Many computer vision frameworks contain a pooling stage that combines local responses at different spatial locations[Boureau et al.(2010)Boureau, Ponce, and LeCun]. This operation is often a sum- or max-pooling mechanism implemented by a wide range of algorithms, e.gin feature descriptors (such as SIFT [Lowe(2004)] and HOG [Dalal and Triggs(2005)]LeCun et al.(1990)LeCun, Boser, Denker, Henderson, Howard, Hubbard, and Jackel, Scherer et al.(2010)Scherer, Müller, and Behnke]. Choosing the correct pooling operator can make a great difference in the performance of a method [Boureau et al.(2010)Boureau, Ponce, and LeCun]. Current standard pooling mechanisms lack the desired generalisation to find equilibrium between frequently-occurring and rare but informative descriptors [Murray and Perronnin(2014)]. Therefore, many computer vision applications can benefit from a more dynamic pooling solution that takes into account the content of pooled signals.
Such pooling operators are commonly used in modelling the phenomenon of colour constancy (i.ea visual effect that makes the perceived colour of surfaces remain approximately constant under changes in illumination [Foster(2011), Hubel(2000)]) both in biological [Carandini and Heeger(2012)] and computational [Land()] solutions. Despite decades of research, colour constancy still remains as an open question [Foster(2011)], and solutions to this phenomenon are important from a practical point of view, e.gcamera manufacturers need to produce images of objects that appear the same as the actual objects in order to satisfy customers. Motivated by above, in this article we propose a contrast-variant-pooling mechanism and investigate its feasibility in the context of computational colour constancy.
1.1 Computational Models of Colour Constancy
Mathematically, the recovery of spectral reflectance from a scene illuminated by light of unknown spectral irradiance is an ill-posed problem (it has infinite possible solutions). The simplest and most popular solution has been to impose some arbitrary assumptions regarding the scene illuminant or its chromatic content. Broadly speaking colour constancy algorithms can be divided into two categories: (i) low-level driven, which reduce the problem to solving a set of non-linear mathematical equations [Land(), Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij, Gao et al.(2015)Gao, Yang, Li, and Li]
, and (ii) learning-based, which train machine learning techniques on relevant image features[Forsyth(1990), Funt and Xiong(2004), Agarwal et al.(2007)Agarwal, Gribok, and Abidi]. Learning-based approaches may offer the highest performance results, however, they have major setbacks which make them unsuitable in certain conditions: (a) they rely heavily on training data that is not easy to obtain for all possible situations, and (b) they are likely to be slow [Gijsenij and Gevers(2011)] and unsuitable for deployment inside limited hardware. A large portion of low-level driven models can be summarised using a general Minkowski expression [Finlayson and Trezzi(2004), Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij] such as:
where represents the illuminant at colour channel ; is the image’s pixel value at the spatial coordinate ; is the Minkowski norm; and is a multiplicative constant chosen such that the illuminant colour,
, is a unit vector.
Distinct values of the Minkowski norm results into different pooling mechanisms. Setting in Eq. 1 reproduces the well known Grey-World algorithm (i.esum-pooling), in which it is assumed that all colours in the scene average to grey [Buchsbaum(1980)]. Setting replicates the White-Patch algorithm (i.emax-pooling), which assumes the brightest patch in the image corresponds to the scene illuminant [Land()]. In general, it is challenging to automatically tune for every image and dataset. At the same time inaccurate values may corrupt the results noticeably [Gijsenij and Gevers(2011)].
The Minkowski framework can be generalised further by replacing in Eq. 1 with its higher-order derivatives [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij]. These non-linear solutions are analogous to centre-surround mechanisms of visual cortex [Land(1986)], which is typically modelled by a Difference-of-Gaussians (DoG) operators where a narrower, positive Gaussian plays the role of the “centre” and a broader, negative Gaussian plays the role of the “surround” [Enroth-Cugell and Robson(1966), Marr and Hildreth(1980)]. Recently, a few biologically-inspired models of colour constancy grounded on DoG have offered promising results [Gao et al.(2015)Gao, Yang, Li, and Li, Parraga and Akbarinia(2016)]
, however their efficiency largely depends on finding an optimum pooling strategy for higher cortical areas. In short, pooling is a crucial component of many colour constancy models driven by low-level features (or even in deep-learning solutions[Barron(2015), Fourure et al.(2016)Fourure, Emonet, Fromont, Muselet, Trémeau, and Wolf]). In the primate visual systems the size of the receptive field varies according to the local contrast of the light falling on it [Shushruth et al.(2009)Shushruth, Ichida, Levitt, and Angelucci, Angelucci and Shushruth(2013)] presenting a dynamic solution dependent on the region of interest.
The low-level models mentioned above are related to the early stages of visual processing, i.ethe primary visual cortex (area V1), that are likely to be involved in colour constancy. Physiological measures suggest that although receptive fields triple in size from area V1 to area V2 [Wilson and Wilkinson(2014)], their basic circuity with respect to surround modulation is similar, i.ekeeping the same size dependency with contrast properties found in V1 [Shushruth et al.(2009)Shushruth, Ichida, Levitt, and Angelucci]. This is consistent with the large body of physiological and psychophysical literature highlighting the significance of contrast in the visual cortex. In computer vision, contrast-dependent models have also shown encouraging results in various applications such as visual attention [Itti and Koch(2001)], tone mapping [Reinhard et al.(2002)Reinhard, Stark, Shirley, and Ferwerda], and boundary detection [Akbarinia and Parraga(2016), Akbarinia and Parraga(2017)], to name a few. From these we can hypothesise the convenience and explanatory value of various “pooling strategies” such as those proposed by previous colour constancy methods. In the rest of this work we will explore the advantages of replacing the different feed-forward (pooling) configurations of some successful colour constancy models [Land(), Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij, Gao et al.(2015)Gao, Yang, Li, and Li] by that of the visual system (as described by current neurophysiological literature [Shushruth et al.(2009)Shushruth, Ichida, Levitt, and Angelucci, Angelucci and Shushruth(2013)]). Our aim in doing so is dual, on the one hand we want to explore the technological possibilities of creating a more efficient algorithm and on the other hand we would like to test the idea that contrast-variant-pooling might play an important role in colour constancy.
1.2 Summary of the Contributions
In the present article we propose a generic contrast-variant-pooling (CVP) mechanism that can replace standard sum- and max-pooling operators in a wide range of computer vision applications. Figure 1 illustrates the flowchart of CVP, which is based on local contrast and therefore it offers a dynamic solution that adapts the pooling mechanism according to the content of region of interest. We tested the feasibility of CVP in the context of colour constancy by substituting the pooling operation of four algorithms: (a) White-Patch [Land()], (b) first-order Grey-Edge [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij], (c) second-order Grey-Edge [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij], and (d) Double-Opponency [Gao et al.(2015)Gao, Yang, Li, and Li] in three benchmark datasets. Results of our experiments show the quantitative and qualitative benefits of CVP over max-pooling.
2.1 Max-pooling Colour Constancy
One of the earliest computational models of colour constancy (White-Patch) is grounded on the assumption that the brightest pixel in an image corresponds to a bright spot or specular reflection containing all necessary information about the scene illuminant [Land()]. Mathematically this is equivalent to a max-pooling operation over the intensity of all pixels:
where represents the estimated illuminant at each chromatic channel ; is the original image and are the spatial coordinates in the image domain.
One important flaw of this simple approach is that a single bright pixel can misrepresent the whole illuminant. Furthermore, the White-Patch algorithm may fail in the presence of noise or clipped pixels in the image due to the limitations of the max-pooling operator [Funt et al.(1998)Funt, Barnard, and Martin]. One approach to address these issues is to account for a larger set of “white” points by pooling a small percentage of the brightest pixels (e.gthe top ) [Ebner(2007)], an operation referred as top-x-percentage-pooling. In this manner, the pooling mechanism is collectively computed considering a group of pixels rather than a single one. This small variant might be a crucial factor in the estimation of the scene illuminant [Joze et al.(2012)Joze, Drew, Finlayson, and Rey]. A similar mechanism has also been deployed successfully in other applications such as shadow removal [Finlayson et al.(2002)Finlayson, Hordley, and Drew].
In practice, given the chosen x-percentage, the top-x-percentage-pooling can be implemented through a histogram-based clipping mechanism [Finlayson et al.(2002)Finlayson, Hordley, and Drew, Ebner(2007)]. Let be the histogram of the input image , and let represents number of pixels in colour channel with intensity (histogram’s domain). The scene illuminant is computed as:
where is the total number of pixels within the intensity range to . The values of are determined from the chosen x-percentage such that:
where is the total number of pixels in the image and is the chosen percentage for colour channel . Within this formulation it is very cumbersome to define a universal, optimally-fixed percentage of pixels to be pooled [Ebner(2007)] and consequently the free variable requires specific tuning for each image or dataset individually. Naturally, this limitation restricts the usability of the top-x-percentage-pooling operator. In the following sections we show how to automatically compute this percentage, based on the local contrast of each image.
2.2 Pooling Mechanisms in the Visual Cortex
We know from the physiology of the cerebral cortex that neurons in higher cortical areas pool information from lower areas over increasingly larger image regions. Although the exact pooling mechanism is yet to be discovered, “winner-takes-all” and “sparse coding” kurtotical behaviour are common to many groups of neurons all over the visual cortex [Carandini and Heeger(2012), Olshausen et al.(1996)], and it is conceivable that a mechanism analogous to max-pooling might be present within the cortical layers. Indeed such behaviour has been discovered across a population of cells in the cat visual cortex [Lampl et al.(2004)Lampl, Ferster, Poggio, and Riesenhuber]
and the activation level of cells with max-like behaviour was reported to vary depending on the contrast of visual stimuli.
Results reported by [Lampl et al.(2004)Lampl, Ferster, Poggio, and Riesenhuber] suggest an inverse relationship between the contrast of a stimulus and the percentage of the signal pooled. When pooling neurons were exposed to low contrast stimuli, their responses shifted slightly away from pure max-pooling (selecting the highest activation response within a region) towards integrating over a larger number of highly activated neurons. In the language of computer vision, this can be regarded as top-x-percentage-pooling, where x assumes a smaller value in high contrast and a larger value in low contrast. Interestingly, the pooling of those neurons remained always much closer to max-pooling than to the linear integration of all neurons (sum-pooling) [Lampl et al.(2004)Lampl, Ferster, Poggio, and Riesenhuber]. Mathematically, this can be interpreted as having a very small (top-x-percentage-pooling) value.
It does not come as a great surprise that the pooling mechanism in the visual cortex depends on the stimulus’ contrast. There is a large body of physiological studies showing that receptive fields (RF) of neurons are contrast variant (for a comprehensive review refer to [Angelucci and Shushruth(2013)]). Quantitative results suggest that RFs in visual area one (V1) of awake macaques double their size when measured at low contrast [Shushruth et al.(2009)Shushruth, Ichida, Levitt, and Angelucci]. Similar expansions have also been reported for the RFs of neurons in extrastriate areas, such as V2 and V4. This implies that a typical neuron in higher cortical areas that normally pool responses from its preceding areas over about three neighbouring spatial locations [Wilson and Wilkinson(2014)] can access a substantially larger region to pool from in the presence of low contrast stimuli. This is in line with the reported pooling mechanism in the cat visual cortex [Lampl et al.(2004)Lampl, Ferster, Poggio, and Riesenhuber].
2.3 Contrast Variant Pooling
In order to model this contrast-variant-pooling mechanism, we first computed local contrast of the input image
at every pixel location by means of its local standard deviation defined as:
where indexes each colour channel; are the spatial coordinates of a pixel; is an isotropic average kernel with size ; and represents the neighbourhood centred on pixel of radius .
To simulate this inverse relation between stimulus contrast and percentage of signal pooled [Lampl et al.(2004)Lampl, Ferster, Poggio, and Riesenhuber] in the top-x-percentage-pooling operator, we determined the percentage in Eq. 4 as the average of inverse local contrast signals:
where is computed from Eq. 5, and is the spatial image domain. In this fashion, instead of defining a fix percentage of signal (pixels) to be pooled (as in [Finlayson et al.(2002)Finlayson, Hordley, and Drew]), we chose an adaptive percentage according to the contrast of image. In terms of colour constancy, this effectively relates the number of pooled pixels to compute the scene illuminant to the average contrast of an image. We illustrated this mechanism of contrast-variant-pooling in the central panel of Figure 1, where red, green and blue signals correspond to the histogram of each chromatic channel. Pixels on the right side of the dashed lines () are pooled. In this example, contrast is higher for the red signal and therefore a smaller percentage of cells are pooled in the red channel.
Bearing in mind that “contrast” is just a fraction in the range – with characterising an absolutely uniform area and representing points with the highest contrast, e.gedges – we can predict that the percentage will be a very small number for natural images where homogeneous regions are likely to form the majority of the scene. Consequently in our top-x-percentage-pooling operator we always pool a small percentage. This is in agreement with observations of [Lampl et al.(2004)Lampl, Ferster, Poggio, and Riesenhuber] which indicate that such pooling mechanism is always much closer to max-pooling than to sum-pooling.
2.4 Generalisation to Other Colour Constancy Models
There is a number of colour constancy models in the literature which are driven by low-level features that require a pooling mechanism on top of their computed feature maps in order to estimate the scene illuminant. In the Double-Opponency [Gao et al.(2015)Gao, Yang, Li, and Li] algorithm this feature map is computed by convolving a colour-opponent representation of the image with a DoG kernel followed by a max-pooling operation. In the Grey-Edge [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij] model, higher-order derivatives of the image are calculated through its convolution with the first- and second-order derivative of a Gaussian kernel. This is complemented by a Minkowski summation, which oscillates between sum- and max-pooling depending on its norm.
Similar to the White-Patch algorithm, the pooling mechanism of these models can also be replaced with our top-x-percentage-pooling operator, where x is computed according to the local contrast of image as explained above. The only difference is that instead of pooling from an intensity image (as in case of the White-Patch algorithm), Double-Opponency and Grey-Edge pool over their respective feature maps. This means that Eq. 3 receives a feature map instead of an intensity image as input.
3 Experiments and Results
In order to investigate the efficiency of our model, we applied the proposed contrast-variant-pooling (CVP) mechanism to four different colour constancy algorithms whose source code were publicly available: White-Patch [Land()], first-order Grey-Edge [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij], second-order Grey-Edge [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij], and Double-Opponency [Gao et al.(2015)Gao, Yang, Li, and Li]. We simply replaced their max-pooling operator with our proposed pooling mechanism. To evaluate each method we used the recovery angular error defined as:
where represents the dot product of the estimated illuminant and the ground truth ; and stands for the Euclidean norm of a vector. It is worth mentioning that this error measure might not correspond precisely to observers’ preferences [Vazquez-Corral et al.(2009)Vazquez-Corral, Parraga, Baldrich, and Vanrell], however, it is the most commonly used comparative measure in the literature. We also computed the reproduction angular error [Finlayson and Zakizadeh(2014)] in all experiments (due to lack of space these results are not reported here). Readers are encouraged to check the accompanying supplementary materials.
We conducted experiments on three benchmark datasets111All source code and materials are available in the supplementary submission., (i) SFU Lab (321 images) [Barnard et al.(2002)Barnard, Martin, Funt, and Coath], (ii) Colour Checker (568 images) [Shi and Funt()], and (iii) Grey Ball (11,346 images) [Ciurea and Funt(2003)]. In Table 1 we have reported the best median and trimean angular errors for each of the considered methods (these metrics were proposed by [Hordley and Finlayson(2006)] and [Gijsenij et al.(2009)Gijsenij, Gevers, and Lucassen] respectively to evaluate colour constancy algorithms since they are robust to outliers). Mean angular errors are reported in the supplementary materials.
|SFU Lab [Barnard et al.(2002)Barnard, Martin, Funt, and Coath]||Colour Checker [Shi and Funt()]||Grey Ball [Ciurea and Funt(2003)]|
|Grey-Edge 1st-order||[Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij]||3.52||4.02||3.72||4.76||5.01||5.80|
|Grey-Edge 2nd-order||[Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij]||3.22||3.65||4.27||5.19||5.72||6.39|
|Double-Opponency||[Gao et al.(2015)Gao, Yang, Li, and Li]||2.37||3.27||3.46||4.38||4.62||5.28|
|CVP Grey-Edge 1st-order||3.29||3.72||2.48||2.79||4.70||5.17|
|CVP Grey-Edge 2nd-order||2.85||3.13||2.59||2.93||4.65||5.05|
Figure 2 illustrates three exemplary results obtained by the proposed contrast-variant-pooling (CVP) operator for two colour constancy models: White-Patch and the first-order Grey-Edge. Qualitatively, we can observe that CVP does a better job than max-pooling in estimating the scene illuminant. This is also confirmed quantitatively for the angular errors, shown at the right bottom side of each computed output.
|Original||Ground Truth||WP||CVP WP||GE1||CVP GE1|
3.1 Influence of the Free Parameters
For each free variable of the tested models we compared the performance of max- to contrast-variant-pooling. White-Patch does not have any free variable, therefore it is exempted from this analysis. In Figure 3 we have reported the impact of different s (receptive filed size) on Double-Opponency algorithm for the best and the worst results obtained by free variable in each dataset (results for all s are available in the supplementary material). We can observe that almost in all cases contrast-variant-pooling outperforms max-pooling. The improvement is more tangible for the Colour Checker and Grey Ball datasets and in low s.
|SFU Lab [Barnard et al.(2002)Barnard, Martin, Funt, and Coath]||Colour Checker [Shi and Funt()]||Grey Ball [Ciurea and Funt(2003)]|
Figure 4 illustrates the impact of different s (Gaussian size) on the first- and second-order Grey-Edge algorithm. We can observe similar patterns as with Double-Opponency (contrast-variant-pooling outperforms max-pooling practically in all cases). This improvement is more significant for low s, for the Colour Checker dataset and for the second-order derivative. It must be noted that the objective of this article was merely to study the performance of max-pooling and CVP on top of the Grey-Edge algorithm. However, the angular errors of our CVP Grey-Edge happen to be in par with the best results reported for Grey-Edge (obtained by using the optimum Minkowski norm for each dataset [Van De Weijer et al.(2007)Van De Weijer, Gevers, and Gijsenij]), with the important caveat that CVP has no extra variables to be tuned, whereas in the Minkowski norm optimisation the value of must be hand-picked for each dataset.
|SFU Lab [Barnard et al.(2002)Barnard, Martin, Funt, and Coath]||Colour Checker [Shi and Funt()]||Grey Ball [Ciurea and Funt(2003)]|
From Figures 3 and 4 we can observe that the greatest improvement occurs in the Colour Checker dataset. We speculate that one of the reasons for this is the larger range of intensity values in the Colour Checker dataset (16-bit) in comparison to the other two datasets that contain 8-bit images, therefore, an inaccurate max-pooling is more severely penalised.
We would like to emphasise that the objective of this article is not to improve state-of-the-art in colour constancy, but to show that contrast-variant-pooling (CVP) almost always produces improvements over max-pooling. Surprisingly, the results we obtained are even competitive with the state-of-the-art. For instance, in the SFU Lab dataset, the lowest reported angular error is 2.1 (obtained by an Intersection-based Gamut algorithm [Barnard(2000)]). This means that our CVP Double-Opponency with an angular error of 2.0 outperforms the state-of-the-art in this dataset. In the Colour Checker and Grey Ball datasets there are a few learning-based models (e.gExemplar-based method [Joze and Drew(2014)]) that obtain lower angular errors in comparison to CVP Double-Opponency, nevertheless our results are comparable with theirs.
Physiological evidence besides, the better performance of CVP can be explained intuitively by the fact that max-pooling relies merely on the peak of a function (or a region of interest), whereas in our model, pooling is defined collectively based on a number of elements near the maximum. Consequently, those peaks that are outliers and likely caused by noise get normalised by other pooled elements. The rationale within our model is to pool a larger percentage at low contrast since in those conditions, peaks are not informative on their own, whereas at high contrast peaks are likely to be more informative and other irrelevant details must be removed (therefore a smaller percentage is pooled).
Although, the importance of choosing an appropriate pooling type has been demonstrated both experimentally [Jarrett et al.(2009)Jarrett, Kavukcuoglu, LeCun, et al., Yang et al.(2009)Yang, Yu, Gong, and Huang], and theoretically [Boureau et al.(2010)Boureau, Ponce, and LeCun], current standard pooling mechanisms lack the desired generalisation [Murray and Perronnin(2014)]. We believe that contrast-variant-pooling can offer a more dynamic and general solution. In this article, we evaluated CVP on the colour constancy phenomenon as a proof-of-concept, however our formulation of CVP is generic (and based on local contrast) and in principle it can be applied to a wider range of computer vision algorithms, such as deep-learning, where pooling is a decisive factor [Scherer et al.(2010)Scherer, Müller, and Behnke].
In our implementation of CVP, we approximated local contrast through local standard deviation (see Eq. 5). There are at least two factors that require a more profound analysis: (i) incorporating more sophisticated models of contrast perception [Haun and Peli(2013)] accounting for extrema in human contrast sensitivity; and (ii) analysing the role of kernel size in the computation of local contrast.
In this article, we presented a novel biologically-inspired contrast-variant-pooling (CVP) mechanism grounded in the physiology of the visual cortex. Our main contribution can be summarised as linking the percentage of pooled signal to the local contrast of the stimuli, i.epooling a larger percentage at low contrast and a smaller percentage at high contrast. Our CVP operator remains always closer to max-pooling rather than to sum-pooling since natural images generally contain more homogeneous areas than abrupt discontinuities.
We tested the efficiency of our CVP model in the context of colour constancy by replacing the max-pooling operator of four algorithms with the proposed pooling. After that, we conducted experiments in three benchmark datasets. Our results show that contrast-variant-pooling outperforms the commonly used max-pooling operator nearly in all cases. This can be explained by the fact that our model allows for more informative peaks to be pooled while suppressing less informative peaks and outliers.
We further argued that the proposed CVP is a generic operator, thus its application can be extended to a wider range of computer vision algorithms by offering a dynamic (and automatic) framework that is based on the local contrast of an image or a pixel. This opens a multitude of possibilities for future lines of research and it remains an open question whether our model can reproduce its excellent results in other domains as well. Therefore, it certainly is interesting to investigate whether our CVP can improve convolutional neural networks.
This work was funded by the Spanish Secretary of Research and Innovation (TIN2013-41751-P and TIN2013-49982-EXP).
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