P2). In BP-based AMI (BPAMI), ultrasonic incident pulses and reflection coefficients are inferred by solving the convex optimization problem of (P3), for which Chen and Donoho proposed two algorithms in [19], the BPDeNoise-Interior and BPDeNoise-Simplex. BPDeNoise-interior, which is designed for dictionaries with fast implicit algorithms, implements a primal-dual logarithmic barrier interior point algorithm with a conjugate gradients solver for linear equations arising in interior point algorithm iterations. BPDeNoise-Simplex is a simple method, where one needs to provide dictionary matrices. In our experiments, we use the ATOMIZER [20] and WAVELAB MATLAB packages for fast implementation of BPDeNoise-Interior and BP-Interior [for solving problem (P2)]. The parameter in (P3) controls the influence of sparseness penalty. It was shown in [19] that the solution of (P3) is closely related to the problem of denoising by thresholding the representation coefficients. A number of threshold estimating methods were proposed [21]–[23] in the past. In BPAMI, is , where set to ‘VisuShrink’ threshold [21], i.e.,
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Fig. 4. Ultrasonic pulses (solid line) measured in a water tank from a planar reflector, respectively, using (a) 230-MHz and (b) 50-MHz transducers and their approximations (dotted line)and (c) the amplitude spectrum of the measure pulse in (a). (d) The amplitude spectrum of the measure pulse in (b).
is a standard deviation of the noise in (1), and is the cardinality of the dictionary. In BPAMI, only dictionaries with fast implicit algorithms are used due to computational complexity. We especially focus on the WP and CP dictionaries, in which the depth parameter is set as big as possible. B. MP-Based AMI MP is a stepwise greedy algorithm, iteratively building up an approximation by adjoining at each stage an atom with the biggest correlation to the current residual. In the following experiments, both WP/CP dictionaries and Gabor dictionaries are used. In MP-based AMI (MPAMI), MP with WP dictionaries are implemented by ATOMIZER [19] and WAVELAB MATLAB packages while we use WAVE++ [17] for the implementation of MP with Gabor dictionaries. The parameter is discretised as follows: • , for , where n is the biggest integer , and can be selected in an power of such that arbitrary way. and • • • , where . Due to frequency downshifting in AMI systems [7], can be set small (in most cases is enough) to reduce computation burden. are disIn [4], it is proven that if the parameters such cretised as indicated previously, there exists an
that the MP algorithm is suboptimal with respect to .
, i.e.,
As seen in (1), the received ultrasonic signal is usually corrupted by noise. To remove the effect of noise on sparse signal representation, before picking up an appropriate echo the softthresholding operation [24] is applied to the decomposed coefficients (13) where threshold
is set to
.
C. BOB-Based AMI BOB is specially designed for WP and CP dictionaries. It finds the orthogonal basis minimizing an additive entropy measure of coefficients. In BOB-based AMI (BOBAMI), for denoising the soft thresholding operation in (13) is applied during selection of the best orthogonal basis with special entropy. VI. SIMULATION RESULTS A. Ultrasonic Signal Simulation The performance of SSRAMI was tested through simulation. Simulated data were produced according to the model of signal formation in (1) and (2), assuming that each signal contains are produced two echoes. The incident ultrasonic pulses by Gabor functions in (12) which model measured ultrasonic pulses very well. Fig. 4 depicts two measured pulses obtained
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Fig. 5. Ultrasonic signal simulation. (a) A simulated ultrasonic signal. (b) The amplitude spectrum. (c) The time-frequency energy distribution.
from a planar reflector in a water tank, respectively, using 230- and 50-MHz transducers on a Sonoscan D9000 system, together with their approximation pulses generated from the Gabor model. All the pulses were normalized. The average root-mean squared error of disparity between the simulated and measured pulses is about 2%. Fig. 4(c) and (4d) gives the amplitude spectrum of the measured pulses, where heavy frequency downshifting is observed due to the frequency-dein a simulated A-scan is pendent attenuation. Each echo obtained by the convolution of the normalized incident pulse with a reflection coefficient function which includes only representing one interface. The one nonzero coefficient simulated A-scan is the superposition of two echoes as (1). Though this simulation might seem simplistic, it turns out to be very useful, allowing investigation of all the main features of the proposed technique. In fact, similar simulations have been widely used in the community of ultrasonic imaging [2], [14], [25]. An example of synthesized ultrasonic A-scans, its amplitude spectrum, and time-frequency energy distribution are illustrated in Fig. 5. For this A-scan, the amplitude of either echo might be affected each other due to overlap in time and frequency, so that TAMI likely produces a contaminated C-scan image when gating either echo. Similarly, FAMI produces incorrect C-scans in these frequencies within the overlapped interval in Fig. 5(b). However, two echoes are clearly separated in the time-frequency plane so that more accurate C-scans can be produced. Various A-scans were simulated by adjusting the parameters , , , and of two echoes (each echo sequence was truncated to 128 samples in width) so that each echo has different waveform, frequency, and phase. The parameter was chosen such that two echoes in the A-scans possessed different degrees of overlap in the time domain. The overlap degree in the frequency domain was tuned by . To make the simulation as close to prac-
tice as possible, the parameters , , and were initialized to the real pulse echo shown in Fig. 4(a). Furthermore, the reflection coefficient which determines the amplitude of each echo was changed to simulate different reflectivity properties of the interrogated interfaces/features. Note that in order to perform reflection coefficient AMI all the incident pulses were normalized during simulation, implying . that B. Selection of Dictionaries A simulation was performed to examine several existing dictionaries which are suitable for SSRAMI. The performance of SSRAMI was measured by three performance criteria: energy error, coefficient error, and amplitude error. The energy error , which has been used in blind source separation [9] is defined as (14) where is the original echo and coefficient error is defined as
is the recovered echo. The
(15) where is the original reflection coefficient, and the estimated reflection coefficient. The coefficient error measures the performance of reflection coefficient AMI. And the amplitude error is defined as (16) where is the peak intensity value of recovered echo by is its theoretical AMI techniques from gated A-scans, and
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TABLE I MPAMI RESULTS WITH SIMULATED A-SCANS (# 1, 2, 3, AND 4) IN VARIOUS CP/WP DICTIONARIES
TABLE II MPAMI RESULTS WITH THE SIMULATED A-SCANS IN TABLE I IN GABOR DICTIONARIES
value. The amplitude error measures the performance of amplitude-polarity AMI. MPAMI was performed on simulated data over WP dictionaries with different mother wavelet and CP dictionaries. The time-frequency window was set to the Heisenberg box of the first echo. The results for four simulated A-scans (#1, 2, 3, and 4) are listed in Table I, from which it can be seen, overall, that the WP dictionary with symmlet 8 is a little superior to other WP dictionaries and CP dictionaries. Table II presents the results of MPAMI with the Gabor dictionaries for the same A-scans in Table I, where is the scale parameter in Section V-B. It is clearly seen that Gabor dictionaries is the most suitable one for SSRAMI among these existing dictionaries. Moreover, the smaller the scale (i.e., the finer partition in , , ), the smaller the AMI error. C. Performance Comparison of Three SSRAMI Methods This simulation was performed to examine the performance of three SSRAMI methods. The WP dictionary with symmlet wavelet was used. Results of quantitative evaluation are listed
in Table III. The interface for C-scan imaging was set to the first echo by setting the time-frequency window to the Heisenberg box of the first echo. The AMI error is presented in Table III. To examine the robustness of the three SSRAMI methods, in the lower part of Table III the results from noisy data are listed, which were produced by adding zero-mean Gaussian white noise into the simulated A-scans in the upper part. From Table III, it can be seen that MPAMI and BOBAMI almost give the same AMI results in our simulation. BPAMI is not always better than MPAMI and BOBAMI. And MPAMI with Gabor dictionaries achieves better performance. It is worth noting that BPAMI, MPAMI, and BOBAMI are different with BP, MP, and BOB. BP is superior to MP and BOB in sparse signal representations, but it is subject to overall decomposed coefficients. For SSRAMI, the AMI error is determined by a single atom which is picked as an appropriate echo. Ideally, the simulated A-scan can be represented by two atoms. In practice, the simulated A-scan is approximated by a lot of atoms because of the mismatch between atoms and echoes. Therefore, although the decomposition coefficients produced
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TABLE III PERFORMANCE COMPARISON
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E
OF THREE SSRAMI METHODS. THE AMI ERROR ( AND POSITION ERROR (IN SAMPLE NUMBER) ARE PRESENTED
by BP have the best sparsity and superresolution overall, we cannot guarantee that the single atom picked by BPAMI will give the best recovery of the expected echo. Another probable reason is the conditions which assure (P1) equivalent to (P2) is not satisfied. In fact, BP is formally equivalent to the probabilistic framework proposed by Lewicki and Olshausen in [26], where the coefficient prior is assumed to be Laplace distribution. But a Laplacian might not be sparse enough to reflect the practical coefficient distribution [26], especially for our simulations where the coefficient distribution is sparser than predicted by the Laplace distribution. In Table III, we also present the position error (in sample number) of estimated echo time position . It is seen that all three methods locate the echoes well, even for noisy signals. D. Performance Comparison Between SSRAMI and Conventional AMI Assuming amplitude-polarity AMI was applied, the imaging performance of TAMI, FAMI, and SSRAMI (MPAMI with Gabor dictionaries was used) was quantitatively evaluated by . Some results of quantitative evaluation are listed in Table IV, where parameters and are the amplitudes determines the frequency of two echoes separately, determines the time separation of two echoes, and separation (in sample number) of two echoes. The interface for
,
c
AND
A
)
TABLE IV AMI RESULTS WITH SIMULATED A-SCANS BY DIFFERENT AMI TECHNIQUES
C-scan imaging was set to the first echo by choosing gates as follows: for TAMI, the gate was chosen to center at , with width equal to 32 samples. The time-frequency window was set to the Heisenberg box of the first echo. For FAMI, the center frequency of the first echo is used as the imaging frequency, and is the spectrum amplitude at this frequency. so From Table IV, it can be seen that SSRAMI is more stable, and gives more accurate results in most cases when compared to TAMI and FAMI.
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TABLE V AMI RESULTS WITH NOISY A-SCANS BY DIFFERENT AMI TECHNIQUES
The robustness of SSRAMI was examined by adding zeromean Gaussian white noise into the above simulated A-scans. Table V tabulates the test results for a simulated A-scan with different noise energy added, which indicates that SSRAMI is more robust than the conventional AMI techniques. Fig. 6 shows dB. We can clearly see TAMI and the results of FAMI are susceptible to noise but SSRAMI is more resistant to noise. For TAMI and FAMI, when SNR is below zero, the AMI is unstable and unreliable, producing very poor C-scan output. E. Discussion Although the attenuation in the coupling medium and package materials is not incorporated explicitly into the AMI model in Section II-A, it has been considered throughout this paper implicitly. The frequency-dependent attenuation in materials affects the frequency, phase, and amplitude of and reflected echo in (1) and (2). each incident pulse Unlike conventional deconvolution techniques where either the reflected echoes were assumed to have similar waveforms or the exact attenuation had to be considered, arbitrary attenuation is assumed in this paper because each reflected echo is assumed to be any frequency, phase, and amplitude. For the proposed SSRAMI, its performance is affected by the overlap degree of two echoes in time-frequency domain. If two echoes have very small differences in both time position and center frequency, the proposed techniques could not resolve them. The resolution limitation of SSRAMI is determined by the used sparse signal representation algorithm. Obviously, there is no limitation to the center frequency of the used transducer for the proposed technique. In other words, the proposed technique is applicable to higher frequency transducers.
VII. EXPERIMENTAL RESULTS Experiments were performed to validate the proposed technique. A real automotive electronic circuit board (ceramic thick-film hybrid circuit) from a manufacturing line was used as the test sample. For the thick-film hybrid circuitry, multiple layers (tracks, conductors, dielectric layers, etc) were printed on one ceramic substrate. This paper focused on a flip-chip package on the ceramic board as illustrated in Fig. 1(a). The C-scan images for the solder bonds of the package are shown in Fig. 2. A-scans at 32 points from different locations were investigated. Selected points are the representatives of three typical areas: dark solder bond area, nonsolder bond area, and the white area in the center of solder balls or bright areas representing disbonds between underfill and chip. At each point we captured A-scans using both a 230-MHz transducer and a 50-MHz transducer. Before SSRAMI, these A-scans were first gated to the interrogated section, and then were extended by zero-padding. For WP and CP to a fixed signal length dictionaries and discrete Gabor dictionaries in Section V, the number of atoms in them is connected to . A short signal will be decomposed in a dictionary with less atoms (see Section V) so that atoms will not be well matched to the structure in the data. However, long signals increase computation load. Hence, an approximate signal length is necessary. The compromise in this paper was to set to 256. Fig. 7 shows example A-scans from a nonsolder area. Fig. 7(a) is acquired by the 230-MHz transducer, in which the first echo is from the chip–solder bond (C-B) interface, the second echo is from the solder bond-thick film (B-F), and the third one is reflected by the other layer of thick film (F-F). At the same point, the measured A-scan using a 50-MHz transducer is displayed in Fig. 7(b), where three echoes cannot be resolved due to the lower resolution of the 50-MHz transducer. Fig. 7(c)–(e) gives sparse signal representations of Fig. 7(b), which are obtained by BOB, MP, BP in WP dictionaries with Symmlet 8 wavelet, respectively. Fig. 7(f) displays the sparse signal representations by MP with Gabor dictionary and scale . Comparing Fig. 7(a) with the resulting SSRAMI results in Fig. 7(c)–(f), it is observed that SSRAMI resolves these echoes in the low resolution 50-MHz A-scan. Thus, SSRAMI can image the individual interfaces, improving the AMI resolution. Fig. 8 shows another case of example A-scans from a dark solder bond area, where all three echoes reflected at C-B, B-F and F-F interfaces are weak. Fig. 9 shows the third case of example A-scans from disbond areas, where the first echo is strong and the second and third echoes are weak. Results in Fig. 8 and Fig. 9 further demonstrate the effectiveness of SSRAMI. It is observed from Figs. 7–9 that MP with Gabor dictionaries achieves the best results. This confirms the importance of overcomplete dictionaries in SSRAMI. It is also seen that BP is better than MP and BOB according to sparsity and superresolution. The effect of dictionaries on SSRAMI was further examined by the following experiment. Fig. 10 shows the sparse representations of an A-scan signal after being decomposed by BP in different WP dictionaries. Basically, the experimental results
ZHANG et al.: ADVANCED ACOUSTIC MICROIMAGING USING SPARSE SIGNAL REPRESENTATION
Fig. 6. AMI results of a simulated A-scan. (a) Noise-free A-scan. (b) Noisy A-scan with spectrum of (b). (d) Sparse representation of (b).
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SNR = 01 dB, produced by adding white noise onto (a). (c) Amplitude
Fig. 8. Example A-scans from a dark solder bond area acquired by (a) 230-MHz transducer and (b) by 50-MHz transducer, and sparse representations of (b) by (c) , (d) , (e) , and (f) .
BOB + WP MP + Gabor dictionary Fig. 7. Example A-scans from nonsolder area acquired by (a) 230-MHz transducer and (b) by 50-MHz transducer, and sparse representations of (b) by (c) , (d) , (e) , and (f) . (g) time-frequency windows in (f).
BOB + WP MP + Gabor dictionary
MP + WP
BP + WP
are in accordance with the foregoing simulation results in Section VI-B. Finally, to examine the effect of time-frequency windows on SSRAMI, the following experiment was performed. Twelve A-scans from nonsolder area acquired by a 50-MHz transducer are firstly decomposed into sparse representations by MP with
MP + WP
BP + WP
Gabor dictionary. A time-frequency window shown in Fig. 7(f) (dotted frame) is applied to the 12 A-scans. Due to variation of center frequency of echoes in A-scans, there is one , and A-scan choosing improper atoms by the window others work perfectly. When we enlarged the time-frequency to [dashed frame in Fig. 7(f)] and applied window to these A-scans, the mis-selected A-scans work properly. Further research will be performed on 3-D RF data, which can be produced by Sonoscan’s new Virtual Rescanning Module
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separating the incident pulses by exploiting their sparse representability. Third, selecting an appropriate echo and producing a C-scan output. The proposed AMI improves the resolution of AMI by sparse signal representation techniques without increasing ultrasonic frequencies. The simulated and experimental results have demonstrated the super resolution and robustness of the novel technique for the evaluation of microelectronic packages.
REFERENCES
Fig. 9. Example A-scans from the white area in the center of a solder ball acquired by (a) 230-MHz transducer and (b) by 50-MHz transducer, and sparse , (d) , (e) , and representations of (b) by (c) (f) .
BOB + WP MP + Gabor dictionary
MP + WP
BP + WP
Fig. 10. Evaluation of time-frequency dictionaries. (a) A-scan. (b) Symmlet. (c) Coiflet. (d) Daubechies. (e) Vaidyanathan. (f) Battle.
(VRM) [1]. VRM allows for the collection of A-scans along with the acoustic image. VIII. CONCLUSION In this paper, a novel acoustic microimaging technique has been presented for the evaluation of modern microelectronic packages. The proposed AMI technique, differing from conventional time-domain AMI and frequency domain AMI, has been implemented by a three-stage process. First, a priori selection of a possibly overcomplete signal dictionary in which the ultrasonic pulses are assumed to be sparsely representable. Second,
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[23] P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using general gaussian and complexity priors,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 909–919, Apr. 1999. [24] D. L. Donoho, “De-noising by soft thresholding,” IEEE Trans. Inf. Theory, vol. 41, no. 3, pp. 613–627, May 1995. [25] D. Adam and O. Michailovich, “Blind deconvolution of ultrasound sequences using nonparametric local polynomial estimates of the pulse,” IEEE Trans. Biomed. Eng., vol. 49, no. 2, pp. 118–131, Feb. 2002. [26] M. S. Lewicki and B. A. Olshausen, “Probabilistic framework for the adaptation and comparison of image codes,” J. Opt. Soc. Amer. A, vol. 16, no. 7, Jul 1999.
Guang-Ming Zhang received the M.Sc. and Ph.D. degrees in mechanical engineering from Xi’an Jiao Tong University, Xi’an, China, in 1996 and 1999, respectively. He joined the Institute of Acoustics, Nan Jing University, Nan Jing, China, in 1999. From 2001 to 2003, he worked at Signals and Systems Group, Uppsala University, Uppsala, Sweden as a Postdoctoral Fellow and then moved to the U.K. in 2003, where he is currently with Liverpool John Moores University, Liverpool. In the past, he has undertaken research in acoustic signal and image processing, ultrasonic data compression, nonlinear acoustic imaging, nonlinear acoustical field simulation, acoustical microimaging, surface acoustic waves motor, ultrasonic nondestructive evaluation of materials, and in the development of ultrasonic imaging systems. His current research interests are in sparse signal representations of acoustic signals in overcomplete dictionaries, advanced acoustical microimaging, fusion of ultrasound image and X-ray image for semiconductor applications, failure analysis of BGA and flip-chip solder bonds using mixed environment reliability testing, and 3-D ultrasonic data compression and FPGA implementations.
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David Mark Harvey was born in Sheffield, Yorkshire, U.K. He received the B.Sc. (Hons.) degree in electrical and electronic engineering and the Ph.D. degree in real-time microprocessor-based analysis of optoelectronic data from Liverpool Polytechnic, Liverpool, U.K., in 1979 and 1984, respectively He is currently with Liverpool John Moores University, where he is the Professor of electronic engineering, and Director of the engineering development center. He spends much of his time helping small companies to innovate by introducing modern R&D methods into their daily psyche. He also works on the design of engineering e-learning courses for industry and world markets. He has previously acted as a consultant to over 200 companies, and leads a team of professional engineers helping companies in the U.K. His research interests are varied and usually link digital signal processing to real engineering or image processing problems. He is at home working on the design and test of various hardware architectures from parallel to asynchronous systems.
Derek R. Braden received the B.Eng. degree in electrical and electronic engineering from Liverpool Polytechnic, Liverpool, U.K., in 1990 and M.Sc. degree in microelectronics and information systems from Liverpool John Moores University, Liverpool, U.K., in 2000. His Master’s thesis was entitled “Verification and validation testing of automotive thick film hybrid powertrain components.” He is currently with Delphi Delco Electronics Systems, where he is the Engineering Manager for the validation engineering group. The group is responsible for the development of environmental tests, test hardware, and continuous product monitoring systems used to validate engine management systems prior to the start of regular production. He has previously worked in the field of intelligent monitoring systems, oceanography, and underwater military products. His research interests are in accelerating the validation test process using mixed environmental reliability testing (MERT) and the nondestructive evaluation of chip-scale packaged (CSP) components subjected to environmental tests using novel techniques.
