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Land’s experiments show that it is possible to recover surface reflectance information from image brightness measurements [50][51]. In the one-dimensional case, a scheme which is called the retinex works well for certain simple scenes [37]. Now let us consider a condition in which the surface orientation does not play a critical role. In that case, scene radiance may be assumed (approximately) to be proportional to the product of the illumination on the object and the reflectance of the surface as

I(x, y) ∝ R(x, y).L(x, y) (2.1)

where R(x, y) and L(x, y) are the reflectance and illumination at point (x, y) of an image I(x, y), respectively. The challenge presented here is how to efficiently and simply extract or estimate R(x, y), the illumination invariant part of an incoming im- age. Fig. 2.1 (a) shows a sample image without a considerable illumination variation. The same subject has been affected by nonuniform lighting from left and right, as depicted in Fig. 2.1 (b) and (c), respectively.

Physiological and biological researches show that human visual system and the response of retina cells to illumination variation is a nonlinear function that can be represented and approximated by the logarithm of the intensity of pixels in an image.

Therefore, the following approximate equality, a common assumption, has been used in several papers, e.g., in [66], based on the relation in (2.1)

I_log0 (x, y) = log(I(x, y)) = log(R) + log(L) = R0_log(x, y) + L0_log(x, y) (2.2)

where I_log0 , R0_log and L0_log denote the logarithm of the image, reflectance and illumi- nation, respectively. As the illumination mostly lies in the low-frequency part of the image I(x, y), a lowpass filter can be used to filter the logarithm image I_log0 (x, y) to extract the illumination variant part (L0_log(x, y)) which is then subtracted from I_log0 (x, y) to obtain R0_log(x, y), resulting in an approximation of R. To the best of our knowledge, there is no analytic evidence to realize how such an approximation may be close to the ideal case. In [110], the authors have addressed a key issue stat- ing that “the logarithm of the luminance is a crude approximation to the perceived brightness, hence logarithm transform can partly reduce the effect of lighting... it is worth pointing out that solving Eq. (2.2) is also an ill-posed problem, absolutely separating key facial structure R and L is very difficult even if under the common as- sumption”. This statement motivates the idea that filtering the frequency subbands, to preserve or to eliminate low- or high-frequency information in an image, should not be confused with the simple separation of R and L as in (2.2). Ignoring the common assumption in (2.2), we have assumed a box consisting of a combination of illumina- tion and reflectance. Subtraction of illumination from this box in our research and in this chapter is different from algebraic subtraction in (2.2).

Our experiments with human face images under a wide range of illumination variation show that an expression that may better fit the illumination variation has an offset from being the exact log function. Although one may find a transformation to artfully represent the phenomenon, we propose the use of a normalized version of the log function controlled by two parameters. That is

I_pro0 (x, y) = ξ log(I(x, y))

where in this notation pro stands for the proposed, and parameters  and ξ can offer a fine-tuning opportunity depending on an application. In [110], a parameter λ is introduced to find a threshold (T) values. Empirically, the effective range of λ in [110] was reported to be from 0.01 to 0.30. Likewise, and in [14], higher recognition rates can be obtained for a wide range of a parameter (λ) from 0.9 to 1.2. Cao et al. proposed the use of two parameters λ1 and λ2 to be applied to training and testing images independently, where 0.9 ≤ λ1 ≤ 1.2 and λ2 > 2. It has been illustrated that the values of parameters to reach the highest recognition rates are λ1 = 0.95 and λ2 > 2 [14].

In Section 2.5, it is shown that the practical range for  and ξ to achieve higher accuracy is 0 <  ≤ 0.1 and 0.6 ≤ ξ ≤ 1. It should be noted that the parameter selection in [110][14] and this chapter is not an automatic process. Depending on the structure of each method, there is an optimum range for parameters which is not a wide range to be a bottleneck. Our early experiments show that local stochastic distribution analysis to evaluate the level of illumination on any input image may lead to a narrower range, however, it computationally expensive to find an optimum value of parameters for each input image independently.

At this point, the main question is how to efficiently extract the illumination in- variant part, R0_pro, of a given image if the proposed approximation is used. In fact, to separate L0_pro from I_pro0 , an efficient frequency information discriminator is required, that is, multiresolution analysis can be a reasonable solution. As stated in the in- troduction, multiresolution analysis or multiscale approximation, recalls the theory, design and application of the transformations such as DWT, multiwavelets, DT-CWT, CVT, HD-DWT, and DD-DTCWT. First proposed by Kingsbury [65], the dual-tree complex wavelet transform (DT-CWT) is a recent enhancement to the DWT, with two important additional properties, that is, the transformation is shift-invariant and directionally-selective in two and higher dimensions which have addressed the directionality problem in DWT. The double-density discrete wavelet transform (DD-

DWT) [77] is another improvement upon the critically sampled [21] DWT, whereas the DD-DWT outperforms the standard DWT in several applications such as de- noising. The transformation can be significantly improved and upgraded in terms of directionality and shift-invariance. In other words, although DD-DWT takes the ad- vantage of more wavelets, it is not entirely directionally-selective. A solution to this problem is provided by Selesnick, introducing the concept of double-density dual-tree complex wavelet transform (DD-DTCWT) which combines the characteristics of the DD-DWT and DT-CWT.

In the next section, the double-density dual-tree complex wavelet transform is briefly introduced. We then show how the directionally- and frequency-selective sub- bands of the transformation are used to filter and extract the illumination invariant part of an image.