Region reconstruction using the planar model

In Section 4.5 we describe the algorithm from [P7], where we introduced seven methods for finding the three positions A, B, C, whose heights in ^LSplane repre-sents the parameters of the^3Hmethod. Figure 5.7.(a) shows comparative results of the seven methods, for an image from the Middlebury dataset. One can see that the M2 method obtains the best results for almost half of the [40, 70] dB range, and very similar results for the rest of the range, while the M5 method has the best results for the other half of the range. The decision, to select these two methods, was taken based on the experiments done on a set of test images. The comparative results for this set can be found on the^PFwebpage [81].

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 40

45 50 55 60 65 70

Bitrate (bpp)

PSNR(dB)

M₁ − min_Q tr(Q^−TRQ⁻¹) M₂ − max_Q |det Q|

M₃ − A,B,C ⊂ bounding box Ω M₄ − Ω ⊂ ∆ ABC M₅ − Setting Q and using GR M₆ − Setting Q and using LP M₇ − baseline method

(a)

0 0.05 0.1 0.15 0.2 0.25

30 35 40 45 50 55 60 65 70 75

Bitrate (bpp)

PSNR(dB)

GSOmPF + APC + PF_M₂ GSOmPF + APC + PF_M₅ GSOm + APC GSOm + CERV P80 SP1

(b)

Figure 5.7: (a) Comparison between the seven methods M1:M7 tested for plane fitting, for the Art image, Middlebury dataset, full-size resolution, left view. (b) Comparison results between the state of the art methods and our algorithms used with and without the PF method.

In Figure 5.7.(b), we compare the two state of the art methods, P80 and SP1, with the previously developed algorithms: ^GSOm is used together with ^CERV or

APC, at which we added the^PF method for extending ^GSOm and for improving the region reconstruction model. Here we selected the^APClossless compressor for the sequence of images, but the ^CERV algorithm can be also used, or any other lossless compressor. In 5.7.(b), one can see that the ^PF method is producing a significant improvement of the results, with a maximum of around 5 dB at the 0.05 bpp bitrate. The improvement is even larger for other images, for more results see the^PFwebpage [81].

Original Contributions and Conclusions

“Nothing is impossible, the word itself says

‘I’m possible’!”

— Audrey Hepburn This final chapter presents in the first section the original contributions of the compilation of articles included in this dissertation in the domains of Signal Pro-cessing (^SP) and Image Processing (^IP). The second section presents the author’s contributions to the seven publications [P1]-[P7], while in the third section we draw the final conclusions for the work presented in this dissertation.

6.1 Original contributions

The goal of this work was to propose algorithms for depth-map image compression.

To achieve this goal, the research was divided into two parts. The first part was focused on developing lossless compression algorithms for depth-map images, which was the subject of Chapter 3. The approach used for compressing a depth-map image was to encode a segmentation and to reconstruct each region of the segmentation. Therefore, we developed algorithms for compressing the contours of an image segmentation: ^CERV-^{Alg. C} [P3] and ^APC [P5]; and algorithms for region reconstruction: ^NCV[P1] and ^CERV-^{Alg. Y}[P3]. In the second part of this research we focused on developing algorithms for lossy compression of depth-map images, which was the subject of Chapter 4. The lossy compression was achieved by the algorithms by tackling different problems: image segmentation (^L-CRS[P2]

and ^GSO [P4]), progressive coding (^P-GSO[P6]), and parameterization of planar models (^PF[P7]).

In our research, the designed coders are mostly asymmetric, because the en-coder performs more time consuming tasks compared to the deen-coder, e.g. context

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tree pruning, anchor points search, contour segments generation, image segment-ation, plane estimsegment-ation, etc.; while the decoder is less complex and is much faster than the encoder, since all received information is just ‘patched’ together to create the reconstructed image.

The main contributions of the dissertation can be summarized in the following list:

(1) Predictive coding [P1]. The predictive coding techniques are part of an im-portant approach in image compression, which was successfully used in the state of the art coders like ^CALIC and ^LOCO-I. Our research presented a mixture of 15 predictors, for both column-wise search and row-wise search, which obtained good results. It was also the first subject of our research.

(2) Image Segmentation [P2, P4]. An important side outcome, obtained in our research, is the development of an efficient image segmentation algorithm.

The^GSOalgorithm offered to the user the possibility to generate a segmenta-tion of an image by creating a lossy image of the initial image, having a given distortion measured in^PSNR. Image segmentation is an important research topic, where many of our results are relevant.

(3) Contour compression [P3, P5]. One of the main problems that we had to solve, for obtaining an efficient lossless coder, was to develop an efficient contour compression algorithm. The main reason, for which we researched the problem, is that the codelength for encoding the contour represents a large percentage (between 70% and 90%) of the final codelength. Two algo-rithms were developed: one is^APC, which is ‘drawing’ very easy and fast the contour using vertex positions, has a complex contour segment generation, and is more suitable for less complex segmentations; the other one is^CERV

-Alg. C, which is more suitable for complex segmentations, since it is able to find deterministic cases and to classify much better the contour information into horizontal contour edges and vertical contour edges.

(4) Region reconstruction using constant model [P3]. One of the most important algorithms, developed in our research, is the^CERV-^{Alg. Y}algorithm. Because of the efficiency of^{Alg. Y}in encoding constant model parameters, the results were improved with around 15%, and the percentage of codelength allocated for the region’s reconstruction stage was decreased to less than 15% of the final codelength, reason why^{Alg. Y}was integrated with most of our coders.

(5) Progressive coding [P6]. Progressive coding is an important functionality for an algorithm, and hence we developed an efficient algorithm to compress a sequence of ^GSOm images. The algorithm takes advantage of the way the images are generated, by searching for anchor points on the contours of the previously decoded image, and by encoding the constant model parameters using a priori information.

(6) Parameterization of planar model [P7]. Our last published algorithm has shown that a good parameterization of the planar model is producing

signif-icant improvements in lossy compression. The Three Heights (^3H) parame-terization is taking advantage of^{Alg. Y}, and introduces an efficient algorithm for encoding depth differences, called^{Alg. D}. The contribution in the param-eterization was to develop methods that select the positions of three heights, in such a way that the obtained distortion is close to the minimum distortion, and the plane parameters are encoded efficiently by^{Alg. D}.