1. Introduction

Contouring, segmentation and recognition of objects rely on a precise edge detection or edge enhancement. Edges evidence the structure and shape of objects as well as fine details of an image, i.e., they correspond to the high spatial frequencies of the Fourier spectrum of the image. Therefore, edge enhancement increases the discrimination capability of the segmentation and recognition systems [1–3].

Edges in gray scale images are defined in an achromatic way as discontinuities in brightness function. Hence, in this kind of images, edge detection is accomplished by means of searching rapid changes in intensity values. Besides considering an extension of this procedure to color images, color edge detection also involves finding discontinuities along the adjacent regions of a color image in a certain 3D color space, e.g. RGB- or HSI-color space, that would capture different color characteristics [3,4]. Both changes in brightness and color between neighboring pixels should be exploited for more efficient color-edge extraction. In fact, Novak and Shafer [5] found that 10% of the edges detected in color images fall into this non-intensity class.

Color edge detection methods can be classified as synthetic and vector ones [6]. The first class typically includes earlier approaches to color edge detection, which basically consist in extensions of gray value algorithms that act on each RGB color channel independently and are combined later following certain logic operation [7,8]. Vector methods on the other hand, take into account three-dimensional structure of the information of each pixel of an image. Among these, statistical methods based on differences in local vector statistics have been widely applied [9–11]. Precision in assessing edge detection comes at the cost of more computation time, which makes these methods difficult to implement in real-time applications [6].

Parallel processing of large images, in real-time, is the main advantage of optical methods compared to digital processing. We have recently proposed two approaches involving liquid crystal displays (LCD) for directional edge enhancement/detection in gray level images [12,13]. The proposed techniques involve several optical components that require relatively careful alignment. In the present work we propose to accomplish directional edge enhancement (i.e., enhancement of first partial derivatives) of color images by means of a simple setup involving only two elements, a LCD and a calcite prism. In the next section we describe our proposal, and in Sect. 3 experimental results are presented.

2. Description of the method

The proposed setup to obtain first partial derivatives of color images is shown in Fig. 1 . It consists of a commercial LCD (e.g., a standard PC monitor) with its own incoherent white light source, an imaging lens (L), a properly oriented piece of calcite crystal (C) placed in front of the lens, an analyzer (A), and a digital color camera (not shown in Fig. 1) to acquire the optically processed images. Our monitor is a twisted-nematic LCD manufactured to work between a crossed polarizer-analyzer pair. For our purpose, we have removed the analyzer glued in front of LCD.

 

Fig. 1 Proposed setup. For illustrative purposes, narrow light beams with (intentionally) exaggerated lateral displacements are shown. But actually, the beams cover the prism aperture and the slightly displaced images $I_{+}$ and $I_{-}$ will overlap.

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In the absence of the calcite crystal, the usual crossed polarizer-analyzer configuration gives us what we will call “positive image” with intensity $I_{+}(x,y)$. For a color image one can write$I_{+}\left(x,y\right)=\left(I_{+}^{\text{R}}\left(x,y\right),I_{+}^{\text{G}}\left(x,y\right),I_{+}^{\text{B}}\left(x,y\right)\right)$, where the superscripts R, G, B stand for the red, green and blue components of light intensity at $(x,y)$. On the other hand, the parallel polarizer-analyzer configuration gives the “negative image” $I_{-}(x,y)$ with RGB information complementary to that of $I_{+}(x,y)$, in other words a complementary-color image. Let $I_0=\left(1,1,1\right)$ be the intensity of the LCD light source. Then, neglecting absorption in the liquid-crystal, it can be easily shown that [12,13]

The calcite crystal (C) is oriented with its optical axis lying on the incidence plane (i.e., the plane of Fig. 1). As it is well-known, a normally incident light wave splits into two partial waves; an ordinary wave with polarization direction orthogonal to the incidence plane, and an extraordinary wave with polarization parallel to it (see e.g [14]). These partial waves will be laterally displaced (on the incidence plane) a distance δ between them. In the next, we will assume that the partial wave polarized in the x-direction is the extraordinary wave and it corresponds to the “negative image” $I_{-}(x,y)$. [Of course, rotating the calcite prism by 90°, it is possible to displace laterally the partial waves along the y-direction. The image actually displaced will be now the “positive” one, in fact, whichever of the images, “positive” or “negative”, could be really displaced but it does not play an essential role in our argumentation (see below).]

Then, at the output plane we get

In our setup we are assuming that the LCD is relatively far from the prism and that we are dealing with paraxial rays, so that the dependence of δ with the incidence angle and the defocus produced by the difference of optical paths traveled by ordinary and extraordinary wave can be disregarded. [This condition can be fulfilled when $s_0>>f$ and the lens aperture is small (see below).] It is easy to demonstrate that, in a first-order approximation, the amount of image displacement ($\Delta x$) will be given by

3. Experimental results

We performed a series of validation experiments using a setup similar to that shown in Fig. 1. The digital images to be processed were displayed on a color-LCD monitor (model LMS560s, AOC), whose original analyzer was removed. The white light source was the original monitor light source. The optically processed images were acquired with a commercial digital color camera (model Pentax K-x) with an objective lens (L) of focal length $f=50$ mm and aperture f/40. As beam displacer (C) we used a calcite prism (BD27, Thorlabs) with $\delta =2.7$mm and a clear aperture of 10mm × 10mm. An extra positive lens (not shown in Fig. 1) with 500 mm focal length was introduced between prism and monitor, and located at a distance $d=400$mm from it. As d is less than the focal length, the acquisition system (digital camera) sees a virtual image located at a distance $s_0\approx 2.5$m (>> f) from the prism, and thus, we can assume that the incident rays on the calcite prism are nearly parallel. From Newton’s lens equation one has $z\approx f^2/s_0\approx 1$mm, and then, from Eq. (4) we can estimate $\Delta x\approx 55$μm (i.e., ~10 pixels of the digital camera).

Actually, commercial LCD monitors are designed to work with a polarizer-analyzer configuration rotated 45° relative to the monitor horizontal (vertical) direction. To simplify the description of setup and experimental results, we have chosen a $(x,y)$-coordinate system rotated 45° with respect to the monitor horizontal direction, and also the images displayed on LCD monitor were rotated through an angle of 45°.

Firstly we performed a series of experiments using a picture with colored geometric figures created by Byron Callas as test image [15]. Figure 2(a) shows the “positive” image obtained when only light polarized along the y-direction (i.e., the ordinary wave of the calcite prism) is incident on the camera, while Fig. 2(b) shows the complementary (“negative”) image obtained when only light polarized in the x-direction, that corresponds to the extraordinary wave, is incident on the camera. Figures 2(c) and 2(d) show both images superimposed, with the “negative” image slightly displaced with respect to the “positive” one along the y - and x -direction, that correspond to the vertical and horizontal direction of the images shown in Fig. 2, respectively. The required (arbitrary) image displacements in the selected directions were achieved by rotating the beam displacing prism, as explained above.

 

Fig. 2 Experimental results using a geometric abstraction image created by Byron Callas as test image: (a) true-color image; (b) complementary-color image; (c) and (d) optically processed images with fist-order derivative in y- and x- direction, respectively.

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We observe that the optically processed images have colored edge lines superimposed on a grayish, featureless background. The horizontal image borders in Fig. 2(c) and the vertical image borders in Fig. 2(d), are depicted in colors with different tones depending on the sign of the transition and the color interface, e.g., blue to yellow depicts a bluish tone and yellow to blue depicts yellowish tone, and thus, the technique distinguishes the sign of the derivative along an arbitrarily selected direction.

We repeated the experiment using a digital version of the picture Frida Kahlo picks flowers for her hair painted by Tasha [16] as test image. Figures 3(a) and 3(b) show the “positive” and “negative” image obtained experimentally, respectively. Figure 3(c) depicts the superposition of both images, but with the “negative” image slightly displaced with respect to the “positive” one along the vertical direction, respectively. In Fig. 3(c) is clearly shown the enhancement of the edges of the original picture: we see that the colors of the enhanced edges depend on the colors of the adjacent regions of the image.

 

Fig. 3 Experimental results using Tascha’s picture Frida Kahlo picks flowers for her hair as test image.

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Figures 4(a)–(d) show the experimental results obtained using Eduardo Urculo’s picture Catprovi as test image [17]. Figures 4(a) and 4(b) are the “positive” and “negative” images, while Fig. 4(c) shows the superposition of both images, when the negative replica had been slightly displaced in vertical direction.

 

Fig. 4 Experimental results using Eduardo Urculo’s picture Catprovi as test image.

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Again, the borders of the objects are enhanced, but, as mentioned above, the color and the magnitude of the enhanced edge depends on the colors and the brightness difference of the adjacent regions of the image.

The optically processed images 2(c), 2(d), 3(c) and 4(c) show clearly a grayish background tone in the regions where the first derivative of the intensity pattern is null, which is consistent with the assumption $c_{-}=c_{+}=1/2$ in the expression (3).

It can be easily demonstrated [18] that an angular error $(\xi )$ in the orientation of calcite principal directions with respect to the x- and y-axis, would generate an additional (non-linear) intensity term in Eq. (3) proportional to $\sin (2\xi )$. In our setup, the calcite prism was mounted on a rotatory stage with a precision better than 1°, and thus, we estimate that the additional spurious terms due to axis misalignment would be smaller than 2% of the total light intensity.

4. Performance analysis

Figure 5 in [19] show a speed comparison of Sobel edge detector implemented in a GPU (NVIDIA Geforce 8800GT (1.5 GHz), 512 MB global memory) and a CPU (Intel Pentium Dual E2160 CPU@1.8GHz, 1 GB RAM) for gray-level image and using different image sizes. In this figure we observe some trends: the absolute runtime of the Sobel algorithm implanted on GPU increases almost proportional with image dimension NxM, e.g., the reported times were ~0.2 ms and ~2.5 ms for 0.26 Megapixels and 4.2 Megapixels image, respectively. For color images the execution times will increase by a factor of three. On the other hand, in the proposed method the processing time only depends on the LCD characteristics, but in principle the processing time does not depend on the image size. Most of the commercially available LCD computer monitors can run at just 60 Hz, 75 Hz and sometime 85 Hz, and have an image size up to 4 Megapixels. Thus, using a simple optical setup with a standard LCDs, one can achieve processing rates 60 to 85 frame/s (i.e., ~16.6-11.7 ms per image), which is a considerable processing speed (comparable with that of GPUs when processing RGB images), sufficient for most practical applications (e.g., traffic lights detection [20]). Next-generation LCD TVs will be refreshed at least 240 times a second, which will allow to process a higher rates (e.g., 4.1 ms per image), i.e. three to four times as quick as today's version [21].

5. Discussion and conclusions

We presented a novel optical method for edge enhancement in color images based on the polarization properties of liquid-crystal displays and the optical implementation of partial first-order derivatives of the image. This operation can be seen as the superposition of two replicas of the original image with complementary colors and laterally displaced between them. The proposed method works with incoherent illumination, so that the technique is very robust and does not require a highly precise alignment, and allows selecting the direction along which the derivatives are taken. We presented validation experiments that show clearly enhanced (colored lines) edges when a small amount of displacement Δx or Δy is introduced, as predicted by expression (3).

To the best of our knowledge, no other incoherent optical system had been proposed which can perform color edge enhancement in a simple way. Unlike other methods, that propose the difficult decomposition (and subsequent recomposition) of color images in the basic RGB-components, our technique deals directly with the color images without decomposition. The proposed optical procedure consists of a superposition operation rather than using gradients and other digital operations which are computationally expensive, and thus, it could be potentially useful for processing large images in real-time applications requiring color edge detection.