The in-line holography has obvious advantages especially in wider spatial bandwidth over the off-axis holography. However, a direct current(DC)-noise and an unwanted twin image should be separated or eliminated in the in-line holography for a high quality reconstruction. An approach for suppressing the twin image is proposed by separating the real and twin image regions in the digital holography. Specifically, the initial region of real and twin images is obtained by a blind separation matrix, and the segmentation mask to suppress the twin image is calculated by nonlinear quantization from the segmented image. For the performance evaluation, the proposed method is compared with the existing approaches including the overlapping block variance and manual-based schemes. Experimental results showed that the proposed method has a better performance at the overlapped region of the real and twin images. Additionally, the proposed method causes less loss of real image than the overlapping block variance-based scheme. Therefore, we believe that the proposed scheme can be a useful tool for high quality reconstruction in the in-line holography.
© 2012 Optical Society of America
The digital holography has been such a great issue these days because it can provide a three dimensional information corresponding to a real object [1, 2]. There is a practical subject coming from the optical configurations for off-axis and in-line holographies. The off-axis holography gives a better reconstruction with negligible noise than the in-line holography. However, the available bandwidth of off-axis holography is restricted by the tilted reference wave [3–5]. On the other hand, the in-line holography has obvious advantages in holographic reconstruction. One of the advantages is wider spatial bandwidth than that of the off-axis holography. However, its applicability is decreased due to the unwanted components during the reconstruction step. Therefore, it is still an open problem in in-line holography for a better numerical reconstruction. The typical unwanted components are twin image (conjugate image) and direct-current (DC) noise [4, 5]. To use wider spatial bandwidth in holographic reconstruction, twin image should be suppressed in in-line holographic configuration.
Numerical twin mage suppression has been researched by several approaches in in-line holography. The twin images suppression based on the segmented filter has been researched [4–6]. The segmented filter is calculated by the depth from focus(DFF) on in-focus twin image. To obtain the filter, the overlapping block variance of in-focus twin image is calculated to find the focused region [4, 5]. For a micro-object, the twin image suppression has been researched based on the iterations of propagation and back propagation [7, 8]. With the interaction, the binary function to suppress the twin image is calculated. Averaging multiple holographic reconstructions has been used to reduce a twin image based on a characteristic of statistically independent speckle field [9, 10]. However, the suppression by the multiple holographic reconstruction has the high complexity for the holographic reconstruction. Similarly, the twin image suppression with phase shifting interferometry(PSI) has been researched by capturing the multiple holograms with phase-shifting interferometer [11–13].
In this work, an efficient way of generating the segmentation mask is presented to suppress the twin image in digital holography. Specifically, the real and twin image regions are separated by a blind separation matrix based on canceling coefficients, and the mask is created by applying the nonlinear quantization function to the output of the blind separation.
The proposed scheme is numerically explained and its practical procedure is experimentally described in Section 2. Next, experimental result is compared with those of the state-of-arts in Section 3. Finally, Section 4 summarizes the algorithm and concludes with some discussions.
2. Proposed approach: segmentation mask generation for suppressing twin image
In this section, the mask generation for a nonlinear segmentation is described. The complex amplitudes of an object and reference waves, respectively, in the digital holography are mathematically defined as
To generate the segmentation mask, the fringe patten, which is released from , is reconstructed into in-focus real and twin images by the Fresnel approximation . Then, to reduce the effect of DC-noise in the generation of segmentation mask, DC-noise in two images is simply suppressed by numerical high-pass filter [3, 15] as shown in Fig. 1.
The procedure of the mask generation follows the block diagram as shown in Fig. 2. Specifically, two images which are in-focus real and twin images are inputted into the proposed scheme. Noise suppression [16, 17] is applied to suppress the speckle noise of two images. Next, real and twin images are separated by the blind separation matrix, which will be explained in Section 2.2, and the separated real image is quantized by a non-linear quantization function. To compensate the side effect occurred when low amplitude region at real image is filtered, inverse value of in-focus real image is generated for the inverse mapping. Then, the quantization function and the compensated image are combined to generate the nonlinear segmentation mask for the twin image suppression.
2.1. preprocessing to reduce the noise effect
The in-focus real image iR1 after DC-noise reduction contains the focused real and unfocused twin images as illustrated in Fig. 1(a), while the in-focus twin image iT1 after reducing the noise contains the focused twin and unfocused real images as shown in Fig. 1(b). Speckle noise occurs both in two images due to the diffusion of coherent light by an optical rough surface . Therefore, we employ the noise suppression scheme to reduce the effect of speckle noise for generating better segmentation mask. The speckle noise reduction has been researched based on multiple bandpass filters, PSI, and numerical filters including mean, Gaussian, and median filters [6, 16, 18]. The multiple bandpass filter and PSI require higher complexity than numerical filters, but a loss of edge information is caused by the numerical filters . In particular, a bilateral filter  was used to suppress the noise with preserving the edge as a simple noise reduction scheme. A bilateral filter Hb consists of a set of spatial and range filters asFig. 3.
2.2. Real and twin image separation by a blind separation matrix
After speckle noise suppression, a blind matrix is estimated for real and twin image separation. The in-focus real image contains both the focused real and unfocused twin images. Similarly, the in-focus twin image contains both the focused twin and unfocused real images. In order to build a simple model which can reflect the above conditions, an observed image should come both from the focused and unfocused images. Therefore, the in-focus real and twin images can be simply modeled by the combination of real and twin images, and the combination is numerically expressed by a mixing matrix A as
To separate the real and twin images from the observation images, the blind separation matrix, which is the inverse matrix of A, is required, and the separated real and twin images can be rewritten by20–22]. Hence, to obtain the matrix without an iteration and restrictions of the source, the blind and weighted separation matrix is computed from the canceling coefficients  in the separation step as
For the coefficients, the two observed images are analyzed by the ratio in the frequency domain asEq. 6, and the in-focus real and twin images are separated into the regions of real and twin images as shown in Fig. 4, respectively. In practice, 3×3 block variance in ℱ(iR2)/ℱ(iT2) is calculated. Then, the first and the second smallest variances are used for the first and second canceling coefficients , respectively.
2.3. Segmentation mask generation by nonlinear quantization and compensation
Through the separation step, the region of real image is exposed in in-focus real image while the region of twin image is suppressed as illustrated in Fig. 4(a). The output of the separation for in-focus real image is used as an initial segmentation mask because the segmentation mask should pass the real image region and block the twin image region for the suppression of twin image. In order to divide the given image into real and twin regions properly, the critical threshold is calculated by the average of the initial mask as
At the initial mask, a higher value than a threshold is fully passed while lower value than the threshold is filtered out to suppress twin image by the segmentation mask. However, in the overlapped region of real and twin images, a binary mask for separating them may be inappropriate, and it is possible to cause serious loss of real image data when the inaccurate mask is applied. Therefore, in the proposed scheme, a nonlinear quantization function is designed by using a sigmoidal nonlinearity , and defined as
By the nonlinear quantization process, the lower region than the threshold is nonlinearly quantized while the higher region than the threshold is fully passed in the filtering. However, the side effect happens when low amplitude at real image is filtered out since the cut-off value for segmentation mask is decided by the threshold. Hence, the inverse map, which represents the low amplitude region in the in-focus real image at the mapping step, is calculated to compensate the side effect as
According to the segmentation mask, the locations whose values in the map are equal to one are fully passed to reduce the side effect. And, the weights in the rest of the segmentation mask, where the mask values are not equal to one, are calculated by the nonlinear quantization function as4, 5] as 4, 5]. Then, the holographic images are reconstructed by the Fresnel approximation  on the hologram. Procedure of the proposed scheme is summarized as Alg. 1.
For the performance evaluation, the proposed approach is compared with the previous methods: the overlapping block variance-based and manually generated approaches. The fringe pattern  was used, and the pattern was reconstructed by the Fresnel approximation with a wave length λ =632.8nm and a pixel pitch Δ=6.8μm .
For the overlapping block variance-based method, block variance was calculated with 81×81 block size and the threshold was manually selected to generate the best performance segmentation mask. Similarly, for the manual-based approach, the best segmentation mask was generated manually. The generated segmentation masks are illustrated in Fig. 5. White and black regions of the segmentation mask denote one and zero values, respectively. Some regions of the twin and real images were selected by the overlapping block variance-based approach as shown in Fig. 5(a). The manually generated mask had the similar outline, zero values in the mask, with the twin image as illustrated in Fig. 5(b). The segmentation mask by the proposed scheme also had the similar outline with the twin image, and the most region of the twin image is filtered out by the mask while the most outside the twin image is passed by the segmentation mask as shown in Fig. 5(c). In the overlapped region of real and twin images, the real image was partially preserved and the twin image was partly suppressed as shown in Fig. 5(c). Note that the proposed segmentation mask had small spots on the real image region, but its spot size is much smaller than the size of spot by the overlapping block variance-based approach.
The in-focus twin image was multiplied with each of the segmentation masks on a pixel-by-pixel multiplication, and the segmented image was delivered to the hologram plane by numerical propagation [4, 5]. Then, the hologram was reconstructed by the Fresnel approximation . The reconstructed images from the twin image suppression schemes were illustrated in Fig. 6. In Fig. 6(a), the twin image is appeared like a white cloud in the reconstructed image since twin image suppression was not considered. When the overlapping block variance-based segmentation mask was applied, the loss of real image and a little suppression of the twin image are caused as shown in Fig. 6(b). Similarly, the manually generated segmentation mask suppressed the twin image much at the cost of losing the real image in the overlapped region of real and twin images as shown in Fig. 6(c). In contrast, when the proposed mask was applied, twin image was suppressed well, and the loss of real image in the overlapped region is less compared with two methods as illustrated in Fig. 6(d). Also, small spots of the segmentation mask on the real image region had little influence on the real image because the size of spots was small, and the mask was nonlinearly quantized.
In order to show the improvement effectively by the segmentation mask under the low speckle noise, bilateral filter  was finally applied both to the reconstructed images with and without the proposed twin-image suppression scheme. This procedure resulted in the two images of the speckle noise suppression as illustrated in Fig. 7. After reducing the speckle noise, the suppression of the twin image by the proposed scheme is much more clearly come into view. The intensity values of the twin image along the white solid line in Fig. 7 were plotted as shown in Fig. 8 for better observation. The black-dotted line and the gray-solid line indicate the intensity values of the reconstructed images without and with the proposed twin-image suppression scheme, respectively. It is easily seen that the intensity values with the proposed method were lower than those without the proposed method. Furthermore intensity value along the line was decreased from 31.71 to 21.48 on average by the proposed scheme.
The scheme for generating a nonlinear segmentation mask was proposed by using a blind separation matrix obtained from the relation between in-focus real and twin images. The proposed method consisted of four blocks: noise suppression; separation; quantization; and mask generation. Specifically, speckle noise was suppressed first. Next, the region of real and twin images were modeled and separated by a blind separation matrix. Then, the separated image was nonlinearly quantized, and the segmentation mask was finally generated with the quantized image and the inverse map for the compensation of the side effects. In order to evaluate the proposed approach, performance comparison was conducted versus the existing algorithms, which are commonly used for twin image suppression in the digital holography. In the overlapped region of real and twin images, the proposed method had a less loss of real image than the compared methods and could even preserve the shape of real image. Additionally, the proposed method did not have a serious loss of the real image compared with the overlapping block variance-based scheme.
Therefore, we believe that the proposed scheme can be a useful tool for finding the regions of the real and twin images and suppressing the twin image for the high quality digital holography.
This work was supported by the IT R&D program of MKE/KEIT. [KI001810039169, Development of Core Technologies of Holographic 3D Video System for Acquisition and Reconstruction of 3D Information].
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