Several existing strategies for estimating the axial intensity derivative in the transport-of-intensity equation (TIE) from multiple intensity measurements have been unified by the Savitzky-Golay differentiation filter - an equivalent convolution solution for differentiation estimation by least-squares polynomial fitting. The different viewpoint from the digital filter in signal processing not only provides great insight into the behaviors, the shortcomings, and the performance of these existing intensity derivative estimation algorithms, but more important, it also suggests a new way of improving solution strategies by extending the applications of Savitzky-Golay differentiation filter in TIE. Two novel methods for phase retrieval based on TIE are presented - the first by introducing adaptive-degree strategy in spatial domain and the second by selecting optimal spatial frequencies in Fourier domain. Numerical simulations and experiments verify that the second method outperforms the existing methods significantly, showing reliable retrieved phase with both overall contrast and fine phase variations well preserved.
©2013 Optical Society of America
Phase measurement plays a prominent role in many fields of physics, such as optics , electron- and X-ray microscopy [2, 3], diffraction . Many samples such as optical elements, biological soft tissues, and cells are phase objects with little intensity variation for conventional brightfield microscopic imaging. Digital holography, as a paradigm of interferometric techniques is the primary quantitative technique for phase reconstruction with sub-wavelength accuracy . Recently, however, direct phase retrieval from intensity measurements using the Transport-of-intensity equation (TIE) [6, 7] derived from the free space Helmholtz wave equation in the paraxial wave approximation has gained increasing attention. The TIE-based method has the advantages of being non-interferometric , not needing phase unwrapping , and applicable with partially coherent beams [9, 10]. Thus phase reconstruction is now available over wide range of light- and electron-beam imaging systems without significant hardware modification and complicated computation [11, 12].
TIE has its basis in the relationship between phase and the first derivative of intensity along the optical axis. Therefore, an accurate estimate of intensity derivative should be obtained in order to solve the partial differential equation. However, the intensity derivative along the optic axis cannot be directly measured. Conventionally, the TIE approach involves recording two out-of-focus images, which usually being recorded symmetrically over- and under-focused by the same focal step from the in-focus image [13–15]. These are then used to approximate the derivative of the image intensity in the direction of the beam propagation. However, the axial intensity derivative approximated by the finite difference method from two intensity measurements only achieves a maximum of second-order precision with respect to the separation of the adjacent intensity measurements under no noise condition [13, 15]. When noise exists which is natural, this method is rather vulnerable, necessitating large defocus distances in order to increase the signal-to-noise ratio (SNR) at the expense of phase resolution . To overcome this difficulty, intensity measurements in multiple planes have been proposed to minimize the effect of the noise on the retrieved phase , reduce the impact of the higher order axial intensity derivatives [15, 17], or both [17, 18]. In addition, these methods have been extended to the situation in which the intensity is measured in unequally-spaced planes [19, 20]. While these strategies have been shown to work well in many situations, their performance, however, depends heavily on the noise level and the characteristics of the experimental data [18, 20]. With a given set of intensity measurements, making an appropriate choice of a suitable algorithm is difficult. Besides, a comprehensive framework to better understand and, potentially, to improve existing derivative estimation algorithms is still lacking.
The aim of this paper is to show that great insight can be gained into the properties, the shortcomings, and the performance of these existing intensity derivative estimation algorithms by unifying them into the category of Savitzky-Golay filter. Specifically, it is shown that all the above mentioned methods can be viewed as special cases of the Savitzky-Golay differentiation filter. Based on these findings, two novel phase retrieval algorithms that extend the applications of the Savitzky-Golay differentiation filters in TIE are introduced. The adaptive-degree Savitzky-Golay differentiation filter method makes possible the choice of the distinct filter degree in different part of the intensity image. The second algorithm, which we call it optimal frequency selection automatically chooses the best parts of the phases obtained from Savitzky-Golay differentiation filters with various degrees by frequency decomposition. Extensive numerical studies and experiments are carried out to confirm and validate the proposed approaches.
2. Problem formulation
2.1 Transport of intensity equation
With the paraxial approximation, the derivative of intensity in the light propagation direction contains phase information that can be retrieved via TIE :13, 14]Eq. (2) can be approximated to the ideal derivative. However, real measurements yield data with noise and discretization which does not allow to be too small. Thus a compromise is made where is chosen to balance the high-order (or non-linearity) error and the noise effect . Specifically, the optimal is dependent on both the maximum physically significant frequency of the object and the noise level . A priori knowledge of these two aspects is difficult not be known in advance.
2.1 Multiple-plane schemes for derivative estimation
To improve the accuracy of derivative estimation, multiple-plane schemes are proposed by taking the non-linearity error or/and noise effect into account. The high-order finite difference method was first used by Ishizuka and Allman  and more recently generalized by Waller et al. . For symmetric differencing, i.e., given a data set The high-order finite difference method using a Taylor expansion has the form22]19]. To do so, one replaces by and by in Eq. (3). By changing , the defocus distance of different plane can be varied. In the following, we only consider the condition that the measured planes are spaced equally and symmetrically from the center plane.
Without considering the effect of noise, the estimation error of Eq. (3) is of order by using images. However, as with the two-plane method, one is faced with the unavoidable problem of noise and the quantitation error. To reduce the effect of noise, Soto and Acosta  proposed a noise-reduction finite difference formula which with coefficients
The coefficients are derived by minimizing the noise effect (more specifically, the noise reduction ratio (NRR)  defined by ) while ignoring the effect of all high-order terms in the Taylor expansion. The high-order finite difference indicates that the planes close to the center have to assume much larger weight than the ones further away, while the noise-reduction method makes the coefficients to be distributed as evenly as possible. Therefore, these two methods are contradictory in a sense. To balance this contradiction, Bie et al.  presented a higher order finite difference with noise-reduction which blends these two methods together, taking both the higher-order terms and noise effect into account. For the data from planes, the high-order terms are cancelled until mth order (), leaving some degree of freedom for noise suppression. In , the author did not give the explicit formula of these coefficients but showed the problem is well-posed when .
Besides these explicit finite difference methods, Waller et al.  recommended using a least-squares fitting method to estimate the first order derivative. Because the least-squares fitting process weights all data equally, and the high-order polynomial fitting conserves the non-linear component of the original data, so both the higher order and noise effect can be treated simultaneously.
2.2 Main connections with Savitzky-Golay filters
After reviewing those multiple-plane schemes, one issue that arises is that it is not easy to determine which method should be used and is there any interrelation between these methods. Let us start with the Waller’s least-squares fitting method . Given data points , the least-squares polynomial of degree () has the formEq. (8) to be reduced to an expression of the form24] showed that fitting a polynomial to a set of input samples and then evaluating the resulting polynomial at a single point within the approximation interval is equivalent to discrete convolution with a fixed impulse response. The filters, obtained by this method are widely known (especially among chemists) as Savitzky-Golay filters . More generally, to evaluate the sth derivative at point using a polynomial of degree on data points, the convolution weight at point can be calculated as 
We are now ready to establish the correspondence between the different finite difference methods and the Savitzky-Golay differentiation filter (SGDF), i.e. least-squares fitting method. For the case that the center point is the evaluated point, the variable is set to zero. Let for the 1st derivative, the weight becomesEquation (13) can be reduced toEq. (4), we then have:
Observation 1: The high-order finite difference method corresponds to the SGDF with degree .
We now let the degree, Eq. (13) can be reduced to
Observation 2: The noise-reduction finite difference method corresponds to the SGDF with degree 1.
So far, we may instinctively perceive that there should be a connection between Bie’s method with Savitzky-Golay filter, since the two parents it inherited from are both particular cases of SGDF. But we cannot just follow the above routine because there is no explicit formula for Bie’s method. Here we give this observation first and detailed prove will be given in Appendix A.
Observation 3: The higher order finite difference with noise-reduction method corresponds to the SGDF with degree ().
Finally, we have to mention that the unequally spaced multi-plane methods [19, 20] fall in the class of SGDF without exception, since the Savitzky-Golay filter has also been generalized for unequally or non-uniformly spaced data as its offspring [23, 25]. Even the two plane method (Eq. (2)) can be viewed as a special case of SGDF.
2.3 Property of Savitzky-Golay differentiation filters
In last subsection, all of the above derivative estimation strategies used in TIE are unified by the SGDF - an equivalent convolution solution for differentiation of data by a least-squares polynomial fitting. Therefore, it is necessary to examine the property of SGDF first. The SGDF have many good properties : (1) the convolution operation is quite straightforward and much easier to implement than the standard least-squares fitting; (2) the coefficients can be easily obtained using a look-up table generated from the explicit solution, or pre-calculated using existing routines; (3) most importantly, they are the optimal differentiation filters that minimize the NRR, subject to moments preservation constraints .
To quantify the frequency domain properties of SGDF, we plot the frequency response of different degree SGDF with data points = 31 in Fig. 1(a) . The SGDF yields equivalent results for central point differentiation using odd polynomial degree and the next highest even degree, e.g., 1/2, 3/4, etc. The coefficients (impulse response) are anti-symmetric with the central weight always null. Therefore, the SGDF belongs to the type III FIR filter  that has a linear phase. With frequency response decomposition, a SGDF can be regarded as a combination of a low-pass filter and an ideal derivative filter:Fig. 1(b), it can be seen the filters have very flat frequency response in their pass bands with modest attenuation in their stop bands. The frequency response of the SGDF is flatter and closer to the ideal derivative filter at low frequencies as the polynomial degree increases. The increase of the flatness is coincident with a relatively higher cut-off frequency or a wider low-pass range. Conversely, as the degree of fitting polynomial decreases, the effect of low-pass filter becomes more evident.
Another important issue is the accuracy of the derivative estimate using SGDF. For this prior information about the ideal signal is necessary to draw a reliable conclusion . The Gauss–Markov theorem states that the mth degree SGDF is an unbiased estimator of the derivative, and can achieve the Cramer-Rao lower bound if the signal can be perfectly modeled by an m-order polynomial and the observed data are with independent Gaussian white noise . However, in practice, the signal order is unknown and hence using a non-optimal degree will inevitably affect the accuracy and the effect of noise reduction. So the problem now we facing is to choose a polynomial with proper degree which fit a data set best. This problem appears new but is in fact similar to one mentioned in subsection 2.2. Selecting a suitable multi-plane method is similar to determining the degree of the SGDF.
3. Applications of SGDF in phase retrieval by TIE
3.1 Derivative estimation using adaptive-degree SGDF
To find the optimal degree of polynomial, one choice is to apply SGDF using various polynomial degrees to the same data set, and then to choose one or some combination. However, it is difficult to decide which polynomial degree yields the optimal data accuracy. Consider the case when the tested sample is a pure-phase object, then Eq. (1) reduces to a standard 2D Poisson equation [17, 27, 28]:29]:29]
3.2 Phase retrieval by optimal frequency selection
The adaptive-degree SGDF can give more accurate derivative estimates by finding a polynomial that fits the data of each pixel best. However, it does not guarantee a more accurate retrieved phase by applying TIE to this derivative estimate. In fact, the adaptive-degree SGDF usually gives an unsatisfactory phase result with severe patch effects - the result resembles a composite image that puts different pieces of estimated phases using different orders together (see Sections 4 and 5 for details), which is obviously not what we want. Because previously we concentrated our attention on the derivative estimation only, without considering the phase retrieval process using TIE. In this subsection, we present an alternative method which solves the optimal phase retrieval problem in spatial frequency domain.
Considering a thin object located at plane z = 0, illuminated with a monochromatic plane wave, the complex wave field in the focused plane can be written as (with no loss of generality, a single transverse dimension is used for notational simplicity)14, 28, 30]Eq. (22) can be reduced to17, 27, 28]. However, an ideal 1st derivative can never be achieved due to the discrete nature of SGDF. Therefore, the actual phase we obtain using SGDF isEq. (16). Equation (25) suggests that the frequency characteristic of the retrieved phase using SGDF is determined by the frequency response of the low-pass filter that is implicit in the SGDF. Now, reconsidering Fig. 1(b), which indicates the effect of low-pass filter decreases with increasing degree of SGDF, one may think that a higher-degree SGDF gives a more intact phase (with wider range of spatial frequencies can be accurately recovered) than a lower degree one. However, the higher degree SGDF results in larger NRR than the one assumes a smaller order. In another word, smaller order filters always give estimates of the low-frequency phase with higher SNR, but they suffer from the problem of not having enough response for high frequency phase variations. The above discussion reaffirmed that using a single degree of SGDF cannot offer the optimal solution and a tradeoff between phase information at high and low spatial frequencies is necessary.
Rather than choosing the optimal filter degree on a pixel-by-pixel basis, here we solve the optimal phase retrieval problem in the spatial frequency domain. The basic idea is to extract the best frequency components of the phase images obtained from SGDFs with various degrees and then recombine them into a composite phase. Considering the phase information lies within a certain bandwidth of spatial frequencies, such that it can be correctly reconstructed by both a smaller degree SGDF and a higher degree one. The smaller degree filter is preferred for its low NRR, or in another word, higher SNR. The frequency components of higher degree SGDF is only used when the smaller degree filter fails to retrieve the phase information reliably. Thus, a rigorous definition of ‘correct reconstruction’ or ‘reliable retrieval’ is necessary.
Most frequently, a 3dB-point is chosen as a boundary in a filter's frequency response at which the signal through the system begins to be attenuated rather than passing through. In our case, we prefer a stricter boundary point at 0.3dB - amplitude of the filter falls to the 96.6% of the pass band. This means all spatial frequencies of the phase below 0.3dB cutoff frequency of the SGDF with one specific degree can be considered as reliable retrieval. The 0.3dB-points for filters with different for odd polynomial orders are displayed in Fig. 2 (remember the SGDF yields equivalent results for the order 1/2, 3/4, 5/6,…). The points marked with * and connected by a blue line are the measured cutoff frequencies for a fixed value of half data length . The values for short length filters are given more precisely in Table 1 . For large , it is impractical to list all the 0.3dB values due to the limited space here. Fortunately, we found in all cases in Fig. 2, varies almost linearly with when with the slope being dependent inversely on , but the curves for tend to deviate from a straight line as increases. However, when is large, e.g. 50 and 100, the linear region of the curve coincides with the range of usable values of , so a nearly linear relation holds over a wide range of . A reasonably accurate approximation to this behavior for the indicated range of parameters isEq. (26) with 1 in the denominator replaced by 2.5 gives more accurate predictions. While a more complicated functional form based on more measurements could provide more accurate predictions over a wider range of m and n, Eq. (26) should be adequate for most applications.
To decompose and reconstruct the retrieved phase in frequency domain according to the cutoff frequency obtained, a complementary filter bank can be designed with the frequency response , and their amplitudes sum to unity. To achieve a flat response in the pass band and a fast roll off, the can be in the form of high-order Butterworth filters or even ideal ones. For the filters are band-pass and have the pass band from to , where represents the 0.3dB cutoff of mth degree SGDF. For the filter is a low-pass with the cutoff frequency and for the filter is a high-pass with the cutoff frequency . The cutoff frequency of SGDF can be obtained by referring to Table 1, or using Eq. (26) for larger . The phase retrieved by the mth order SGDF is then filtered byFig. 3 . Note usually the final reconstructed phase is not necessarily a composite of all possible orders because of the sampling effect and the limited physically significant frequency of the object. Therefore, we recommend the summation of Eq. (28) begins from , ,… and stops when the mean absolute of is below a given value, or the cutoff frequency with respect to (i.e. ) exceeds the maximal range of the frequency coordinate in the Fourier domain corresponding to . Finally, to reduce the computation load when is large; the filter order can be increased with a step of four rather than two without sacrificing the accuracy too much.
To test the performance of the algorithm, we simulated the propagation of a complex field of a pure phase profile of the letters ‘TIE’ (shown in Fig. 4(a) ) which was defined on a grid with 256×256 pixels with a pixel size of 2μm×2μm. The wavelength is 632.8nm. Fifty one intensity images separated by 10μm were generated with defocused range between and −250μm ~250μm. To simulate noise effect, each defocused image was corrupted by Gaussian noise with standard deviation of 0.002. The Poisson noise case has been also tested on the same data. The results did not differ significantly from the case of Gaussian and hence not discussed further in this work.
The phase retrieval results of different methods are shown in Fig. 4. The metric used to measure the accuracy of phase retrieval is given by the root mean square error (RMSE), which quantify the overall difference between the ideal reference phase and the retrieved phase. The traditional TIE method estimated the derivative using two images separated by 10μm (Figs. 4(b)). The low-frequency artifacts superimposed on the acquired phase significantly obscures the object itself. The 1st-order SGDF (Soto’s noise-reduction finite difference method) showed a much cleaner result with far fewer low-frequency artifacts. However, as is clearly shown in the magnified area, the fine structural details were blurred. It is expected because a lower polynomial degree is more effective in removing noise, but at the expense of distorting small details of the signal too much. Conversely, the 31st order SGDF showed a cloudy image but highlights the high frequency sharp object edges. The results presented in Figs. 4(c) and 4(d) indicate an improper choice of polynomial degree can lead to reduction in phase details if the polynomial degree is too low or sub-optimal removal of low-frequency noise if the polynomial degree chosen is unnecessarily high. To compromise between these opposing trends, we performed a brute-force search and found using a degree between 5 and 11 gave overall visually acceptable results both in resolution and contrast. The lowest RMSE with fixed order SGDF achieved is 0.0092, when the order was 7 (Figs. 4(e)). Similarly, we found that using two images separated by 100μm gave lowest RMSE (0.0115) for the two-plane method (Figs. 4(f)). But by inspecting their error images (shown in Figs. 5(a) and 5(b)), we can see that they still introduced some edge blurring and/or low-frequency artifacts. The adaptive-degree SGDF method gave a result with clean background and fine phase details (Fig. 4(g)). However, as mentioned earlier, the patch-like artifacts deteriorated the result significantly, resulted in a high RMSE. However, the proposed OFS method provided a faithful result with uniform background and sharpest edges (Figs. 4(h)). The residual error was rather low so that we can hardly perceived in Fig. 5(c). These results are verified by the RMSE of each image.
To better explain the low error achieved by the OFS method, we show the RMSE curve versus the filter degree in Fig. 6 . A video sequence (Media 1) is also presented to show the evolution of the phase reconstruction process. It can be seen the phase resolution gradually improved by adding high-frequency information corresponding to the optimal frequency bands in the higher order results. The reconstruction process converged when degree = 31, so the final result was a composite of phases from the 1st to 31st orders. For simplicity, an ideal band-pass filter bank was used to decompose and extract the optimal spatial frequency component of each phase recovered. The Gibbs phenomenon combined with the diffraction ringing effect was clearly visible during the first few steps in reconstruction as ripples extending beyond the object region, but it faded out rapidly as the filter degree increases. The final reconstructed result shows a rather clean phase with fairly high resolution, which proves that our method automatically sieve the best parts of result from different orders without any over-fitting or under-fitting problems.
To test the noise adaptability of the proposed OFS method, we varied the standard deviation of the noise from 0 and 0.005, and the final RMSE curves versus standard deviation of the noise for different methods are shown in Fig. 7 . For each noise level, the simulation was repeated 20 times and then the RMSE values were averaged to reduce data uncertainty. It can be seen when there is no noise, the higher the order of SGDF, the less the RMSE achieved because the derivative non-linearity is the only source of error. In the presence of noise, even though relatively small, the error of high-order SGDF (e.g. m = 23, 31) rise dramatically, indicating they are very sensitive to noise. The lower order SGDFs preform comparatively better, with varying degrees of noise resistibility. When the non-linearity error and the noise effect blend together, a compromise between the two opposing trends is more desirable. As the noise level further increases, the phase error is dominated by noise effect, so the lower degree SGDFs perform better. The 1st degree filter shows a rather good resistibility with an almost flat curve.
The OFS method always performs better than other methods using a fixed order because it chooses the frequency-bands optimally with highest SNR in the results of different degrees. Besides, the curve of OFS is also rather flat despite of a slight increase with the noise variance, showing its great adaptability in noisy situations. From these results, we can safely conclude that the OFS method removes the guesswork in the choice of polynomial degree and significantly outperforms the fixed-order method even with the best selection of polynomial degree.
The experimental test setup is shown in Fig. 8(a) . A He-Ne laser ( = 632.8nm) source is expanded and collimated and then illuminates the object under test. The object is reimaged onto a CCD via a 4f system - two lenses of focal length f = 25 mm separated by the distance 2f, and the distance from the object to the first lens is f. The camera is set on a translation stage in order to modify the defocus distance. The phase object under test is a geometry pattern etched on PMMA substrate, which is also shown in Fig. 8(a). Fifty-one images were captured with an equal separation of 50μm (Media 2). Some data samples from the intensity stack are shown in Fig. 8(b). All these images are recorded by a monochrome CCD imaging device (The Imaging Source DMK 41AU02, 4.65μm pixel size) and digitally processed using MATLAB.
The phases recovered by different methods are shown in Fig. 9 , which are similar with our simulation results. Smaller-order filters effectively suppress noise, at the expense of distorting smaller scale phase features. Higher-order filters preserve smaller scale features well, with the disadvantage of being less effective at suppressing low-frequency noise. By subjective evaluation of the reconstructed phase, it is noticed that the 5th degree SGDF produces best visual effect with regard to the contrast and resolution (Fig. 9(c)). Similarly, the optimum distance for traditional TIE (two-plane method) was found to be 700μm (Fig. 9(d)). However, either the background was too noisy or the phase resolution was sacrificed in their estimated phase (see Figs. 9(b)-(f)). The adaptive-degree SGDF method composited the phase in spatial domain, resulting in an unnatural result with contrast reversal (Figs. 9(g)). Figure 9(h) shows the reconstructed result by OFS, with the evolution process animated in Media 3. The final phase of OFS combined the results of different degrees up to 27, indicating that not only the tiny details were detected but also the overall contrast of the phase was well preserved, despite slight non-uniform distribution in the background (which may be partially caused by the dust particles on the sample and/or the lens).
To better quantify the phase retrieved by the proposed method, we also measured the same sample using a digital holography (DH) microscope based on the Michelson interferometric configuration with a 4x objective (NA = 0.13) . Figure 10 shows the hologram (a) and its spectrum (b). The phase demodulation was done by first filtering one of the first orders (the red circle area) from the hologram spectrum and then reconstructing the hologram in the traditional manner . The phase curvature was physically compensated by introducing the same spherical phase curvature in the reference beam . Figure 10(c) shows the recovered phase. It can be seen that the reconstructed phase surface using the OFS method agrees well with the phase measurements made by DH. To clearly compare the phase profiles, a magnification of the corresponding area from Fig. 9(h) is also shown in Fig. 10(c), in the red dot line rectangle. Figure 10(d) shows a comparison of phase profiles taken across the sample edge showing the phase jump obtained with holography and with our TIE approach identified by blue continuous and red dashed lines, respectively. Both two methods were capable of following the large phase step and fine phase variations (e.g. the vertical line nicks results from the imperfection in fabrication) of the sample quite accurately.
6. Conclusions and discussions
The contribution of this paper is two-fold. First, several existing strategies for estimating the axial intensity derivative in TIE from multiple intensity measurements have been unified by the SGDF, which not only provides different views and insights on known derivative estimation methods but, more important, it also suggests a new way of improving solution strategies from an angle of digital filter design in signal processing. Second, two advanced applications of SGDF in TIE have been proposed. The adaptive-degree SGDF method based on the statistical testing allows the automatic choice of the proper degree of SGDF in different parts of the intensity image. The OFS method, more desirably, decomposes, processes, and then recombines the phases using the SGDF with various degrees in spatial frequency domain. The introduction of OFS provides a powerful tool which, in all of our numerical simulations and experiments, outperforms both optimally chosen and sub-optimally chosen fixed-degree methods that are commonly used.
Although the OFS improves the noise resistibility and the phase resolution of TIE significantly, its performance also shares a common limitation of TIE based method - lower spatial frequencies are more sensitive to noise as compared with the higher spatial frequencies. Under severe noisy conditions, the low frequency noise is difficult to be entirely eliminated, because the inverse Laplacian operation is ill-conditioned at near zero-frequency. Figure 11 show the simulation results when the noise was increased to significant level (standard deviation 0.01). As shown in Fig. 11(a), the underlying intensity signal was submerged in the noise, so that even the 7th fixed degree SGDF can no longer render a visible result (Fig. 11(b)). The OFS reduced the noise significantly, but the residual low-frequency noise remained obvious (Fig. 11(c)). In this situation, proper regularization treatments [33, 34] can help to reduce the low frequency artifacts at the cost of attenuation in slow variations of the underlying true phase signal. By combining the Tikhonov-regularization with OFS, the low frequency noise further reduced, accompanied by some dark ‘halos’ appeared near edges (Fig. 11(d)). Another possible countermeasure is to increase the defocusing range of the intensity stack so that the cut-off frequency of 1st degree SGDF can be narrowed to near zero-frequency. Of course, this point is beyond the scope of the present work and needs more detailed investigation.
Appendix A: Proof of Observation 3
The optimization problem in higher order finite difference with noise-reduction method is to minimize NNR , under constraints that (1) ; (2) ; (3) . In , the authors solve this problem by matrix partition. Here we rewrite the problem as a quadratic programming problem:Eq. (29) using Legendre multiplier:Eq. (7). Therefore,Eq. (8), and extracts the second row from . Thus, the Observation 3 is established.
This project was supported by the Research Fund for the Doctoral Program of Ministry of Education of China (No. 20123219110016) and the Research and Innovation Plan for Graduate Students of Jiangsu Higher Education Institutions, China (No. CXZZ11_0237). The author Chao Zuo gratefully acknowledges the financial support from China Scholarship Council (No. 201206840009).
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