A) Seam carving

SC [Reference Avidan and Shamir1, Reference Rubinstein, Shamir and Avidan2] prunes seams classified as insignificant in an image. In the original SC technique, a dynamic programming approach is used to decide seams which can be pruned. A cumulative cost M(i, j) for each seam is calculated by the following equation:

(1)$$M\lpar i\comma \; j\rpar = \min \left(\matrix{M\lpar i-1\comma \; j-1\rpar + C_L\lpar i\comma \; j\rpar \cr M\lpar i-1\comma \; j\rpar + C_U\lpar i\comma \; j\rpar \cr M\lpar i-1\comma \; j+1\rpar + C_R\lpar i\comma \; j\rpar } \right)\comma \; \eqno\lpar 1\rpar$$

where

$$\eqalign{C_L\lpar i\comma \; j\rpar &= \vert I\lpar i\comma \; j + 1\rpar - I\lpar i\comma \; j - 1\rpar \vert \cr &\quad + \vert I\lpar i-1\comma \; j\rpar - I\lpar i\comma \; j - 1\rpar \vert \comma \; \cr C_U\lpar i\comma \; j\rpar &= \vert I\lpar i\comma \; j + 1\rpar - I\lpar i\comma \; j - 1\rpar \vert \comma \; \cr C_R\lpar i\comma \; j\rpar &= \vert I\lpar i\comma \; j + 1\rpar - I\lpar i\comma \; j - 1\rpar \vert \cr &\quad + \vert I\lpar i-1\comma \; j\rpar - I\lpar i\comma \; j + 1\rpar \vert .}$$

The cost function is called forward energy [Reference Rubinstein, Shamir and Avidan2]. Finally, the seam with the smallest M(H _I, j) is pruned. With this criterion, a retargeted image is forced to avoid the large transition of pixel values between the left and right pixels of the pruned seam. The process is iterated until the desired image width is reached. For other details, please refer to [Reference Avidan and Shamir1,Reference Rubinstein, Shamir and Avidan2].

B) CAIC

So far, several methods were reported in CAIC [Reference Nguyen, Yang and Cai9–Reference Tanaka, Hasegawa and Kato11, Reference Décombas, Capman, Renan, Dufaux and Pesquet-Popescu13–Reference Deng, Lin and Cai15]. In this subsection, we briefly review the related works.

Image coding with selective data pruning (SDP) [Reference Vō, Solé, Yin, Gomila and Nguyen16] is classified as one of the “decimation-then-compression” image coding approach. The SDP method has a very simple architecture for reducing an image size since it is based on line-based downsampling. That is, rows and/or columns in an image are simply pruned and then the pruned image is interpolated on the synthesis side. The pruned lines are located predominantly across low-frequency regions in the image and thus the interpolated image has few artifacts if a small number of lines are pruned. In the comparison of image coding without data pruning, SDP-based image compression improves the peak signal-to-noise ratio (PSNR) at low bitrates. However, when many lines are pruned, the pruned lines most likely cut across salient regions. As a result, the interpolated image could have artifacts. A similar method to SDP was also presented in [Reference Wang and Urahama17].

Incorporating SC into image coding was investigated by Nguyen et al. [Reference Nguyen, Yang and Cai9]. In their method, SPI and the pixel values of seams as well as the core image are encoded. The method can reconstruct the original image losslessly if the core image, the SPI and the pixel values of seams are losslessly encoded and transmitted. However, it demands a large amount of bitrates for SPI, and the overhead bitrate significantly affects the entire image coding performance.

The authors proposed an approach for CAIC called image concentration and dilution [Reference Tanaka, Hasegawa and Kato10,Reference Tanaka, Hasegawa and Kato11], denoted as CD-based image coding hereafter. In the method, first, insignificant regions in the original image are discarded by the SC-like method and then the core image is compressed by some image coding standard. At the same time, the discarded pixel positions are stored. On the decoder side, the core image and the side information are synthesized to reconstruct the image with an arbitrary resolution. The discarded pixels, i.e., non-ROI pixels, are interpolated from the decoded core image. Unfortunately, reconstructed images after CD-based image coding sometimes have the interpolation artifacts at non-ROI.

The SC-based content-aware video coding was proposed in [Reference Décombas, Capman, Renan, Dufaux and Pesquet-Popescu13]. It reduces side information bitrate, i.e., for SPI, by employing key points of seam paths and a seam merging process. In low bitrate video coding, the method succeeded to reduce total bitrates for the equivalent video quality. However, it is also non-perfect-reconstruction video coding as well as SDP- and CD-based image coding. That is, seams should be interpolated from reconstructed core video frames.

Frequency-domain SC and its image coding application were reported in [Reference Deng, Lin and Cai14, Reference Deng, Lin and Cai15]. It calculates optimal seam paths on the LL subband of DWT. The SPI is then used to determine the scanning order of SPIHT. The seams are scanned and transmitted in the energy descending order (high-energy seams are firstly encoded). In full resolution, seams in the LL subband result block-based seams and therefore the decomposition level should not be too large (practically, the decomposition level will be less than or equal to 3 [Reference Deng, Lin and Cai15]). Therefore, its image coding performance at full resolution is substantially inferior to the original SPIHT.Footnote ²

Briefly speaking, the following two main problems should be resolved for CAIC:

1. (1) Reducing side information for CAIR while preserving CAIR performance.

2. (2) Maintaining image coding performance at full resolution.

For both encoding and decoding, the side information, especially SPI, is usually required to obtain a retargeted image with an arbitrary resolution. However, the original SC used in [Reference Nguyen, Yang and Cai9] is not designed to encode SPI effectively. Therefore, the amount of side information is quite large if we do not consider designing a SPI encoding method.

Another problem is the image coding performance. Although the core image can be compressed effectively by any existing standards, non-ROI is difficult to compress with a good quality since the positions of those pixels are not structured and highly depend on images.

In the following sections, we introduce the RD-SC and the modified tree structure of SPIHT, which try to resolve these problems.

A) Piecewise function

Let s be a seam path. In general, it can be defined as follows:

(2)$${\bi s} = \lcub j \in {\open N}\colon j = f\lpar i\rpar \comma \; 0\leq i \leq H_I-1\rcub \comma \; \eqno\lpar 2\rpar$$

where f(i) is a function to determine a seam path. A specific coordinate of the seam path is represented as (i, s(i)). According to the purpose of SC [Reference Avidan and Shamir1, Reference Rubinstein, Shamir and Avidan2], the following conditions are usually imposed on f(i):

• • Connectivity: |f(i) − f(i + 1)| ≤ 1.

• • Rectangular restriction: 0 ≤ f(i) ≤ W _I − 1.

The first condition is the requirement that a seam path should be a set of eight-connected pixels. The second one is imposed to preserve a rectangular shape of the resized image.

Usually s has a complex structure since SC calculates s pixel-by-pixel and thus, it is difficult to compress efficiently. In this paper, we propose an approximation method of s with piecewise predefined functions. We define an approximated seam path a as follows:

(3)$${\bi a}_k = \lcub j \in {\open N}\colon j = f_{k}\lpar i_{k}\rpar \comma \; 0 \leq i_{k} \leq L\lpar k\rpar -1\rcub \comma \; \eqno\lpar 3\rpar$$

where a_k (k = 0, 1, …) is the kth portion of a, L(k) defines the interval for a_k and f _k(i _k) is an approximation function. a_k is calculated within the interval $\left[\sum_{p=0}^{k-1} L\lpar p\rpar \comma \; \sum_{p=0}^{k} L\lpar p\rpar -1\right]$. The relationship between s and a is illustrated in Fig. 3.

Fig. 3. Relationship between s and a. Blue solid and red dashed lines represent s and a, respectively.

B) Candidates of f _k(i _k)

Any functions can be candidates of f _k(i _k) as long as they follow the conditions mentioned in the previous subsection. We consider to reduce the bitrate for SPI by choosing one function from several templates. A template set of f _k(i _k), shown as f _{k, g}(i _k), is heuristically defined as follows:

(4)$$f_{k\comma 0} \lpar i_{k}\rpar = C_{k}\comma \; \eqno\lpar 4\rpar$$

(5)$$f_{k\comma \pm 1}\lpar i_{k}\rpar = \pm i_{k} + C_{k}\comma \; \eqno\lpar 5\rpar$$

(6)$$f_{k\comma \pm 2}\lpar i_{k}\rpar = \pm \left[{1\over 3} i_{k} \right]+ C_{k}\comma \; \eqno\lpar 6\rpar$$

(7)$$f_{k\comma \pm 3}\lpar i_{k}\rpar = \pm \left[{i_{k}^{2} \over 2L\lpar k\rpar } \right]+ C_{k}\comma \; \eqno\lpar 7\rpar$$

(8)$$f_{k\comma \pm 4}\lpar i_{k}\rpar = \pm \min\left(\left[{1 \over 2}\sqrt{L\lpar k\rpar \cdot i_{k}} \right]\comma \; i_{k}\right)+ C_{k}\comma \; \eqno\lpar 8\rpar$$

where C _k = a _k(0) and [·] is the rounding operator. f _{k, g}(i _k) are shown in Fig. 4. Clearly, (4) is the constant function. The upper and lower bounds of seam paths from the connectivity condition are represented in (5). In (8), the minimum operator is used since the square-root functions violate the connectivity condition without the operator.

Fig. 4. f _{k, g}(i _k) where L(k) = 32. The number in legend is corresponding to those in (4–8).

We append one extra condition as follows:

(9)$$C_{k} = a_k\lpar 0\rpar = \left\{\matrix{s_{o}\lpar 0\rpar \hfill & \hbox{if}\ k=0\comma \; \hfill \cr a_{k-1}\lpar L\lpar k-1\rpar - 1\rpar \hfill & \hbox{otherwise}\comma \; \hfill} \right. \eqno\lpar 9\rpar$$

where s_o is the original seam path to be approximated. This means the initial position of a_k is the same as the terminated position of a_k−1. With this restriction, we can reduce the connection information between a_k−1 and a_k.

As a result, the SPI for a_k consists of a selected index among (5–8) and the interval L(k). Furthermore, the overall initial position s _o(0) = a(0) is required for each a.

C) Recursive optimization of approximated seams

The optimal L(k) should not be determined as a constant. For some k, L(k) can be large if s_o goes through very smooth regions or it is well approximated by one of the templates (4–8). Whereas L(k) should be small at the textured region where s_o has a complex structure. Hence, we present the algorithm for recursive optimization of L(k).

We assume s_o is calculated before the approximation. To estimate L(k) effectively, we define the simplified forward energy criterion as follows:

(10)$$C_{SFE} \lpar {\bi s}\rpar = {1 \over s_L} \sum_i \left\vert I\lpar i\comma \; s\lpar i\rpar - 1\rpar - I\lpar i\comma \; s\lpar i\rpar + 1\rpar \right\vert\comma \; \eqno\lpar 10\rpar$$

where s _L is the length of s. This criterion is similar to the forward energy of the improved SC [Reference Rubinstein, Shamir and Avidan2] but nevertheless is less complex. In the optimization process, we use (10) since many repetitive calculations are required.

Algorithm 1 presents the detailed optimization process of L(k) for the kth interval of I:

$$I_{sub} = I \left(\sum_{p = 0}^{k - 1} L\lpar p\rpar \colon \sum_{p = 0}^k L\lpar p\rpar - 1\comma \; \colon \right).$$

Algorithm 1 Optimization of L(k) with piecewise functions

In Algorithm 1, s_{o, sub} is a original sub-seam path for I _sub, T > 0 is a threshold and a_sub is the approximated seam path for I _sub. Moreover,

$${\bi a}_{k\comma g} = \lcub j\colon j = f_{k\comma g}\lpar i_{k}\rpar \comma \; 0\leq i_{k} \leq L\lpar k\rpar -1\rcub .$$

It is clear that the approximated seam path a_{k, g} can be completely recovered from the returned parameters. The simplified structure of this algorithm is shown in Fig. 5. In applications shown in this paper, the maximum row height of I _sub (denoted as L _max) is limited to 32. As a result, each L(k) is selected from {32, 16, 8, 4}.

Fig. 5. Recursive seam approximation process. The red lines represent the division points of s_{o, sub}.

D) Iterative optimization

In Algorithm 1, the optimal a_k associated with L(k) is obtained for a given threshold T. As a further extension, the optimal threshold T is sought to control the bitrate for the SPI and the CAIR quality. The optimization is realized by an iterative optimization of Algorithm 1. The object of RD-SC is to find the nearest seam a_opt to s_o under a bitrate constraint for the SPI, which is represented as follows:

(11)$${\bi a}_{opt} = \hbox{arg} \min_{\bi a} \lpar D_{PA} \lpar {\bi a}\rpar + \lambda R_{PA} \lpar {\bi a}\rpar \rpar . \eqno\lpar 11\rpar$$

where λ ≥ 0 is the Lagrange multiplier. In this approach, we define D _PA(a), shown below, based on the reconstruction error by the simple interpolation.

(12)$$D_{PA}\lpar {\bi a}\rpar = \hbox{MSE} \lpar I\lpar \colon \comma \; {\bi s}_o\rpar \comma \; {\bar I}\lpar \colon \comma \; {\bi s}_o\rpar \rpar - \hbox{MSE} \lpar I\lpar \colon \comma \; {\bi a}\rpar \comma \; {\bar I}\lpar \colon \comma \; {\bi a}\rpar \rpar \comma \; \eqno\lpar 12\rpar$$

where MSE is a mean squared error between the original and interpolated seams and Ī is defined by a linear interpolation from six available neighbors as

(13)$$\bar{I}\lpar i\comma \; v\lpar i\rpar \rpar = {1 \over 6}\sum_{u_0 = -1}^1 \sum_{u_1 \in \lcub -1\comma 1\rcub } I\lpar i + u_0\comma \; v\lpar i\rpar + u_1\rpar \comma \; \eqno\lpar 13\rpar$$

which is depicted in Fig. 6. The calculation method of R _PA (a) is shown in the following subsection (Section III-E).

Fig. 6. Illustrative example of (13). White and black dots represent core (remaining) and seam pixels, respectively.

Finally, a_opt in (11) is sought by gradually changing T shown in Algorithm 2, where ${\bi a}_{N_{cur}}$ is the approximated seam path at the current iteration and T _PA contains N _T threshold values and they are monotonically increasing. In this paper, we use T_PA = [1, 5, 10, …, 50].

Algorithm 2 Iterative optimization of a

It is a similar strategy to the rate-distortion optimization for video coding [Reference Ortega and Ramchandran18]. Instead of the brute-force optimization, our approach is climbing up the R-D slope as shown in Fig. 7 to reduce computational complexity.

Fig. 7. Illustrative example of Algorithm 2.

E) Bitrate calculation

In Algorithm 2, R _PA (a) should be properly calculated from the approximated seam a. As previously mentioned, the function index g _opt in Algorithm 1 and the set of L(k) are the side information for SPI. Since Algorithm 1 is based on the division of I _sub into two halves, we can represent a set of L(k) with a simple binary representation. That is, 0 is stored if I _sub is not divided, whereas 1 is stored if I _sub is divided. As a result, a binary vector b_div can be obtained for the division information, which corresponds to L(k).

By our preliminary experiment, codewords for function indices g _opt are determined as shown in Table 2. The required bitrate for the approximation functions is calculated based on the assigned codewords.

Table 2. Codeword Assignment for f _{k, g}(i) of RD-SC

Finally, R _PA (a) can be represented as follows:

(14)$$R_{PA}\lpar {\bi a}\rpar = \#\lpar {\bi b}_{div}\rpar + \sum b_{g_{opt}}\comma \; \eqno\lpar 14\rpar$$

where #(b_div) is the number of elements in b_div and $b_{g_{opt}}$ is the number of bits to encode g _opt.

F) Weighted cost matrix

Each rate-dependent seam path can be calculated by RD-SC mentioned previously. However, a reconstructed image sometimes presents visible artifacts due to seam pruning from almost the same spatial positions. To alleviate this problem, we consider balancing the positions of the seams. For this purpose, we modify the cost of the original SC M(i, j) in (1). When an already pruned seam is defined as $\hat{\bi a}_{m} \lpar m = 0\comma \; 1\comma \; \ldots\rpar $, a new cost map is represented as follows:

(15)$$M_b\lpar i\comma \; a\lpar i\rpar \rpar = w \sum_{m} \delta_{\hat{a}_m\lpar i\rpar \comma a\lpar i\rpar }\comma \; \eqno\lpar 15\rpar$$

(16)$$\hat{M} \lpar i\comma \; j\rpar = M\lpar i\comma \; j\rpar + M_b\lpar i\comma \; j\rpar \comma \; \eqno\lpar 16\rpar$$

where w and δ_{i, j} are the weight and the Kronecker delta, respectively. Briefly speaking, the extra cost map M _b(i, j) appends w for the conventional cumulative cost M(i, j) when a seam candidate crosses (or touches) a pruned seam path. We use ${\hat M}\lpar i, j \rpar$ instead of M(i, j) in (1) for calculating a seam path. As a result, the updated cost ${\hat M}\lpar i, j \rpar$ avoids pruning a seam adjacent to already pruned seams.

Furthermore, ${\hat M}\lpar i, j \rpar$ is utilized to determine the maximum number of seams. If $\min\lpar {\hat M}\lpar H_{I}\comma \; j\rpar \rpar \gt \tau$, where τ is an arbitrary threshold, we stop pruning seams since the remaining retargeted image is sufficiently condensed and no more seams should be pruned from the image.

G) Multiple seam pruning

In the original SC, just one seam is pruned at each seam calculation. Hence, pruning many seams increases the computational complexity. Therefore, we reuse a primal seam path for pruning the seams around it. In this paper, we use the seam-by-seam calculation for s_o to clarify our contribution. To further reduce the computational complexity, we could use an improved SC method, such as [Reference Chiang, Wang, Chen and Lai19], which uses one saliency map for pruning several seams. It is able to speed up the calculation of s_o.

We present an algorithm to prune multiple seams with a primal seam path a_p in Algorithm 3. In the algorithm, ${\bi B} = \lcub {\bi b}_{-\lpar N_{m} - 1\rpar /2}\comma \; \ldots\comma \; {\bi b}_{0}\comma \; \ldots\comma \; {\bi b}_{\lpar N_{m}-1\rpar /2}\rcub $ is a set of seams to be pruned at the same time, N _m is the (odd) number of pruned seams and T _m is the threshold to determine whether multiple seams can be pruned.

Algorithm 3 Multiple seam pruning

A) Lifting-based seamlet

For preserving the image coding performance at full resolution, the pixel values of seams should be encoded as well as the core image. Before constructing a full multiresolution image, seams are transformed by the lifting-based discrete wavelet transform (DWT).

The wavelet transform along seam paths, named seamlet, was proposed by Conger et al. [Reference Conger, Radha and Kumar20]. It generates coarse and detail coefficients similar to the conventional DWT, but its downsampling positions are determined by seam paths. Though it is able to use any wavelet filters, the original seamlet recommended Haar wavelet. However, we focus on the image coding application using CAIR. The detail coefficients should be as small as possible for image coding. Thus, lifting-based 5/3 DWT is employed for seamlet in this paper. The detail coefficient is calculated as follows:

(17)$$\eqalign{d\lpar i\comma \; a\lpar i\rpar \rpar &= I\lpar i\comma \; a\lpar i\rpar \rpar \cr &\quad - \left[\left(I\lpar i\comma \; a\lpar i\rpar - 1\rpar + I\lpar i\comma \; a\lpar i\rpar + 1\rpar \right)/{2}\right]\comma \; } \eqno\lpar 17\rpar$$

where d(i, a(i)) is the detail coefficient at (i, a(i)). This lifting-based transformation is shown in Fig. 8. In our approach, the coarse coefficients remain unchanged different from the original seamlet. That is, the five-tap lowpass DWT filtering is not performed, since these unfiltered original pixels in the core image are finally transformed by the conventional DWT. Finally, d(i, a(i))'s are collected and one column of detail coefficients is yielded. The process is recursively performed for the number of seams.

Fig. 8. Lifting-based seamlet. Top: transformation of I(i, a(i)). Bottom: illustrative example of seamlet for one seam.

In this paper, first vertical seams are calculated and transformed by the lifting-based seamlet and then the horizontal seams are calculated and transformed, shown in Fig. 9. Finally, the core image is transformed by the multilevel DWT to construct a fully decomposed multiresolution image. Note that this can be easily extended to the optimal order seam pruning [Reference Rubinstein, Shamir and Avidan8] for 2-D CAIR; however, we choose this simple 2-D CAIR because of the computational complexity of the optimization of the seam pruning sequence. A retargeted image with an arbitrary resolution is easily obtained by interpolating seam paths.

Fig. 9. Encoder structure of the proposed CAIC.

B) Pre-quantization of detail seam coefficients

It is clear that pixel values of the seams are not as important as the core image since prominent regions are preserved in the core image through RD-SC. Therefore, we consider to quantize detail seam coefficients before SPIHT encoding. It is represented as follows:

(18)$$d_q\lpar i\comma \; a\lpar i\rpar \rpar = \hbox{sgn}\lpar d\lpar i\comma \; a\lpar i\rpar \rpar \rpar \left[{\vert d\lpar i\comma \; a\lpar i\rpar \rpar \vert + f_q \over \Delta}\right]\comma \; \eqno\lpar 18\rpar$$

where d _q(i, a(i)) is the pre-quantized detail seam coefficient, f _q is the rounding control parameter, and Δ is the quantization step size. Clearly, no quantization is performed when Δ = 1. The pre-quantized signal is reconstructed by the following equation:

(19)$$d^{\prime} \lpar i\comma \; a\lpar i\rpar \rpar = \Delta d_q\lpar i\comma \; a\lpar i\rpar \rpar . \eqno\lpar 19\rpar$$

This pre-quantization is based on the scalar quantization with offset used in H.264/AVC [Reference Wedi and Wittman21] and we use f _q = Δ/3 similar to the intra-coding of H.264/AVC. By using this pre-quantization, we can control the qualities between the core image and seams.

Modified tree structure of SPIHT for CAIC

For the proposed multiresolution image representation, we introduce a new parent–children relationship between the transformed coefficients in the core image and detail coefficients of seams for SPIHT encoding [Reference Said and Pearlman12].

The ordinary parent–children relationship in [Reference Said and Pearlman12] is defined to connect a pixel to four pixels in the same subband for the next lower decomposition (higher frequency) level. In this paper, the relationship is used for transformed coefficients of the core image. Let O(q, r) be a set of children indices of the parent index (q, r). The relationship is defined as

(20)$$\eqalign{O\lpar q\comma \; r\rpar &=\lcub \lpar 2q\comma \; 2r\rpar \comma \; \lpar 2q\comma \; 2r+1\rpar \comma \; \lpar 2q+1\comma \; 2r\rpar \comma \; \cr &\qquad \lpar 2q+1\comma \; 2r+1\rpar \rcub .}\eqno\lpar 20\rpar$$

Besides the ordinary relationship, we introduce the new relationship between the core image and seams. We define (q _c, r _c) as the index of a transformed coefficient of the core image after DWT in the HL subband for the lowest decomposition (the highest frequency) level, corresponding to I(i, a(i) − 1). Furthermore, (q _s, r _s) is defined as the index of a detail coefficient of a seam corresponding to d _q (i, a(i)).

The new parent–children relationship is represented as follows:

(21)$$O\lpar q_c\comma \; r_c\rpar \ni \left\{\matrix{\lpar q_s\comma \; r_s\rpar \hfill & \hbox{if}\, \lpar q_s\comma \; r_s\rpar \, \hbox{exists}\comma \; \hfill \cr \phi \hfill & \hbox{otherwise}. \hfill} \right. \eqno\lpar 21\rpar$$

The new parent–children relationship is illustrated in Fig. 10. This relationship means that detail coefficients of seams are related to the nearest core pixels. The case of horizontal seams is similarly defined as well.

Fig. 10. New parent–children structure of the proposed CAIC. An arrow with a solid line represents the relationship.

A) Effects of parameter setup

The theoretical optimization of parameters shown in Table 4 is quite difficult since the optimization process could be highly non-linear and the best parameter set is certainly depending on the one who views the retargeted and full resolution images. Therefore, all of them are set heuristically as previously mentioned. Here, we describe the effect of parameter settings. In the following experiments in this subsection, one parameter is changed and the remaining ones are the same as Table 4 unless specified. Since effects of a few parameters are relatively clear, i.e., those of L _max and τ are analogous, we describe about λ, ω, T _m, and N _m. It is an interesting future work to determine optimal parameters automatically. As a simple approach, we could append a parameter tuning step by a user with some specific test images.

1) λ

It controls bitrates for SPI and the accuracy of seam paths. In the experiment, W _I/4 vertical seams and H _I/4 horizontal seams are calculated regardless of τ. The SPI of SC is encoded by arithmetic coding similar to [Reference Nguyen, Yang and Cai9]. Figure 12 is a plot of the bitrate of SPI as a function of λ. As expected, the bitrate decreases as λ increases and the bitrates are much lower than those of SC.

Fig. 12. Bitrates for side information. The solid lines represent required bitrates of SPI by SC.

Figure 13 shows the calculated seams of Pisa Tower for various λ. The seams represent different paths depending on λ. The seams move finely like SC for a small λ, whereas for a large λ, the seam paths are a coarse approximation of the original ones.

Fig. 13. Approximated seams by RD-SC. The enlarged portions of Pisa Tower are shown. Black lines represent approximated seam paths.

2) ω

It is used for balancing seam positions. ω = 0 means there are no restrictions for the positions. Figure 14 shows approximated seam paths of Crew for various ω. The seam paths are significantly changed within the range [0, Reference Guo, Liu, Shi, Zhou and Gleicher5]. Without the restriction, i.e., ω = 0, seams are mainly located on the smooth areas and the positions are concentrated similar to the original SC. Whereas for the large ω, such as ω = 5.0 or 10, the seams are distributed uniformly. Seams for the large ω are sometimes passing through the prominent regions of the image, e.g., faces of astronauts.

Fig. 14. Approximated seams by RD-SC for various ω. A total of 128 seams are removed regardless of τ for comparison purpose.

For the retargeting purpose, ω should be small; which yields similar seam positions to those of the original SC (Fig. 14(a)). Whereas for the image coding purpose, seam paths should be distributed uniformly. As a simple comparison, Table 5 presents PSNRs of the interpolated seams by linear interpolation. It is clear that ω = 0, i.e., the most concentrated case, shows the worst interpolation result since the small ω requires interpolating wide areas from few remaining pixels. It is also observed that ω should not be too large. Therefore, as a trade off issue, we picked up ω = 0.5.

Table 5. Linear interpolation results for Crew image.

3) N _m and T _m

N _m determines the maximum number of seams removed at the same time. The large N _m permits removing many seams by one seam path calculation step. The seam paths for N _m = 3 and 11 are shown in Fig. 15. As expected, many seams share one seam paths in Fig. 15(b). PSNRs of interpolated seams by linear interpolation and SPI bitrates are compared in Fig. 16. It seems that the large N _m is recommended anytime. However, as shown in Fig. 17, the large N _m significantly affects the retargeted image quality. In the figure, the straight line in the original Crew image was broken in the retargeted image with N _m = 11. Hence, we picked up N _m = 3, which seems a good trade-off as shown in Fig. 16.

Fig. 15. Approximated seams by RD-SC for different N _m (T _m = 4). A total of 128 seams are removed regardless of τ for comparison purpose.

Fig. 16. PSNRs of interpolated seams and bitrates for SPI (Crew image, 128 seams).

Fig. 17. Enlarged portions of retargeted Crew image.

T _m has a similar effect to N _m, since it decides whether N _m seams can be pruned or not (see Algorithm 3). If N _m > 255² (possible upper bound of MSE), the algorithm always choose to prune N _m seams regardless of MSE. Based on the similar experiment about T _m, we experimentally set T _m = 4. The approximated seams for a few T _m are shown in Fig. 18.

Fig. 18. Approximated seams by RD-SC for different T _m (N _m = 1). A total of 128 seams are removed regardless of τ for comparison purpose.

B) CAIR performance

The second comparison is for CAIR performance. As far as we know, few numerical measures exist to compare CAIR performances. Hence, we present all of the retargeted images for visual quality comparison. Table 3 summarizes the sizes of the core images. Figure 19 shows CAIR results by using SC and RD-SC. For comparison purpose, scaled images by imresize function of MATLAB are also shown. All images are resized to have the same size of the retargeted images by RD-SC. From the figures, RD-SC presents very similar retargeted images to those obtained by SC.

Fig. 19. Resizing results. The left, center, and right columns correspond to the resized images by scaling, SC, and RD-SC, respectively. From top to bottom: Park Joy, Pisa Tower, Beach, and Crew.

C Computational complexity

It is worth noting that the computational complexity in the seam approximation process of RD-SC is higher than those of SC due to the recursive estimation. It is difficult to present the complexity theoretically due to the highly image-adaptive computations of RD-SC. Thus, the execution time is compared. All algorithms are implemented on a MATLAB (R2011a) platform. The simulation is run on the desktop computer with Intel 3.4-GHz CPU (Corei7-2600), 16-GB RAM. Table 6 shows the execution time of the average of five executions for Beach image. In the table, the seam calculation time by SC (the column of “SC”) is also shown for comparison. The computation time is proportional to the number of seams to be removed. Seam path approximation time is longer than that of the original SC. However, Table 6 also presents the seam path decoding time of RD-SC. Obviously, it is significantly faster than the seam calculation of SC. It is beneficial for the traditional client–server model, where the decoder will be a mobile device which often has poor computation power.

Table 6. Execution time (s) of Beach image.

D) Application to image coding

Here, we compare image coding performance. As the conventional encoders, SPIHT, the method in [Reference Nguyen, Yang and Cai9] (denoted as SC-SPIHT) and CD-based SPIHT [Reference Tanaka, Hasegawa and Kato11] (denoted as CD-SPIHT) are used. Our CAIC method is referred to as CA-SPIHT in this section. Note that SC-SPIHT does not include pixel values of seams; that is, the compressed pruned image is interpolated by linear interpolation since the bitrate for SPI already requires a high percentage of the entire bit budget. All methods apply five-level 9/7 DWT for core imagesFootnote ³ before performing SPIHT.

The R-D curves, encoded with Δ = 1 in (18), are shown in Fig. 20. SC-SPIHT is inferior to the other methods due to its large amount of bitrates for SPI. Furthermore, CD-SPIHT is worse than SPIHT and CA-SPIHT especially for high bitrates since its non-ROI is just interpolated from the core image at the decoder. SPIHT and CA-SPIHT are comparable, but SPIHT slightly outperforms the CA-SPIHT. More precisely, in comparison to SPIHT, CA-SPIHT increases bitrates less than or equal to 10 %. However, the bitrate increase would be a permissible penalty for users demanding both of a good CAIR quality and image coding performance at full resolution. In addition, reconstructed images at full resolution are shown in Figs. 21–24. Similar to the R-D curves, the reconstructed images by SC-SPIHT and CD-SPIHT have interpolation artifacts, whereas our CA-SPIHT significantly reduces those artifacts and the reconstructed images are very similar to those encoded by SPIHT.

Fig. 20. R-D curves of reconstructed image at full resolution. From left to right, top to bottom: Park Joy, Beach, Pisa Tower, and Crew.

Fig. 21. Reconstructed Park Joy images. Each image is encoded at 1.0 bpp.

Fig. 22. Reconstructed Pisa Tower images. Each image is encoded at 1.0 bpp.

Fig. 23. Reconstructed Beach images. Each image is encoded at 1.0 bpp.

Fig. 24. Reconstructed Crew images. Each image is encoded at 1.0 bpp.

Moreover, Fig. 25 represents PSNRs of core images for a few step sizes of the pre-quantization in Section IV-B. As expected, if Δ in (18) is large, the quality of the core image is improved especially for high bitrate coding. In addition, PSNR gains from Δ = 1 depend on the test images. For example, seam paths for Park Joy pass through the very smooth regions, the PSNR gains due to the pre-quantization is marginal, whereas for Beach, the PSNR gain is significant since vertical seams pass through the highly textured sand area.

Fig. 25. R-D curves of reconstructed core images for various Δ.