Acoustic Holographic Recording and Reproduction System Using Meta Material Layers

Abstract

Holographic sound is recorded and reproduced by way of a single monaural recording per left and right ear recorded. This is accomplished by determining the phase shift of frequencies recorded after dividing the sound into discrete frequencies in a recording device having resonators, each resonating at a different frequency, placed in a circular arrangement and divided into discrete channels by non-resonant material. The resonators are placed in a pseudo-randomized arrangement within the recording device and the circle of resonators is in front of a microphone which records the sound monaurally. Playback is then by way of arranging speakers or transducers into micro perforated sheets which amplify the sound, the arrangement of speakers/transducers around a central point. The sound is then played back directionally based on the position where the sound originally was recorded from and the position of the particular transducer around the central point.

Claims

1-10. (canceled)

11. A device for receiving and reproducing sound in three dimensions, comprising: a plurality of resonators, each which resonates at a different frequency, arranged in a substantially circular arrangement; a microphone situated between said plurality of resonators on at least one plane of a three dimension plane; a tangible storage medium storing a digital version of said sound after applying compressed sensing to output from said microphone; a plurality of flat-surfaced piezo-drivers arranged equi-distant from a center point on a sound reproduction device having at least eight equal sides, playing back an uncompressed version of said output from said microphone with output to specific piezo-drivers of said plurality of piezo-drivers based on a determined phase shift of at least some frequencies within said output and an orientation of said specific piezo-driver of said plurality thereof.

12. The device of claim 11, wherein a directionality of sound recorded by said microphone is reproduced by said piezo-drivers which cause vibrations through acoustic meta material layers attached to said sound reproduction device.

13. The device of claim 11, comprising an additional and substantially identical set of said plurality of resonators, said microphone, and said plurality of piezo-drivers such that output from said microphone and an additional said microphone records sound in binaural or stereo and playback by said plurality of piezo-drivers and additional said plurality of piezo-drivers allows for detection of a position of sound picked up by said microphone and said additional said microphone to be aurally determined in three-dimensional space relative to said plurality of piezo-drivers and said additional piezo-drivers.

14. The device of claim 11, wherein said plurality of piezo-drivers are each attached to a single said sound reproduction device which amplifies sound of said piezo-drivers.

15. The device of claim 14, wherein spacing of said piezo-drivers has no correspondence with spacing of said plurality of resonators.

16. The device of claim 11, wherein said plurality of resonators are pseudo-randomly arranged in said substantially circular arrangement.

17. The device of claim 16, wherein said plurality of resonators are bifurcated by a plurality of equi-spaced non-resonant material arranged within said circular arrangement.

18. The device of claim 11, wherein said microphone is monaural and said storage medium stores sound received therefrom in a single channel of data in said compressed version.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0032] FIG. 1 shows a perspective view of a monaural recording device used in embodiments of the disclosed technology.

[0033] FIG. 2 shows a plan view of a monaural recording device used in embodiments of the disclosed technology.

[0034] FIG. 3 shows a high level diagram of a sound acquisition or recording device used in embodiments of the disclosed technology.

[0035] FIG. 4 shows a cross section of the sound acquisition or recording device of FIG. 3 turned 90 degrees relative to FIG. 3.

[0036] FIG. 5 shows a playback device used to exhibit holographic sound recorded using embodiments of the disclosed technology.

[0037] FIG. 6 shows a first schematic representation of a recording device of embodiments of the disclosed technology.

[0038] FIG. 7 shows a second schematic representation of a recording device of embodiments of the disclosed technology.

[0039] FIG. 8 shows a high level flow chart of steps taking to carry out methods of the disclosed technology.

[0040] FIG. 9 shows a high level block diagram of measuring and reconstructing holographic waves.

[0041] FIG. 10 is a high level block diagram showing devices on which embodiments of the disclosed technology may be carried out.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE DISCLOSED TECHNOLOGY

[0042] Holographic sound is recorded and reproduced by way of a single monaural recording per left and right ear recorded. This is accomplished by determining the phase shift of frequencies recorded after dividing the sound into discrete frequencies in a recording device having resonators, each resonating at a different frequency, placed in a circular arrangement and divided into discrete channels by non-resonant material. The resonators are placed in a pseudo-randomized arrangement within the recording device and the circle of resonators is in front of a microphone which records the sound monaurally. To achieve stereo recording, two such circular arrangements of resonators each associated with a single monaural microphone are used. Playback is then by way of arranging speakers or transducers into micro perforated sheets which amplify the sound, the arrangement of speakers/transducers around a central point. The sound is then played back directionally based on the position where the sound originally was recorded from and the position of the particular transducer around the central point.

[0043] To understand embodiments of the disclosed technology, it is necessary to explain the discoveries about the human ear and recording which have been made by the inventor.

[0044] The human ear cochlea can be viewed as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and traveling wave theories. The spectrum-analysis architecture used by the biological cochlea is extremely efficient: analysis time, power and hardware usage all scale linearly with, the number of output frequency bins, versus N log(N) for the Fast Fourier Transform. A graded bank of uncoupled harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and that resonances gives rise to similar frequency responses, group delays, and traveling wave velocities as observed by experiment. A tone produces global, near-simultaneous forcing in a graded bank of coupled resonators and this causes an apparent traveling wave. The band-pass filter mechanism is simulated by a bank of randomized Helmholtz resonators or subwavelength resonators in the present patent.

[0045] Acoustic meta materials (AMM) (defined as a device used to absorb sound and reduce sound intensity comprised of, or consisting of, a thin flat plate less than, or equal to, 2 mm thick, with at least one hole or a series of spaced-apart holes.) combine geometrically placed spacing and resonance effects. For example, a conventional base material is modified so as to have a regular structure containing holes, channels, resonators or scattering sites leading to a material exhibiting negative effective density (.sub.eff) and bulk modulus (K.sub.eff), whereas normally they are both positive, resulting in a negative acoustic refractive index ((.sub.eff<0). A simple AMM resonator is Helmholtz resonator used extensively in various applications. Helmholtz resonator, a well known acoustic resonator, which is small compared to the wavelength while presenting relatively low losses due to viscous damping. The latter consists of a rigid container embedding a volume V, terminated by an open-ended neck of length l and cross section S. An acoustically reflecting surface can be designed to acquire hybrid resonances which becomes impedance-matched to airborne sound at tunable frequencies, such that no reflection is generated. The Helmholtz resonator shown here is used as a baseline conventional element to explain the AMM concept. When compared to a single resonator, a duct with several identical resonators exhibits a unique attenuation characteristic caused by structural periodicity, and may, if carefully designed, provide a much broader noise attenuation bands. This behavior of multiple resonators alludes to Bloch (i.e., Bragg) wave phenomena which is also known as phononic crystal behavior. It may be mentioned that phononic crystals are a sub-class of acoustic meta-materials. A Helmholtz resonator used as acoustic resonator or scatterer can be tuned to a single design frequencies with side band gaps introduced due to Bloch waves.

[0046] Since the resonant frequency of a unit depends only on its inertia (for example, the mass) and the restoring force (for example, that of the spring), the relevant wavelength at the resonant frequency can be orders of magnitude larger than the physical dimension of the resonant unit. This sub-wavelength characteristic is therefore a common feature of all types of meta materials, which also includes sub-wavelength structures with functionalities not found in nature. The meta material properties are independent of the material used and are dependent only on the geometry of the structure and the medium that fills it. Therefore these structures are ideal for the realization of tunable negative bulk modulus. Sub-wavelength control of acoustic waves has not been studied as much as in electromagnetic, but there have been few attempts in order to realize super-resolution imaging based on canalization or hyperlens.

[0047] Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficient acquisition and reconstruction of a signal, by finding solutions to underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Shannon-Nyquist sampling theorem. There are two conditions under which recovery is possible. The first one is sparsity which requires the signal to be sparse in some domain. The second one is incoherence which is applied through the isometric property which is sufficient for sparse signals. Compressed sensing is advantageous whenever signals are sparse in a known basis; measurements (or computation at the sensor end) are expensive; but computations at the receiver end are cheap. Being able to recover images from incomplete data is very important: less time spent on imaging or other sensing technologies, relieves storage requirement, because only incomplete data is needed to recover all that we need and conserves energy. The signal of interest is sampled by taking small number of linear random projections of the signal which contain most of the vital information about the signal. It basically relies on two major assumptions about the signal i.e. sparsity and incoherence. Sparsity depends upon the signal of interest and incoherence depends upon the sensing modality. Sparsity means that the amount of information present in the signal is much less than the total bandwidth acquired by the signal. Most of the natural signals are sparse in nature. On the other hand, incoherence means that, signals that can be represented sparsely should be spread out in the domain in which they are acquired. It is interesting to note that sparsity lives in audio signals, radar, statistical models, PDE solutions and much more.

[0048] Acoustic meta materials are artificially fabricated materials designed to control, direct, and manipulate sound waves. Since the acoustic meta materials are one of the branch of the meta materials, the basic principle of the acoustic meta materials is similar to the principle of meta materials. These meta materials usually gain their properties from structure rather than composition, using the inclusion of small inhomogeneities to enact effective macroscopic behavior. Control of the various forms of sound waves is mostly accomplished through the bulk modulus K.sub.eff mass density .sub.eff, and chirality. Acoustic meta materials can be generally divided into two main areas. Resonant materials usually consist of a matrix material in which is embedded periodic arrangements of inhomogeneities such as resonators, rigid spheres or cylinders with spacing less than a wave-length. The embedded structures cause wave scattering and resonant behavior which creates stop bands and refraction effects. Non-resonant acoustic meta materials are designed to control the propagation of acoustic waves through fluids and materials. Both resonant and non-resonant meta material designs are used in the present patent. An acoustically reflecting surface can acquire hybrid resonances and becomes impedance-matched to airborne sound at tunable frequencies, such that no reflection is generated. Each resonant cell of the meta surface is deep-subwavelength in all its spatial dimensions, with its thickness less than the peak absorption wavelength by two orders of magnitude. As there can be no transmission, the impedance-matched acoustic wave is hence either completely absorbed at one or multiple frequencies, or converted into other form(s) of energy, such as an electrical current.

[0049] Acoustic signals, such as, speech, music, etc. are information rich signals which has become the primary means of communication among humans. Digitizing real world signals helps to achieve more compact representations and provides better utilization of available resources. Also, inverse problems abound in many application areas of signal/image processing: remote sensing, radar imaging, tomographic imaging, microscopic imaging, astronomic imaging, digital photography, etc. Image restoration is one of the earliest and most classical linear inverse problems in imaging, dating back to the 1960s. Signal processing theory states that the rate at which signals must be sampled in order to capture all of the information of that signal is equal to twice the Fourier bandwidth of the signal (Nyquist rate). This sampling method produces a large amount of data with a large amount of redundant information. Traditionally, to recover a signal, enough samples need to be taken to avoid aliasing and reconstruct with good accuracy. The Nyquist-Shannon sampling theorem states that to restore a signal exactly and uniquely, you need to have sampled the signal with at least twice its frequency, Of course, this theorem is still valid; if you skip one byte in a signal or image of white noise, you can't restore the original. But most interesting signals and images are not white noise. When represented in terms of appropriate, basis functions, such as trigonometric functions or wavelets, many signals have relatively few non-zero coefficients. In compressed (or compressive) sensing terminology, they are sparse. Shannon's theorem (also called the Shannon-Nyquist sampling theorem) then says that the resolution of an image is proportional to the number of measurements. If you want to double the resolution, number of pixels needs to be doubled.

[0050] With compressive sensing, the Nyquist criteria can be violated, as long as certain conditions such as signal sparsity and incoherence can be used to compensate for the limited measurements and, thus, still recover the signal accurately. It has been observed that many important signals have this property of sparseness, thus allowing the number of samples required to capture all of the signal's information to be reduced. A signal is called sparse in nature if it has only a few significant (large in magnitude) components and a greater number of insignificant (close to zero) components.

[0051] In compressive sensing or sampling (CS) signal sparseness allows signals to be under sampled without the loss of information. CS is defined as a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to under-determined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Shannon-Nyquist sampling theorem. There are two conditions under which recovery is possible. The first one is sparsity which requires the signal to be sparse in some domain. The second one is incoherence which is applied through the isometric property which is sufficient for sparse signals. In the compressed-sensing view of the world, the achievable resolution is controlled primarily by the information content of the image. A signal is said to be compressible if it is sparse in nature. An image with low information content can be reconstructed perfectly from a small number of measurements. Once the requisite number of measurements have been made, it doesn't help to add more.

[0052] Compressive sensing is a technique in signal processing that allows for accurate reconstruction of sparse signals given a limited number of measurements and an under-determined linear system. However compressive sensing shows that this rule does not need to hold if we know that the signal is sparse, and the system is incoherent, which means that the system should be able to spread out the sparse signal in the measurement. When the system is coherent, or fails to spread out the original signal appropriately, this technique fails. Since sparsity is the main principle behind CS, effective sparse representations of signals play a major role in the success of CS based applications. Compressed sensing (CS) technology, for example, has recently shown that tomographic images can be well retrieved from far less sample data than the Shannon-Nyquist criterion.

[0053] Ground truth involves the collection of measurements and observations about the type, size, condition and any other physical or spectral properties believed to be of importance concerning the acoustic sources and frequency content that are being sensed. CS comprises a collection of methods of representing a signal on the basis of a limited number of measurements and then recovering the signal from these measurements.

[0054] The signal to be acquired may be represented as follows:

s=x,

[0055] where s is signal to be acquired, is sparsifying matrix and x is real valued column vector.

[00002]$\begin{array}{c}y=.Math.{}_s=.Math..Math.x,\\ =.Math.\mathrm{Ax}=A_k.Math.{}_k\end{array}$

where y is compressed samples and is sensing matrix.
The solution to above equation is:

[00003]$X_k=(\begin{array}{c}{A}_k^T.Math.A_k\\ {-1}^{}.Math.A_k^T.Math.y\end{array}$

[0056] Above is an under-determined problem i.e. projection of an n-dimensional vector into an M dimensional space i.e. Number of equations<Number of Unknowns.

[0057] To solve this kind of problems, the concept of Norms is used. Norms assign strictly positive length to vectors in a vector space. Norms are of following types:

[0058] a. L0 Norm: It simply counts the number of non-zero components in a vector

[0059] b. L1 Norm: It is given by the following equation:

[00004]${.Math.x.Math.}_i=\underset{i=1}{\overset{N}{.Math.}}.Math..Math.x_i.Math.$

[0060] L.sub.2 Norm: It is given by following equation:

[00005]${.Math.x.Math.}_2={\left(\underset{i=1}{\overset{N}{.Math.}}.Math.{.Math.x_i.Math.}^2\right)}^{\frac{1}{2}}$

[0061] The resonant behavior of a Helmholtz resonator is due to the oscillation of the mass of the gas column located in the neck, while the cavity behaves as the restoring force of the harmonic oscillator. The resonant frequency of the Helmholtz resonator may therefore be simply written as:

[00006]$f=\frac{c}{2.Math.}.Math.\sqrt{\frac{S}{\mathrm{VI}}}$

[0062] The resonance frequency can occur in the low frequency range where the operating wavelength is much larger than the resonator dimension. By selecting the resonance frequencies properly, it is possible to simultaneously achieve negative density and negative compressibility over a finite range of frequencies. A system of independent resonators can produce traveling wave with zero transfer of energy in the direction of the travel.

[0063] In Helmholtz resonators, the effective bulk modulus, rather than the effective mass density, becomes frequency-dependent. The resonance-induced anomalous effective bulk modulus K.sub.eff can be achieved by a waveguide shunted by a chain of Helmholtz resonators. Helmholtz resonance is characterized by the oscillation of the fluid in the neck section under the restoring force provided by the compression and expansion of the fluid in the cavity. The sample is sub-wavelength in its dimension. Negative bulk modulus, caused by the frequency dispersion of the local resonances, is obtained. The hidden source for a Helmholtz-resonator-based meta material is the extra air volume injected from the resonator cavity. An AMM meta surface with sub-wavelength scale unit cells that is impedance-matched to airborne sound at tunable frequencies is achieved by coupling different resonators and generating a hybrid resonance mode.

[0064] Designing a Sensing Matrix: Following conditions need to be strictly satisfied while designing a sensing matrix so that, the signal is recovered faithfully:

[0065] Universal Incoherence condition: The value of cross correlation between two channels or column vectors of a sensing matrix must be minimum.

[0066] Data Independence: The construction of a random matrix does not depend upon any prior knowledge of data.

[0067] Robustness: Transmission of randomly projected coefficients is robust to packet loss in the network.

[0068] Incoherence condition: The sensing matrix should be as different from the sparsifying matrix. Time and frequency basis are maximally incoherent. Following equation signifies the incoherence condition:

<1/(2K1)

[0069] Embodiments of the disclosed technology will become clearer in view of the following description of the figures.

[0070] Referring first to FIGS. 1-3, these figures show a sound acquisition system using acoustic meta material layers forming resonators and non-acoustic separation barriers separating groups of resonators into discrete channels. The exterior shape of the sound acquisition device can be spherical, conical, or formed from a regular polygon extended in three dimensional space. The distances given (non-bold numbers) in FIGS. 1 and 2 are in millimeters showing the size of the sound recording or acquisition system in a particular embodiment. It should be understood that the size can vary and the sizes given are for exemplary purposes only.

[0071] Referring first to FIGS. 3 and 4, FIG. 3 shows a high level diagram of a sound acquisition or recording device used in embodiments of the disclosed technology. FIG. 4 shows a cross section of the sound acquisition or recording device of FIG. 3 turned 90 degrees relative to FIG. 3. Two such devices 300 each form one monaural recording device, such as for left and right sides. Each comprises a plurality of acoustic resonators 310, each resonator resonating at specific discrete frequencies and surrounding a (monaural and omni-directional, in some embodiments) microphone 390. The left half of the system records sound for the left recording/sound representing a left ear and the right half for the right recording/sound (i.e., right ear). The distance d1 between the centers of two recording devices 300 can be set to binaural or stereo recording requirements. The distance d1 for binaural can be the distance between the ears the intended listened (about 21.5 cm). The distance d1 for stereo recording and playback via two playback devices can be between about 1 meters and 25 meters, by way of example. Each AMM recording system has several channels, defined by the resonators 210 placed between each layer of non-resonant material 315. Each channel comprises resonators with randomized resonances and a recording microphone at the center of the system. After the sound is received by the microphone 390, it is sent to, in some embodiments, a pre-amplification device 410 which amplifies the sound and then a data acquisition system 520 which manipulates the data and stores the data to a tangible storage medium.

[0072] Embodiments of the disclosed technology will become more clear in view of the following discussion of the figures.

[0073] FIG. 1 shows a perspective view of a monaural recording device used in embodiments of the disclosed technology. As described above, in embodiments two such monaural recording devices 300 are used. Each device 300 has a plurality of Helmholtz/acoustic sub-wavelength resonators of varying volumes (relative sizes to each other). Some are larger and some are smaller in order to vary at different frequencies, each. The resonators 310 are placed randomly or pseudo-randomly in position around a center point of the device 300 to provide maximum resonant dispersion. A face plate over the resonators has a specific percent open area (POA), thickness and hole diameter which covers the resonators in each sound acquisition device 300 to provide acoustic impedance matching for incoming acoustic signals. Similar randomized dispersion in resonator channels can also be achieved by varying hole diameter and keeping the volume (size) of resonators constant, as better described when viewing FIG. 2.

[0074] FIG. 2 shows a plan view of a monaural recording device used in embodiments of the disclosed technology. The resonating dispersion of each channel (resonators between non-resonant layers 315 which acoustically separate a group of resonators 310 from one another into the discrete channels) has a plurality of Helmholtz (acoustic) resonators distributed into channels separated by layers 315. The resonances are randomized or pseudo-randomized and contribute to a measurement matrix that supports compressive sensing, as described above. The randomized modulation from all of the channels scrambles the original omni-directional measurement modes of the single sensor which an omni-directional microphone 390 in embodiments of the disclosed technology, placed in the center of a ring of resonators 310 and channels. As a result, the measurement modes are complex in both the spatial and spectral dimensions.

[0075] Skipping now to FIG. 9, FIG. 9 shows a high level block diagram of measuring and reconstructing holographic waves. Assuming the resonances in each channel (the group of resonators 310 between each non-resonant section 315) are distributed sparsely over the frequency range of interest and only first-order filtering responses dominate, the overall frequency modulation of a waveguide can be approximated by the multiplication of the individual responses of the resonators:

T.sub.i()=.sub.jT.sub.ij()

[0076] For a source located at r.sub.k, frequency response can be derived by propagating the waveguide responses from each waveguide aperture .sub.i to the source location .sub.k:

[00007]$G\left(,{\stackrel{.fwdarw.}{r}}_i,{\stackrel{.fwdarw.}{r}}_k\right).Math.T_i\left(\right).Math.R\left(,{\stackrel{.fwdarw.}{r}}_i,{\stackrel{.fwdarw.}{r}}_k\right)$$P_c\left(,{\stackrel{.fwdarw.}{r}}_k,S_0\right)=a\left(\right).Math.S_0\left(\right).Math.\underset{i=1}{\overset{N}{.Math.}}.Math.$

[0077] where S.sub.0() is the spectrum of the audio signal from the source, R(,{right arrow over (r)}.sub.i,{right arrow over (r)}.sub.k) is the AMM channel radiation pattern which is mostly determined by the shape of the channel, aperture, and

[00008]$G(,{\stackrel{.fwdarw.}{r}}_i,{\stackrel{.fwdarw.}{r}}_k=\begin{array}{c}{\stackrel{.fwdarw.}{r}}_i,-{\stackrel{.fwdarw.}{r}}_k\\ {\stackrel{.fwdarw.}{r}}_i,-{\stackrel{.fwdarw.}{r}}_k\\ e^{-\mathrm{jk}}.Math..Math./.Math.\end{array}$

is the Green's function from the location .sub.i of the aperture of the i.sup.th channel to the location .sub.k. The coefficient () includes all other factors such as sensor and speaker responses that are uniform for different source locations and audio signals.

[0078] Each column of the measurement matrix

[00009]$H=\left[\begin{array}{ccc}{h}_{11}.Math.h_{12}& \mathrm{.Math.}& h_{1.Math.N}\\ \mathrm{.Math.}& & \mathrm{.Math.}\\ h_{M.Math..Math.1}.Math.h_{M.Math..Math.2}& \mathrm{.Math.}& h_{\mathrm{MN}}\end{array}\right]$

[0079] represents the discretized Fourier components of source emitting the music signal from one of the speakers on the recording stage. The number of columns of the matrix is N=KP, where K is the possible speaker locations on the stage and P is the size of the audio segments.

[0080] Each row H.sub.mn of the measurement matrix (i.e., H) represents a test function for the object vector at one frequency, because a measurement value in the measurement data vector is sampled in the way defined by the test function as g.sub.m=f,H.sub.m>, where the angle bracket denotes the inner product. The randomization of the measurement matrix for the AMM acoustic sensing system is contributed by the carefully designed AMM channel responses T.sub.i().

[0081] An element in the measurement matrix may be expressed as:

[00010]$G\left({}_m,{\stackrel{.fwdarw.}{r}}_i,{\stackrel{.fwdarw.}{r}}_k\right).Math.T_i\left({}_m\right).Math.R\left({}_m,{\stackrel{.fwdarw.}{r}}_i,{\stackrel{.fwdarw.}{r}}_k\right)$$h_{\mathrm{mn}}=P_c\left({}_m,{\stackrel{.fwdarw.}{r}}_k,S_p\right)=a\left({}_m\right).Math.S_p\left({}_m\right).Math.\underset{i=1}{\overset{q}{.Math.}}.Math.$

[0082] Thus P.sub.c(,{right arrow over (r)}.sub.k,S.sub.p) represents frequency spectra (amplitude and frequency content) of the sources, each at different (.sub.k, location, and can be determined through h.sub.mn.

[0083] A problem now arises of how to effectively recover the original signal from the compressed data, an is solved by the present technology. Basis pursuit (BP) is a popular mathematical optimization problem which is based on constrained 11 norm minimization, and the split Bregman method is an effective technique for solving a variety of L1-regularized optimization problems. Several reconstruction algorithms based on constrained Lp norm minimization with p<1 have also been proposed. Furthermore, a signal reconstruction algorithm based on the optimization of a smoothed approximate L0 norm (SL0) is studied in where simulation results are compared with corresponding results obtained from several existing algorithms. The results favor the use of the approximate L0 norm.

[0084] The Two-Step Iterative Shrinkage Thresholding (TwIST) is an algorithm that provides solutions to inverse linear problems. The TwIST algorithm is known in the art and described, at the time is writing, for example at http://www.1x.it.pt/bioucas/TwIST/TwIST.htm which is quoted here in full:

[0085] Many approaches to linear inverse problems define a solution (e.g., a restored image) as a minimizer of the objective function where y is the observed data, K is the (linear) direct operator, and F(x) is a regularizer. The intuitive meaning of f is simple: minimizing it corresponds to looking for a compromise between the lack of fitness of a candidate estimate x to the observed data, which is measured by yKx2, and its degree of undesirability, given by F(x). The so-called regularization parameter 1 controls the relative weight of the two terms.

[0086] State-of-the-art regularizers are non-quadratic and non-smooth; the total variation and the lp norm are two well known examples of such regularizers with applications in many statistical inference and signal/image processing problems, namely in deconvolution, MRI reconstruction, wavelet-based deconvolution, Basis Pursuit, Least Absolute Shrinkage and Selection Operator (LASSO), and Compressed Sensing.

[0087] Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to the minimization of f, with F(x) a non-quadratic, maybe non-smooth regularizers. It happens that the convergence rate of IST algorithms depends heavily on the linear observation operator, becoming very slow when it is ill-conditioned or ill-posed. Two-step iterative shrinkage/thresholding TwIST algorithms overcome this shortcoming by implementing a nonlinear two-step (also known as second order) iterative version of IST. The resulting algorithms exhibit a much faster convergence rate than IST for ill conditioned and ill-posed problems.

[0088] An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them. TwIST has been used to solve many image restoration and compressed sensing problems. The recent approach for restoration of images is the use of wavelets in a two step process, the TwIST. The two steps, in TwIST are Iterative Shrinkage and Thresholding. TwIST algorithm produces faster convergence compared to conventional IST algorithms even for ill conditioned problems.

[0089] The two-step iterative shrinkage-thresholding (TwIST) algorithm based on a second-order approach is used to improve convergence performance. In an iterative optimization process based on a Fourier space, the TwIST algorithm shows a convergence rate better than that of other first-order methods. When compared to other Iterative Shrinkage/Thresholding (IST) algorithms TwIST is more effective since its convergence is based on both past and present iterations. The iterative shrinkage-thresholding (IST) algorithm is derived from a consideration of the L1-norm of the proximal gradient method and is now a common tool for image recoveryone that is based on the principle of CS. A soft-thresholding filtering algorithm using a pseudo-inverse of a discrete difference transform demonstrates a good image recovery.

[0090] The TwIST algorithm can be used, in step 210, to handle highly ill posed de-noising problems. In an inverse problem, the goal is to estimate an unknown original signal/image x from a (possibly noisy) observation y, produced by an operator K applied to x. For the linear system of ill-conditioned problems

y=Kx

[0091] where, for different values of x the image is observed.

[0092] In TwIST, the approach is to solve the minimization problem:

f(x)=yAx.sup.2+(x),

[0093] where is a constant weight factor, N is the noise vector, y is the measurement, and A is the system matrix, is a regularization function, and is an energy matching coefficient.

[0094] Many approaches to linear inverse problems define a solution (e.g., a restored image) as a minimizer of the objective function

f(x)=yKx.sup.2+(x),

where y is the observed data, K is the (linear) direct operator, and F(x) is a regularizer. The intuitive meaning of f is simple: minimizing it corresponds to looking for a compromise between the lack of fitness of a candidate estimate x to the observed data, which is measured by yKx2, and its degree of undesirability, given by F(x). The so-called regularization parameter 1 controls the relative weight of the two terms. Two-step iterative shrinkage/thresholding TwIST algorithms by implements a nonlinear two-step (also known as second order) iterative version of IST. The resulting algorithms exhibit a much faster convergence rate than IST for ill conditioned and ill-posed problems.

[0095] The TwIST method aims at keeping the good de-noising performance of the IST scheme, while still being able to handle ill-posed problems as efficiently as the IST algorithm. In this method a new class of iterative methods, called TwIST, which have the form of Two-step Iterative Shrinkage/Thresholding (TwIST) algorithms has been used. The update equation depends on the two previous estimates (thus, the term two-step), rather than only on the previous one. This class contains and extends the Iterative Shrinkage/Thresholding (IST) methods.

[0096] The sensing system of the present technology has randomized placement (or pseudo-randomized placement) of Helmholtz resonators with a general sampling model as g=Hf, where g is the vector form of the measured data (measurement vector); f is the object vector to be estimated. The measurement matrix H, which represents the forward model of the sensing system, is formed by stacking rows of linear sampling vectors [also known as test functions] at sequentially indexed frequencies. This matrix is randomized by the physical properties of the meta materials to generate highly uncorrelated information channels for sound wave from different azimuths and ranges. The level of randomization of the matrix determines the supported resolution and the multiplexing capability of the sensing system.

[0097] Now referring to the A multi-speaker system provides acoustic signals being played through several speakers on the stage. A Fourier component of the collected signal can be expressed as the superposition of the responses from all of the waveguides at this frequency:

[00011]$\underset{}{F_i\left(\right)}$$F_c\left(\right)=\underset{i}{\overset{N}{.Math.}}.Math.$

is the response from the i.sup.th AMM channel.

[0098] The measured data vector to be used for reconstruction is

[00012]$g=[\underset{T^{}}{F_c\left({}_1\right).Math.F_c\left({}_2\right).Math..Math.\mathrm{.Math.}.Math..Math.F_c\left({}_M\right)},$

and the object vector f is a scalar vector containing N=KP elements (K is the number of the possible locations and P is the size of the finite audio library). Because of the sparsity of f (only several elements are nonzero, corresponding to the activated sources), the sensing process is an ideal fit for the framework of compressive sensing. L1-norm regularization is performed with the Two-step Iterative Shrinkage/Thresholding (TwIST) algorithm to solve the ill-posed inverse problem.

[0099] In an inverse problem, the goal is to estimate an unknown original signal/image x from a possibly noisy observation y, produced by an operator K applied to x. For the linear system of ill-conditioned problems

y=Kx

[0100] Where, for different values of x the image is observed.

[0101] Now, in step 210 (still referring to FIG. 9), from the TwIST method for linear system, a linear function Ax=B is considered where the matrix A is split to C and R given below (in step 230):

A=CR,

[0102] Taking C=I+Dt and R=IKTK in the above equation (step 240)

A=Dt+KTK,

[0103] The two-step iteration for linear system Ax=B becomes (step 250)

Xt+1=(1a)xt1+()xt1+(1)xt+(xt)A=Dt+KTK,

[0104] The process of TwIST is performed (step 260):

X1=(x0)

Xt+1=(1)xt1+()xt+(xt)

[0105] The different values of and are set as follows (step 270):

=2+1,

=2/(m+1)

[0106] where he value of p is given as (step 280):

=(1{square root over (k)})/(1+{square root over (k)})<1

If convergence (step 290) is proved the iteration is stopped (step 295), otherwise the process is reiterative and steps 260-290 are carried out again.

[0107] The non-resonant acoustic meta material (AMM) impedance system using micro-perforated panels (MPP) periodically arranged within porous layers and air gaps used in embodiments of the disclosed technology layered device are optimized for acoustic impedance in addition to sound absorption. Traditional micro-perforates are tuned to certain frequencies, as done for Helmholtz resonators, whereas in the present technology, AMM devices are tuned over a frequency range of 20-20000 Hz. In embodiments of the disclosed technology, non-resonant acoustic meta material layers which utilizes periodic arrangement of meta material MPP sheets and sound absorptive layers as well as air gaps are used. The thickness and material properties of absorptive layers and design parameters of micro-perforated sheets, such as hole diameter, hole spacing etc., are optimized using the meta material approach [Ref]. The AMM impedance matching is essentially frequency independent and may be tailored by the geometry of the acoustic meta material speaker system.

[0108] Now discussing FIG. 5, FIG. 5 shows a playback device used to exhibit holographic sound recorded using embodiments of the disclosed technology. Relative size is shown in the non-bolded numerals, such a in millimeters. In this high level drawing of a polyhedral speaker 500 (as shown, 10-sided) piezoelectric actuators 510 (a single one is shown, but each section/side of the polyhedron has same in embodiments of the disclosed technology) reproduce an acoustic hologram. Multiple piezoelectric actuators may be bonded to the surfaces of the polyhedron speaker. In another embodiment, multiple conventional loudspeakers can are used to reproduce an acoustic hologram. The speakers must be arranged in three dimensions around a center point, the center point equivalent to the position of the microphone 390 used to record the holographic sound. The reproduction functions by way of exhibiting the sound at the direction (three dimensional vector in space relative to the recording point) from the center point from where the sound emanated. In this manner, the sound is reproduced and exits, in the example of the polyhedral speaker 500, in directions 1 and 2 transverse to the direction of sound propagation from the resonator plates 510. The resonators 510 can be acoustic meta-materials vibrated by, and amplifying the sound from, piezoelectric transducers. One such sound playback device 500 or set of speakers can be used for each recorded holographic sound, such as one for a left ear and one for a right ear.

[0109] FIG. 8 shows a high level flow chart of steps taking to carry out methods of the disclosed technology. In step 110, acoustic resonators are arranged into discrete (individual and separated) channels in a pseudo-random circular arrangement. A microphone, such as a monaural omni-directional microphone placed in front of or at the end of the acoustic resonators of the recording device and centered there-between each of, a plurality of, some of, or most of the circular cross sections of the acoustic resonators of the recording device. Output sound from two or more sources or points in three dimensional space (meaning, having different distances from the microphone in at least one, two, or three distances on an X, Y, and Z coordinate plane) is created in step 120. The output sound from the ambient space around the recording device causes the resonators to resonate. Each is tuned to a specific frequency such as at 1 Hz, 2 Hz, 5 Hz, or 20 Hz difference from another, but pseduo-randomly arranged within the circular arrangement and/or within the discreet channels. This sound is then picked up by the microphone after being reflected off of the resonators in addition to being picked up by the microphone directly without the reflection, in some embodiments. Based on the time difference between the sound hitting the microphone directly and being reflected off of the resonator for a particular frequency, and knowing how far the resonator is from the microphone, one can calculate, as above, the direction of the source of the sound, at least partially. By combining this with the myriad of other frequencies, latency, and position of the resonator for the particular sound, one then calculates, in step 340 the phase shift of the sound waves to enough of a degree to determine the position of each sound relative to the microphone.

[0110] The received sound into the microphone is then compressed and stored in a single channel (monaural sound recording) of output from the circular arrangement (recording device) in step 160. Two recording devices can be used in order to create a stereo recording, and as such, one monaural channel is created from each recording device. The recording devices are then, in this embodiment, duplicative such that there are two circular arrangements with discrete channels and with their own monaural microphone, the output of which are each recording into a single monaural recording (step 160). Together, this forms a stereo recording.

[0111] Steps 170 and 180 are the steps taken for playback. Playback can be in real-time (as soon as allowed by the processors, networked devices, and transmission devices between the recording device and playback device) or at a later time from the recording. To do so, in step 170, piezo-drivers or speakers are arranged equidistant or substantially equidistant from a center point for each channel. Sound is then played back through specific piezo-drivers, in step 180, based on the previously determined phase shift as stored within each or the sole monaural channel.

[0112] Referring back to FIGS. 6 and 7, schematic representations of recording devices of embodiments of the disclosed technology are shown. FIG. 6 shows a first schematic representation of a recording device of embodiments of the disclosed technology. FIG. 7 shows a second schematic representation of a recording device of embodiments of the disclosed technology. The top of the page has lower frequencies while the bottom of the page has higher frequencies. As this is a section/wedge of the circle (see FIG. 1), one is looking at a cross-section of the circle with lower frequencies requiring larger resonators, while higher frequencies require smaller resonators and are typically more towards the outside of the circle. Each insulative layer 315 on the outer side and a face plate 319, having a plurality of holes 317 through which sound passes in to the sub-wavelength resonators on the inner side of the recording device in embodiments of the disclosed technology. Flipped 90 degrees, one can see the depth of the resonators 310 in the smaller figures to the right and below the cross section.

[0113] FIG. 10 shows a high-level block diagram of a device that may be used to carry out the disclosed technology. Device 600 comprises a processor 650 that controls the overall operation of the computer by executing the device's program instructions which define such operation. The device's program instructions may be stored in a storage device 620 (e.g., magnetic disk, database) and loaded into memory 630, when execution of the console's program instructions is desired. Thus, the device's operation will be defined by the device's program instructions stored in memory 630 and/or storage 620, and the console will be controlled by processor 650 executing the console's program instructions. A device 600 also includes one, or a plurality of, input network interfaces for communicating with other devices via a network (e.g., the Internet). The device 600 further includes an electrical input interface. A device 600 also includes one or more output network interfaces 610 for communicating with other devices. Device 600 also includes input/output 640, representing devices which allow for user interaction with a computer (e.g., display, keyboard, mouse, speakers, buttons, etc.). One skilled in the art will recognize that an implementation of an actual device will contain other components as well, and that FIG. 4 is a high level representation of some of the components of such a device, for illustrative purposes. It should also be understood by one skilled in the art that the method and devices depicted in FIGS. 1 through 9 may be implemented on a device such as is shown in FIG. 10.

[0114] Further, it should be understood that all subject matter disclosed herein is directed, and should be read, only on statutory, non-abstract subject matter. All terminology should be read to include only the portions of the definitions which may be claimed. By way of example, computer readable storage medium is understood to be defined as only non-transitory storage media.

[0115] While the disclosed technology has been taught with specific reference to the above embodiments, a person having ordinary skill in the art will recognize that changes can be made in form and detail without departing from the spirit and the scope of the disclosed technology. The described embodiments are to be considered in all respects only as illustrative and not restrictive. All changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope. Combinations of any of the methods and apparatuses described hereinabove are also contemplated and within the scope of the invention.