## Skyless polarimetric calibration and visibility enhancement

Optics Express, Vol. 17, Issue 2, pp. 472-493 (2009)

http://dx.doi.org/10.1364/OE.17.000472

Acrobat PDF (2446 KB)

### Abstract

Outdoor imaging in haze is plagued by poor visibility. A major problem is spatially-varying reduction of contrast by airlight, which is scattered by the haze particles towards the camera. However, images can be compensated for haze, and even yield a depth map of the scene. A key step in such scene recovery is subtraction of the airlight. In particular, this can be achieved by analyzing polarization-filtered images. This analysis requires parameters of the airlight, particularly its degree of polarization (DOP). These parameters were estimated in past studies by measuring pixels in sky areas. However, the sky is often unseen in the field of view. This paper derives several methods for estimating these parameters, when the sky is not in view. The methods are based on minor prior knowledge about a couple of scene points. Moreover, we propose blind estimation of the DOP, based on the image data. This estimation is based on independent component analysis (ICA). The methods were demonstrated in field experiments.

© 2009 Optical Society of America

## 1. Introduction

3. R. C. Henry, S. Mahadev, S. Urquijo, and D. Chitwood “Color perception through atmospheric haze,” J. Opt. Soc. Am. A **17**, 831–835 (2000). [CrossRef]

4. J. S. Jaffe, “Computer modelling and the design of optimal underwater imaging systems,” IEEE J. Oceanic Eng. **15**, 101–111 (1990). [CrossRef]

7. P. C. Y. Chang, J. C. Flitton, K. I. Hopcraft, E. Jakeman, D. L. Jordan, and J. G. Walker, “Improving visibility depth in passive underwater imaging by use of polarization,” Appl. Opt. **42**, 2794–2803 (2003). [CrossRef] [PubMed]

21. K. Tan and J. P. Oakley, “Physics-based approach to color image enhancement in poor visibility conditions,” J. Opt. Soc. Am. A **18**, 2460–2467 (2001). [CrossRef]

23. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt. **42**, 511–525 (2003). [CrossRef] [PubMed]

24. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi,“Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A **19**, 687–694 (2002). [CrossRef]

25. V. Gruev, A. Ortu, N. Lazarus, J. V. der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express **15**, 4994–5007 (2007). [CrossRef] [PubMed]

3. R. C. Henry, S. Mahadev, S. Urquijo, and D. Chitwood “Color perception through atmospheric haze,” J. Opt. Soc. Am. A **17**, 831–835 (2000). [CrossRef]

14. J. S. Tyo, M. P. Rowe, E. N. Pugh Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. **35**, 1855–1870 (1996). [CrossRef] [PubMed]

32. C. F. Bohren and A. B. Fraser, “At what altitude does the horizon cease to be visible?,” American Journal of Physics **54**, 222–227 (1986). [CrossRef]

23. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt. **42**, 511–525 (2003). [CrossRef] [PubMed]

*sky*by the

*horizon*(even automatically [22]). In this paper we refer to that method as

*sky-based*. For example, a hazy scene is shown [36

36.
For clarity of display, the images shown in this paper have undergone the same standard contrast stretch. This operation was done only towards the display. The algorithms described in the paper were run on raw, unstretched data. The data had been acquired using a Nikon D-100 camera, which has a linear radiometric response. The mounted zoom lens used with the camera was set to focal length of ≈ 200*mm*, except for Fig. 9 in which it was ≈ 85*mm*. The camera was pointed at or slightly below the horizon.

*sky-based*dehazing result is shown in Fig. 1(b).

23. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt. **42**, 511–525 (2003). [CrossRef] [PubMed]

*blindly*separates the airlight radiance (the main cause for contrast degradation) from the object’s signal. The parameter that determines this separation is the degree of airlight polarization. It is estimated without any user interaction. The method exploits mathematical tools developed in the field of blind source separation (BSS), also known as independent component analysis (ICA).

37. H. Farid and E. H. Adelson, “Separating reflections from images by use of independent component analysis,” J. Opt. Soc. Amer A **16**, 2136–2145 (1999). [CrossRef]

## 2. Theoretical background

**42**, 511–525 (2003). [CrossRef] [PubMed]

*L*

^{object}the object radiance as if it was taken in a clear atmosphere, without scattering in the line of sight (LOS). Due to atmospheric attenuation [23

**42**, 511–525 (2003). [CrossRef] [PubMed]

*direct transmission*

*z*between the object and the camera, and on the atmospheric attenuation coefficient

*β*, where ∞ >

*β*> 0. The second component is the

*path radiance*(

*airlight*). It originates from the scene illumination (e.g., sunlight), a portion of which is scattered into the LOS by the haze. Let

*a*(

*z*) be the contribution to airlight from scattering at

*z*, accounting for attenuation this component undergoes due to propagation in the medium. The aggregate of

*a*(

*z*) yields the airlight

*A*

_{∞}is the value of airlight at a non-occluded horizon. It depends on the haze and illumination conditions. Contrary to the direct transmission, airlight increases with the distance and dominates the acquired image irradiance

*best state*of the polarizer. Denote this airlight component as

*A*

^{min}. There is another polarizer orientation (perpendicular to the former), for which the airlight contribution is the strongest, and denoted as

*A*

^{max}. The overall airlight given in (Eq. 3) is given by

*D*is equally split among the two polarizer states. Hence, the overall measured intensities at the polarizer orientations mentioned above are

*A*is given in Eq. (3). For narrow FOVs, this parameter does not vary much. In this work we indeed use a narrow FOV, hence assume that

*p*is laterally invariant. Eq. (7) refers to the aggregate airlight, integrated over the LOS. Is

*p*invariant to distance? Implicitly, this would mean that the DOP of

*a*(

*z*) is unaffected by distance. This is not strictly true. Underwater, it has recently been shown [18

18. N. Shashar, S. Sabbah, and T. W. Cronin, “Transmission of linearly polarized light in seawater: implications for polarization signaling,” J. Exper. Biology , **207**, 3619–3628 (2004). [CrossRef]

*z*may decay with

*z*. Such depolarization [43

43. R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. **44**, 2490–2495 (2005). [CrossRef] [PubMed]

*D*. For simplicity, we neglect the consequence of multiple scattering (including blur), as an effective first order approximation, similarly to Refs. [22, 23

**42**, 511–525 (2003). [CrossRef] [PubMed]

*A*from the object’s direct transmission

*D*. The airlight is estimated as

*D*. Subsequently, Eq. (1) is inverted based on an estimate of the transmittance (following Eq. 3)

*p*. Secondly, compensation for attenuation requires the parameter

*A*

_{∞}. Both of these parameters are generally unknown, and thus provide the incentive for this paper. In past studies that used polarization for dehazing, these parameters were estimated based on pixels which correspond to the sky near the horizon.

**42**, 511–525 (2003). [CrossRef] [PubMed]

*two*sets, each having a distinct class of similar objects. Actually, sometimes scenes do not have two such classes. Moreover, to ensure a significant difference between the classes, one should be darker than the other. However, estimating the parameter based on dark objects is prone to error caused by noise. Therefore, in practical tests, we found the theoretical skyless possibility mentioned in Ref. [23

**42**, 511–525 (2003). [CrossRef] [PubMed]

## 3. Skyless parameter calibration and dehazing

*p*and

*A*

_{∞}when the sky is not in view. The method presented in Sec. 3.1 requires the use of just a single class of objects residing at different distances. The consecutive methods assume that the parameter

*p*is known. This parameter can be

*blindly*derived by a method described in Sec. 4. Consequently, there is reduction of the information needed about objects and their distances. The method presented in Sec. 3.2 only requires the relative distance of two areas in the FOV, regardless of their underlying objects. The method described in Sec. 3.3 requires two similar objects situated at different, but not necessarily known distances. Table 1 summarizes the requirements of each of these novel methods.

### 3.1. Distance-based dehazing

*p*and

*A*

_{∞}based on known distances to similar objects in the FOV. An idea to estimate atmospheric parameters by marking selected scene points was suggested in [20]. Motivated by this idea, suppose we can mark two scene points (

*x*),

_{k},y_{k}*k*= 1,2, which, in the absence of scattering, would have a similar (unknown) radiance. For example, these can be two similar buildings which have an unknown radiance

*L*

^{build}. The points, however, should be at different distances from the camera

*z*

_{2}>

*z*

_{1}. For example, the two circles in Fig. 1(a) correspond to two buildings, situated at known distances of 11km and 23km. Using Eqs. (1,2,3,6), the image values corresponding to the object at distance

*z*

_{1}are

*z*

_{2},

*C*

_{2}>

*C*

_{1}. Note that

*C*

_{1}and

*C*

_{2}are known, since

*I*

^{max}

_{k}and

*I*

^{min}

_{k}constitute the acquired data at coordinates in the FOV.

*G*(

*V*) be at

*V*

_{0}. We now show that based on this

*V*

_{0}, it is possible to estimate

*p*and

*A*

_{∞}. Then, we prove the existence and uniqueness of

*V*

_{0}.

*p*and

*A*

_{∞}. Two known distances of similar objects in the FOV are all that is required to extract parameters used for polarization-based dehazing, when the sky is not available.

*V*

_{0}.

*G*|_{V=0}> 0, since*C*_{2}>*C*_{1}.*G*|_{V=1}= 0. This root of*G*is not in the domain.- The function
*G*(*V*) has only one extremum. The reason is that its derivativeis null only when - This extremum is a minimum. It can be shown that
*∂*^{2}*G*/*∂V*^{2}> 0.

*V*

_{0}∈ (0,1). Typical plots of

*G*(

*V*) are shown in Fig. 3. Due to the simplicity of the function

*G*(

*V*), it is very easy to find

*V*

_{0}using standard tools (e.g., Matlab).

*V*

_{0}can be found even when

*z*

_{1}and

*z*

_{2}are only

*relatively*known, i.e., it is possible to estimate the parameters

*A*

_{∞}and

*p*based only on the relative distance

*z*̃ =

*z*

_{2}/

*z*

_{1}, rather than absolute distances. For example, in Fig. 1(a),

*z*̃ = 2.091. Denote

*V*̃

_{0}∈ (0,1). Hence, deriving the parameters is done similarly to Eqs. (25,26,27). Based on

*V*̃,

*A*

_{∞}is estimated as

*p̂*.

*Â*(

*x,y*) and

*t̂*(

*x,y*). Then

*L̂*

^{object}(

*x,y*) is derived using Eq. (13), for the entire FOV. This dehazing method was applied to Scene 1, as shown in Fig. 1(e). There is a minor difference between Figs. 1(b) and 1(e). This is discussed in Sec. 7.

*z*

_{1},

*z*

_{2}or their ratio

*z*̃ can be determined in various ways. One option is to use a map (this can be automatically done using a digital map), assuming the camera location is known. Relative distance can be estimated using the apparent ratio of two similar features that are situated at different distances. Furthermore, the absolute distance can be estimated based on the typical size of known objects.

### 3.2. Distance-based dehazing, with known p

*z*

_{1}<

*z*

_{2}, regardless of the underlying objects. Therefore, having knowledge of two distances of arbitrary areas is sufficient. The approach assumes that the parameter

*p*is known. This knowledge may be obtained by a method we describe in Sec. 4, which is based on ICA. Based on a known

*p*and on Eq. (11),

*Â*is derived for every coordinate in the FOV. As an example, the estimated airlight map

*Â*corresponding to Scene 1 is shown in Fig. 4. The two rectangles represent two regions, situated at distances

*z*

_{1}and

*z*

_{2}. Note that unlike Sec. 3.1, there is no demand for the regions to correspond to similar objects.

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}) having respective distances

*z*

_{1}and

*z*

_{2}, Eq. (34) can be written as

*Â*is known via Eq. (11), since here

*p*is known. Hence,

*α*can be calculated. Since

*z*

_{1}<

*z*

_{2}, then

*α*> 0.

*V*

_{0}∈ (0,1). We now prove the existence and uniqueness of

*V*

_{0}.

*G*|_{p}_{V=0}< 0, since (-2*α*) < 0.*G*|_{p}_{V=1}= 0. This root of*G*is not in the domain._{p}- The function
*G*(_{p}*V*) has only one extremum: its derivative is null only once. - This extremum is a maximum. It can be shown that
*∂*^{2}*G*/_{p}*∂V*^{2}< 0.

*V*

_{0}∈ (0,1). Typical plots of

*G*(

_{p}*V*) are shown in Fig. 5. Based on Eq. (35),

*A*

_{∞}based only on the relative distance

*z*̃ =

*z*

_{2}/

*z*

_{1}, rather than absolute ones. Then,

### 3.3. Feature-based dehazing, with known p

*A*

_{∞}, based on identification of two similar objects in the scene. As in Sec. 3.1, these can be two similar buildings which have an unknown radiance

*L*

^{build}. Contrary to Sec. 3.1, the distances to these objects are not necessarily known. Nevertheless, these distances should be different. As in Sec. 3.2, this method is based on a given estimate of

*p̂*, obtained, say, by the BSS method of Sec. 4. Thus, an estimate of

*Â*(

*x,y*) is at hand.

**42**, 511–525 (2003). [CrossRef] [PubMed]

*I*

^{total}and

*A*. According to Eq. (43),

*Î*

^{total}as a function of

*Â*forms a straight line. Such a line can be determined using two data points. Extrapolating the line, its intercept yields the estimated radiance value

*L̂*

^{build}. Let the slope of the fitted line be

*S*

^{build}. We can now estimate

*A*

_{∞}as

*̂A*

_{∞}and

*p̂*, we can recover

*L̂*

^{object}(

*x,y*) for all pixels, as explained in Sec. 3.1. As an example, the two circles in Fig. 1(a) mark two buildings residing at different distances. The values of these distances are ignored, as if they are unknown. The corresponding dehazing result is shown in Fig. 1(c).

## 4. Blind estimation of *p*

*p*is known. In this section, we develop a method for blindly estimating

*p*. First, note that Eq. (13) can be rewritten as

*I*

^{max}and

*I*

^{min}, since they appear in the denominator, rather than just superimposing in the numerator. However, the image model illustrated in Fig. 2 has a linear aspect: in Eqs. (4,10), the sum of the two acquired images

*I*

^{min},

*I*

^{max}is equivalent to a linear mixture of two components,

*A*and

*D*. This linear interaction makes it easy to use tools that have been developed in the field of ICA for linear separation problems. This section describes our BSS method for hazy images. The result of this BSS yields

*p̂*.

### 4.1. Facilitating linear ICA

*A*(

*x,y*) from

*D*(

*x,y*). ICA relies on independence of

*A*and

*D*. Thus, we describe a transformation that enhances the reliability of this assumption. From Eq. (9), the two acquired images constitute the following equation system:

*p*is unknown, then the mixing matrix

**M**and separation matrix

**W**are unknown. The goal of ICA in this context is: given only the acquired images

*I*

^{max}and

*I*

^{min}, find the separation matrix

**W**that yields “good”

*Â*and

*D̂*. A quality criterion must be defined and optimized. Typically, ICA would seek

*Â*and

*D̂*that are statistically independent (see [44–47

44. S. J. Bell and T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Computation **7**, 1129–1159 (1995). [CrossRef] [PubMed]

*A*and

*D*. However, the airlight

*A*always increases with the distance

*z*, while

*D*tends to fall, in general, with

*z*. Thus, there is a negative correlation between

*A*and

*D*. To observe this, consider the hazy

**Scene 2**, shown in Fig. 6. The negative correlation between

*A*and

*D*, corresponding to this scene is seen in Fig. 7. There are local effects that counter this observation, in places where the inherent object radiance

*L*

^{object}increases with

*z*. Thus, the significant negative correlation mentioned above occurs mainly in the

*lowest*spatial frequency components:

*D*decays with the distance only

*roughly*. On the other hand, in some frequency components we can expect significant independence (Fig. 7).

*D*. Here

*c*denotes the sub-band channel, while 𝓦 denotes the linear transforming operator. Similarly, define the transformed version of

*A*,

*Â*,

*D̂*,

*I*

^{max}and

*I*

^{min}as

*A*,

_{c}*Â*

_{c},

*D̂*

_{c},

*I*

^{max}

_{c}and

*I*

^{min}

_{c}, respectively (see example in Fig. 7). Due to the commutativity of linear operations,

**W**is the same as defined in Eq. (50).

**W**, from which we derive

*p*. Based on

*p*, the airlight is estimated, and can then be separated from

*D*(

*x,y*), as described in Sec. 2.

### 4.2. Scale insensitivity

46. A. Hyvärinen, J. Karhunen, and E. Oja, *Independent Component Analysis*, John Wiley and Sons, New York (2001). [CrossRef]

*scale*: if two signals are independent, then they remain independent even if we change the scale of any of them (or both). Thus, ICA does not reveal the true scale of the independent components. A special case of scale ambiguity is the

*sign*ambiguity, for which the scale is -1. This scale (and sign) ambiguity can be considered both as a problem, and as a helpful feature. The problem is that the estimated signals may be ambiguous. However, in our case, we have a

*physical model*behind the mixture formulation. As we shall see, this model eventually disambiguates the derived estimation. Moreover,

*we benefit*from this scale-insensitivity. As we show in Sec. 4.3, the fact that ICA is insensitive to scale simplifies the intermediate mathematical steps we take [50].

### 4.3. Optimization criterion

*Â*

_{c}and

*D̂*

_{c}can be expressed as (see for example [51

51. T. M. Cover and J. A. Thomas, *Elements of Information Theory*, John Wiley and Sons, New York (1991). [CrossRef]

_{Â̂c}and 𝓗

_{D̂̂c}are the marginal entropies of

*Â̂*

_{c}and

*D̂̂*

_{c}, respectively, while 𝓗

_{Â̂c,D̂̂c}is their joint entropy. However, estimating the joint entropy from samples is an unreliable calculation. Therefore, it is desirable to avoid joint entropy estimation. In the following, we bypass direct estimation of the joint entropy, and in addition we describe other steps that enhance the efficiency of the optimization.

**W**(Eq. 50). Its structure implies that up to a scale

*p*, the estimated airlight

*Â*is a simple difference of the two acquired images. Denote

*Ã*as an estimation for the airlight component

_{c}*Â*, up to this scale

_{c}*D̂*up to a scale

_{c}*p*, where here

*D̂*and

_{c}*Â*is

_{c}*Â*and

_{c}*D̂*means that

_{c}*Ã*and

_{c}*D̃*should minimize their dependency too. We thus minimize the MI of

_{c}*D̃*and

_{c}*Ã*,

_{c}46. A. Hyvärinen, J. Karhunen, and E. Oja, *Independent Component Analysis*, John Wiley and Sons, New York (2001). [CrossRef]

_{Imaxc,Iminc}is the joint entropy of raw frames. As such, its value is a constant set by the raw data, and hence does not depend on

**W**̃. For this reason, we ignore it in the optimization process. Moreover, note from Eq. (54), that

*Ã*does not depend on

_{c}*w*

_{1},

*w*

_{2}. Therefore, 𝓗

_{Ẫc}is constant and can also be ignored in the optimization process. Thus, the optimization problem we solve is simplified to

**W**̃ (Eq. 57) has essentially only one degree of freedom

*p*, since

*p*dictates

*w*

_{1}and

*w*

_{2}. Would it be simpler to optimize directly over

*p*? The answer is no. Such a move implies that

*p*= (

*ŵ*

_{1}+

*ŵ*

_{2})/2. This means that the scale of

*ŵ*

_{1}and

*ŵ*

_{2}is fixed to the true unknown value, and so is the scale of the estimated sources

*D̂*and

*Â*. Hence scale becomes important, depriving us of the ability to divide

**W**̃ by

*p*. Thus, if we wish to optimize the MI over

*p*, we need to explicitly minimize Eq. (53). This is more complex than Eq. (60). Moreover, this requires estimation of 𝓗

_{Âc}, which is unreliable, since the airlight

*A*has very low energy in high-frequency channels

*c*. Thus, minimizing Eq. (60) while enjoying the scale insensitivity is preferable to minimizing Eq. (53) over

*p*.

### 4.4. Back to polarization calibration

*ŵ*

_{1}and

*ŵ*

_{2}. We now use these values to derive an estimate for

*p*. Apparently, from Eq. (56),

*p̂*is simply the average of

*ŵ*

_{1}and

*ŵ*

_{2}. However, ICA yields

*ŵ*

_{1}and

*ŵ*

_{2}up to a global scale factor, which is unknown. Fortunately, the following estimator

*p̂*is derived, it is used for constructing

**W**in Eq. (50). Then, Eq. (49) separates the airlight

*Â*and the direct transmission

*D̂*. This recovery is

*not*performed on the sub-band images. Rather, it is performed on the raw image representation, as in prior sky-based dehazing methods.

*physics-based method*, not a pure signal processing ICA. We use ICA only to find

*p̂*, and this is done in a way (Eq. 61) that is scale invariant.

### 4.5. Efficient optimization using a probability model

*ŵ*

_{1}and

*ŵ*

_{2}, from which

*p̂*is subsequently derived. In this section, we take steps that further simplify the estimation of the cost function (60). This would allow for more efficient optimization.

*sparse*. In other words, almost all the pixels in a sub-band image have values that are very close to zero. Hence, the probability density function (PDF) of a sub-band pixel value is sharply peaked at the origin. A PDF model which is widely used for such images is the generalized Laplacian (see for example [49])

*ρ*∈ (0,2) and

*σ*are parameters of the distribution. Here

*μ*(

*ρ, σ*) is a normalization constant. The scale parameter

*σ*is associated with the standard deviation (STD) of the distribution. However, we do not need this scale parameter. The reason is that ICA recovers each signal up to an arbitrary intensity scale, as mentioned. Thus, optimizing a scale parameter during ICA is meaningless. We can thus set a fixed unit scale (

*σ*= 1) to the PDF in Eq. (62). This means that whatever

*D̃*(

_{c}*x,y*) is, its values are

*implicitly*re-scaled by the optimization process to fit this unit-scale model. Therefore, the generalized Laplacian in our case is

53. P. Bofill and M. Zibulevsky, “Underdetermined blind source separation using sparse representations,” Signal Processing **81**, 2353–2362 (2001). [CrossRef]

54. M. Zibulevsky and B. A. Pearlmutter, “Blind source separation by sparse decomposition in a signal dictionary,” Neural Computation archive **13**, 863–882 (2001). [CrossRef]

51. T. M. Cover and J. A. Thomas, *Elements of Information Theory*, John Wiley and Sons, New York (1991). [CrossRef]

*N*is the number of pixels in the image, while

*ν*(

*ρ*) = log [

*μ*(

*ρ*)]. Note that

**ν**(

*ρ*) does not depend on

*D̃*, and thus is independent of

_{c}*w*

_{1}and

*w*

_{2}. Hence,

*ν*(

*ρ*) can be ignored in the optimization process. The generalized Laplacian model simplifies the optimization problem to

*D̃*(

_{c}*x,y*) in the sub-band images.

*ρ*= 1. We explain in the appendix that this approximation is unimodal and convex. Furthermore, the appendix discusses why this approximation is reasonable. Thus, Eq. (67) is the core of our ICA optimization. For convex problems such as this, convergence speed is enhanced by use of local gradients. See [48] for the differentiation of the absolute value function.

### A note about channel voting

*p*should be independent of the wavelet channel

*c*. However, in practice, the optimization described above yields, for each wavelet channel, a different estimated value

*p̂*. The reason is that some channels better comply with the independence assumption of Sec. 4.1, than other channels. Nevertheless, there is a way to overcome poor channels. Channels that do not obey the assumptions yield a random value for

*p̂*. On the other hand, channels that are “good” yield a consistent estimate. Hence the optimal

*p̂*is determined by voting. Moreover, this voting is constrained to the range

*p̂*∈ [0,1], due to Eq. (8). Any value outside this range is ignored. As an example, the process described in this section was performed on Scene 1. The process yielded a set of

*p̂*values, one for each channel. Fig. 8 plots the voting result as a histogram per color channel. The dominant bar in each histogram determines the selected values of

*p̂*.

## 5. Inhomogeneous distance

*A*from

*D*using ICA, both must be spatially varying. Consider the case of a spatially constant

*A*. This occurs when all the objects in the FOV are at the same distance

*z*from the camera. In this case, 𝓗

_{Ac}and 𝓘(

*A*) are null, no matter what the value of the constant

_{c},D_{c}*A*is. Hence, ICA cannot derive

*p*here. Therefore, to use ICA for calibrating the DOP, the distance

*z*must vary across the FOV. Distance nonuniformity is also necessary in the other methods (not ICA-based), described in Sec. 3, for estimating

*p*and

*A*

_{∞}. We note that scenarios having laterally inhomogeneous

*z*are the most common and interesting ones. In the special cases where

*z*is uniform, dehazing by Eq. (13) is similar to rather standard contrast stretch: subtracting a constant from

*I*

^{total}, followed by global scaling.

## 6. Additional experiments and comparisons

**42**, 511–525 (2003). [CrossRef] [PubMed]

*p*

_{sky}and

*A*

^{sky}

_{∞}, respectively. We have done so also in the experiments shown in this paper. Then, we compare

*p*

^{sky}with the value of

*p̂*obtained by each of the methods and experiments described in this paper, per color channel. This comparison is shown in Table 2. A similar comparison is done with respect to

*Â*

_{∞}in Table 3. Per scene and per color, the values of

*p̂*are quite similar to one another and to

*p*

^{sky}. The absolute errors in the DOP are ≈ 1 – 3%. Since the DOP itself is given in percent, the relative deviations in the DOP are typically ≈ 5%. The relative deviations in

*Â*

_{∞}are ≈ 8%.

*same*underlying radiance

*L*

^{object}. However, this is a rough assumption. It can be expected that buildings at different geographical places built at different years would have a somewhat different reflectance. The deviation from equality of the object reflectance propagates to the numerical estimation of the airlight parameters.

*p*

^{sky}and

*A*

^{sky}

_{∞}are based on an assumption. These sky values correspond to objects at an infinite distance from the camera. This assumes that atmospheric conditions and scattering effects are effectively uniform to

*infinity*. However, along an infinite LOS, atmospheric parameters

*do*change. Eventually, this LOS passes beyond the atmosphere, in outer space, due to the Earth curvature. The direct sky measurement method thus assumes that most effects accumulate within a finite

*effective*distance. Hence, the sky-based measurement, which was used in the prior art [20, 23

**42**, 511–525 (2003). [CrossRef] [PubMed]

*p̂*and

*Â*

_{∞}in any estimator deviate from the true value and from the results obtained by other estimators. However, the results are rather close to each other, given the uncontrolled, outdoor field conditions.

*p̂*and

*Â*

_{∞}affect the visual result of

*L̂*

^{object}, as seen in Figs. 1,6,9,10 and 11. Nevertheless, in each experiment, the result of each dehazing method is remarkably better than the original hazy image

*I*

^{min}, which was taken using the best polarizer state. The contrast and color of distant objects (trees, red roofs) are recovered from their dull bluish data. Hence, the enhancement achieved by dehazing is significant, in any of the calibration methods. Dehazing is thus tolerant to parameter uncertainties that arise from small deviations from the assumptions.

## 7. Discussion

18. N. Shashar, S. Sabbah, and T. W. Cronin, “Transmission of linearly polarized light in seawater: implications for polarization signaling,” J. Exper. Biology , **207**, 3619–3628 (2004). [CrossRef]

_{p}. This was done by taking a mathematical approach (ICA), which has solid foundations. To complete the automation, the estimation of

*A*

_{∞}should be blind. This is an important direction. It is worth pursuing adaptations of this work to other scattering modalities, such as underwater photography [6

6. Y. Y. Schechner and N. Karpel, “Recovery of underwater visibility and structure by polarization analysis,” IEEE J. Oceanic Eng. **30**, 570–587 (2005). [CrossRef]

## A. A convex formulation

*D̃*(

_{c}*x,y*) is a convex function of

*w*

_{1}and

*w*

_{2}, as seen in the linear relation given in Eq. (55). Moreover, the term [- log |

*w*

_{2}+

*w*

_{1}|] in Eq. (66) is a convex function of

*w*

_{1}and

*w*

_{2}, in the domain (

*w*

_{2}+

*w*

_{1}) ∈ 𝓡

^{+}. The optimization search can be limited to this domain. The reason is that following Eq. (56), (

*ŵ*

_{2}+

*ŵ*

_{1}) = 2

*κp*, where

*κ*is an arbitrary scale arising from the ICA scale insensitivity. If

*κ*> 0, then (

*w*

_{2}+

*w*

_{1})∈ 𝓡

^{+}since by definition

*p*≥ 0. If

*κ*< 0, we may simply multiply

*κ*by -1, thanks to this same insensitivity. Hence, the overall cost function (66) is convex, if |

*Dį*|

_{c}^{ρ}is a convex function of

*D̃*.

_{c}*D̃*|

_{c}^{ρ}convex occurs only if

*ρ*≥ 1. Apparently, we should estimate

*ρ*at each iteration of the optimization, by fitting the PDF model (Eq. 62) to the values of

*D̃*(

_{c}*x,y*). Note that this requires estimation of

*σ*as well. Such parameter estimation is computationally complex, however. Therefore, we preferred using an approximation and

*set*the value of

*ρ*, such that convexity is obtained. Note that

*ρ*< 1 for sparse signals, such as typical sub-band images. The PDF representing the sparsest signal that yields a convex function in Eq. (66) corresponds to

*ρ*= 1. Thus we decided to use

*ρ*= 1 (see also [53–55

53. P. Bofill and M. Zibulevsky, “Underdetermined blind source separation using sparse representations,” Signal Processing **81**, 2353–2362 (2001). [CrossRef]

*ρ*= 1 is an

*approximation*. How good is this approximation? To study this, we sampled 5364 different images

*D̃*(

_{c}*x,y*). These images were based on various values of

*p, c*and on different raw frames. Then, the PDF model (Eq. 62) was fitted to the values of each image. The PDF fit yielded an estimate of

*ρ*per image. A histogram of the estimates of

*ρ*over this ensemble is plotted in Fig. 12. Here,

*ρ*= 0.9 ± 0.3. It thus appears that the approximation is reasonable.

## Acknowledgments

## References and links

1. | N. S. Kopeika, |

2. | R. T. Tan, N. Pettersson, and L. Petersson, “Visibility enhancement for roads with foggy or hazy scenes,” In Proc. IEEE Intelligent Vehicles Symposium 19–24 (2007). |

3. | R. C. Henry, S. Mahadev, S. Urquijo, and D. Chitwood “Color perception through atmospheric haze,” J. Opt. Soc. Am. A |

4. | J. S. Jaffe, “Computer modelling and the design of optimal underwater imaging systems,” IEEE J. Oceanic Eng. |

5. | D. M. Kocak, F. R. Dalgleish, F. M. Caimi, and Y. Y. Schechner, “A focus on recent developments and trends in underwater imaging,” MTS Journal |

6. | Y. Y. Schechner and N. Karpel, “Recovery of underwater visibility and structure by polarization analysis,” IEEE J. Oceanic Eng. |

7. | P. C. Y. Chang, J. C. Flitton, K. I. Hopcraft, E. Jakeman, D. L. Jordan, and J. G. Walker, “Improving visibility depth in passive underwater imaging by use of polarization,” Appl. Opt. |

8. | D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,” In Proc. SPIE |

9. | S. G. Demos and R. R. Alfano, “Optical polarization imaging,” Appl. Opt. |

10. | X. Gan, S. P. Schilders, and M. Gu, “Image enhancement through turbid media under a microscope by use of polarization gating method,” J. Opt. Soc. Am. A |

11. | S. Harsdorf, R. Reuter, and S. Tönebön, “Contrast-enhanced optical imaging of submersible targets,” In Proc. SPIE |

12. | M. J. Raković, G. W. Kattawar, M. Mehrübeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. |

13. | S. P. Schilders, X. S. Gan, and M. Gu, “Resolution improvement in microscopic imaging through turbid media based on differential polarization gating,” Appl. Opt. |

14. | J. S. Tyo, M. P. Rowe, E. N. Pugh Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. |

15. | J. S. Tyo, “Enhancement of the point-spread function for imaging in scattering media by use of polarization-difference imaging,” J. Opt. Soc. Am. A |

16. | K. M. Yemelyanov, S. S. Lin, E. N. Pugh Jr., and N. Engheta, “Adaptive algorithms for two-channel polarization sensing under various polarization statistics with nonuniform distributions,” Appl. Opt. |

17. | G. Horváth and D. Varjù, |

18. | N. Shashar, S. Sabbah, and T. W. Cronin, “Transmission of linearly polarized light in seawater: implications for polarization signaling,” J. Exper. Biology , |

19. | R. Wehner, “Polarization vision a uniform sensory capacity?,” J. Exper. Biology |

20. | F. Cozman and E. Kroktov, “Depth from scattering,” In Proc. IEEE CVPR, 801–806 (1997). |

21. | K. Tan and J. P. Oakley, “Physics-based approach to color image enhancement in poor visibility conditions,” J. Opt. Soc. Am. A |

22. | E. Namer and Y. Y. Schechner, “Advanced visibility improvement based on polarization filtered images,” In Proc. SPIE |

23. | Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt. |

24. | D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi,“Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A |

25. | V. Gruev, A. Ortu, N. Lazarus, J. V. der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express |

26. | N. Gupta, L. J. Denes, M. Gottlieb, D. R. Suhre, B. Kaminsky, and P. Metes, “Object detection with a field-portable spectropolarimetric imager,” Appl. Opt. |

27. | C. K. Harnett and H. G. Craighead, “Liquid-crystal micropolarizer array for polarization-difference imaging,” Appl. Opt. |

28. | J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. |

29. | J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. |

30. | J. Wolfe and R. Chipman, “High speed imaging polarimeter,” In Proc. SPIE |

31. | L. B. Wolff, “Polarization camera for computer vision with a beam splitter,”J. Opt. Soc. Am. A |

32. | C. F. Bohren and A. B. Fraser, “At what altitude does the horizon cease to be visible?,” American Journal of Physics |

33. | D. K. Lynch, “Step brightness changes of distant mountain ridges and their perception,” Appl. Opt. |

34. | E. J. McCartney, |

35. | S. K. Nayar and S. G. Narasimhan, “Vision in bad weather,” Proc. IEEE ICCV , 820–827 (1999). |

36. |
For clarity of display, the images shown in this paper have undergone the same standard contrast stretch. This operation was done only towards the display. The algorithms described in the paper were run on raw, unstretched data. The data had been acquired using a Nikon D-100 camera, which has a linear radiometric response. The mounted zoom lens used with the camera was set to focal length of ≈ 200 |

37. | H. Farid and E. H. Adelson, “Separating reflections from images by use of independent component analysis,” J. Opt. Soc. Amer A |

38. | S. Shwartz, M. Zibulevsky, and Y. Y. Schechner, “Fast kernel entropy estimation and optimization,” Signal Processing |

39. | S. Umeyama and G. Godin, “Separation of diffuse and specular components of surface reflection by use of polarization and statistical analysis of images,” IEEE Trans. PAMI |

40. | D. Nuzilland, S. Curila, and M. Curila, “Blind separation in low frequencies using wavelet analysis, application to artificial vision,” In Proc. ICA, 77–82 (2003). |

41. | S. Shwartz, E. Namer, and Y. Y. Schechner, “Blind haze separation,” In Proc. IEEE CVPR |

42. | E. H. Adelson, “Lightness perception and lightness illusions,” in |

43. | R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. |

44. | S. J. Bell and T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Computation |

45. | J.-F. Cardoso, “Blind signal separation: statistical principles,” Proc. IEEE |

46. | A. Hyvärinen, J. Karhunen, and E. Oja, |

47. | D. T. Pham and P. Garrat, “Blind separation of a mixture of independent sources through a quasi-maximum likelihood approach,” IEEE Trans. Signal Processing , |

48. | P. Kisilev, M. Zibulevsky, and Y. Y. Zeevi, “Multiscale framework for blind source separation,” J. Machine Learning Research |

49. | E. P. Simoncelli, “Statistical models for images: Compression, restoration and synthesis,” In Proc. Conf. Sig. Sys. and Computers, 673–678 (1997). |

50. |
An additional ICA ambiguity is |

51. | T. M. Cover and J. A. Thomas, |

52. | T. Treibitz and Y. Y. Schechner, “Instant 3Descatter,” In Proc. IEEE CVPR 1861–1868 (2006). |

53. | P. Bofill and M. Zibulevsky, “Underdetermined blind source separation using sparse representations,” Signal Processing |

54. | M. Zibulevsky and B. A. Pearlmutter, “Blind source separation by sparse decomposition in a signal dictionary,” Neural Computation archive |

55. | Y. Li, A. Cichocki, and S. Amari, “Analysis of sparse representation and blind source separation,” Neural Computation |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(100.3020) Image processing : Image reconstruction-restoration

(150.1488) Machine vision : Calibration

(110.5405) Imaging systems : Polarimetric imaging

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 3, 2008

Revised Manuscript: August 3, 2008

Manuscript Accepted: August 7, 2008

Published: January 7, 2009

**Citation**

Einav Namer, Sarit Shwartz, and Yoav Y. Schechner, "Skyless polarimetric calibration and visibility enhancement," Opt. Express **17**, 472-493 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-2-472

Sort: Year | Journal | Reset

### References

- N. S. Kopeika, A System Engineering Approach to Imaging, SPIE Press, Bellingham (1998).
- R. T. Tan, N. Pettersson and L. Petersson, "Visibility enhancement for roads with foggy or hazy scenes," In Proc. IEEE Intelligent Vehicles Symposium 19-24 (2007).
- R. C. Henry, S. Mahadev, S. Urquijo, and D. Chitwood "Color perception through atmospheric haze," J. Opt. Soc. Am. A 17, 831-835 (2000). [CrossRef]
- J. S. Jaffe, "Computer modelling and the design of optimal underwater imaging systems," IEEE J. Oceanic Eng. 15, 101-111 (1990). [CrossRef]
- D. M. Kocak, F. R. Dalgleish, F. M. Caimi and Y. Y. Schechner, "A focus on recent developments and trends in underwater imaging," MTS Journal 42, 52-67 (2008).
- Y. Y. Schechner and N. Karpel, "Recovery of underwater visibility and structure by polarization analysis," IEEE J. Oceanic Eng. 30, 570-587 (2005). [CrossRef]
- P. C. Y. Chang, J. C. Flitton, K. I. Hopcraft, E. Jakeman, D. L. Jordan, and J. G. Walker, "Improving visibility depth in passive underwater imaging by use of polarization," Appl. Opt. 42, 2794-2803 (2003). [CrossRef] [PubMed]
- D. B. Chenault, J. L. Pezzaniti, "Polarization imaging through scattering media," In Proc. SPIE 4133, 124-133 (2000).
- S. G. Demos and R. R. Alfano, "Optical polarization imaging," Appl. Opt. 36, 150-155 (1997). [CrossRef] [PubMed]
- X. Gan, S. P. Schilders and M. Gu, "Image enhancement through turbid media under a microscope by use of polarization gating method," J. Opt. Soc. Am. A 16, 2177-2184 (1999). [CrossRef]
- S. Harsdorf, R. Reuter, and S. Tönebön, "Contrast-enhanced optical imaging of submersible targets," In Proc. SPIE 3821, 378-383 (1999).
- M. J. Raković, G. W. Kattawar, M. Mehrübeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, "Light backscattering polarization patterns from turbid media: theory and experiment," Appl. Opt. 38, 3399-3408 (1999). [CrossRef]
- S. P. Schilders, X. S. Gan, and M. Gu, "Resolution improvement in microscopic imaging through turbid media based on differential polarization gating," Appl. Opt. 37, 4300-4302 (1998). [CrossRef]
- J. S. Tyo, M. P. Rowe, E. N. PughJr., and N. Engheta, "Target detection in optically scattering media by polarization-difference imaging," Appl. Opt. 35, 1855-1870 (1996). [CrossRef] [PubMed]
- J. S. Tyo, "Enhancement of the point-spread function for imaging in scattering media by use of polarizationdifference imaging," J. Opt. Soc. Am. A 17, 1-10 (2000). [CrossRef]
- K. M. Yemelyanov, S. S. Lin, E. N. Pugh, Jr., and N. Engheta, "Adaptive algorithms for two-channel polarization sensing under various polarization statistics with nonuniform distributions," Appl. Opt. 45, 5504-5520 (2006). [CrossRef] [PubMed]
- G. Horváth and D. Varjú, Polarized Light in Animal Vision, Springer-Verlag, Berlin (2004).
- N. Shashar, S. Sabbah, and T. W. Cronin, "Transmission of linearly polarized light in seawater: implications for polarization signaling," J. Exper. Biology, 207, 3619-3628 (2004). [CrossRef]
- R. Wehner, "Polarization vision a uniform sensory capacity?," J. Exper. Biology 204, 2589-2596 (2001).
- F. Cozman and E. Kroktov, "Depth from scattering," In Proc. IEEE CVPR, 801-806 (1997).
- K. Tan and J. P. Oakley, "Physics-based approach to color image enhancement in poor visibility conditions," J. Opt. Soc. Am. A 18, 2460-2467 (2001). [CrossRef]
- E. Namer and Y. Y. Schechner, "Advanced visibility improvement based on polarization filtered images," In Proc. SPIE 5888, 36-45 (2005).
- Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, "Polarization-based vision through haze," Appl. Opt. 42, 511-525 (2003). [CrossRef] [PubMed]
- D. Miyazaki, M. Saito, Y. Sato and K. Ikeuchi,"Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths," J. Opt. Soc. Am. A 19, 687-694 (2002). [CrossRef]
- V. Gruev, A. Ortu, N. Lazarus, J. V. der Spiegel and N. Engheta, "Fabrication of a dual-tier thin film micropolarization array," Opt. Express 15, 4994-5007 (2007). [CrossRef] [PubMed]
- N. Gupta, L. J. Denes, M. Gottlieb, D. R. Suhre, B. Kaminsky, and P. Metes, "Object detection with a fieldportable spectropolarimetric imager," Appl. Opt. 40, 6626-6632 (2001). [CrossRef]
- C. K. Harnett and H. G. Craighead, "Liquid-crystal micropolarizer array for polarization-difference imaging," Appl. Opt. 41, 1291-1296 (2002). [CrossRef] [PubMed]
- J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, "Review of passive imaging polarimetry for remote sensing applications," Appl. Opt. 45, 5453-5469 (2006). [CrossRef] [PubMed]
- J. S. Tyo and H. Wei, "Optimizing imaging polarimeters constructed with imperfect optics," Appl. Opt. 45, 5497-5503 (2006). [CrossRef] [PubMed]
- J. Wolfe, R. Chipman, "High speed imaging polarimeter," In Proc. SPIE 5158, 24-32 (2003).
- L. B. Wolff, "Polarization camera for computer vision with a beam splitter,"J. Opt. Soc. Am. A 11, 2935-2945 (1994). [CrossRef]
- C. F. Bohren and A. B. Fraser, "At what altitude does the horizon cease to be visible?," American Journal of Physics 54, 222-227 (1986). [CrossRef]
- D. K. Lynch, "Step brightness changes of distant mountain ridges and their perception," Appl. Opt. 30, 3508-3513 (1991). [CrossRef] [PubMed]
- E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles, John Willey & Sons (1975).
- S. K. Nayar and S. G. Narasimhan, "Vision in bad weather," Proc. IEEE ICCV, 820-827 (1999).
- For clarity of display, the images shown in this paper have undergone the same standard contrast stretch. This operation was done only towards the display. The algorithms described in the paper were run on raw, unstretched data. The data had been acquired using a Nikon D-100 camera, which has a linear radiometric response. The mounted zoom lens used with the camera was set to focal length of ¼ 200mm, except for Fig. 9 in which it was ¼ 85mm. The camera was pointed at or slightly below the horizon.
- H. Farid and E. H. Adelson, "Separating reflections from images by use of independent component analysis," J. Opt. Soc. Amer A 16, 2136-2145 (1999). [CrossRef]
- S. Shwartz, M. Zibulevsky, and Y. Y. Schechner, "Fast kernel entropy estimation and optimization," Signal Processing 85, 1045-1058 (2005). [CrossRef]
- S. Umeyama and G. Godin, "Separation of diffuse and specular components of surface reflection by use of polarization and statistical analysis of images," IEEE Trans. PAMI 26, 639-647 (2004). [CrossRef]
- D. Nuzilland, S. Curila, and M. Curila, "Blind separation in low frequencies using wavelet analysis, application to artificial vision," In Proc. ICA, 77-82 (2003).
- S. Shwartz, E. Namer, and Y. Y. Schechner, "Blind haze separation," In Proc. IEEE CVPR 2, 1984-1991 (2006).
- E. H. Adelson, "Lightness perception and lightness illusions," in The New Cognitive Neuroscience, 2nd ed. ch. 24 339-351, MIT Preess, Cambridge (2000).
- R. A. Chipman, "Depolarization index and the average degree of polarization," Appl. Opt. 44, 2490-2495 (2005). [CrossRef] [PubMed]
- S. J. Bell and T. J. Sejnowski, "An information-maximization approach to blind separation and blind deconvolution," Neural Computation 7, 1129-1159 (1995). [CrossRef] [PubMed]
- J.-F. Cardoso, "Blind signal separation: statistical principles," Proc. IEEE 86, 2009-2025 (1998). [CrossRef]
- A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley and Sons, New York (2001). [CrossRef]
- D. T. Pham and P. Garrat, "Blind separation of a mixture of independent sources through a quasi-maximum likelihood approach," IEEE Trans. Signal Processing, 45, 1712-1725 (1997). [CrossRef]
- P. Kisilev, M. Zibulevsky, and Y. Y. Zeevi, "Multiscale framework for blind source separation," J. Machine Learning Research 4, 1339-1364 (2004).
- E. P. Simoncelli, "Statistical models for images: Compression, restoration and synthesis," In Proc. Conf. Sig. Sys. and Computers, 673-678 (1997).
- An additional ICA ambiguity is permutation, which refers to mutual ordering of sources. This ambiguity does not concern us at all. The reason is that our physics-based formulation dictates a special form for the matrix W, and thus its rows are not mutually interchangeable.
- T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley and Sons, New York (1991). [CrossRef]
- T. Treibitz and Y. Y. Schechner, "Instant 3Descatter," In Proc. IEEE CVPR 1861-1868 (2006).
- P. Bofill and M. Zibulevsky, "Underdetermined blind source separation using sparse representations," Signal Processing 81, 2353-2362 (2001). [CrossRef]
- M. Zibulevsky and B. A. Pearlmutter, "Blind source separation by sparse decomposition in a signal dictionary," Neural Computation Archive 13, 863 - 882 (2001). [CrossRef]
- Y. Li, A. Cichocki, and S. Amari, "Analysis of sparse representation and blind source separation," Neural Computation 16, 1193-1234 (2004). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

« Previous Article | Next Article »

OSA is a member of CrossRef.