## 3 The Decoding-Complexity, Rate, and Distortion Relationship
### The Decoding-Complexity, Rate and Distortion Behaviour  
In order to determine the behavior of decoding-complexity, rate, and distortion, the parameter space created by (2) must first be determined. To achieve this, an experimental sweep of the space created by \(\lambda_{r}\in[0,\infty)\) and \(\lambda_{c}\in[0,\infty)\) was performed on six different test sequences (with representative and varying spatial and temporal characteristics). Empirical data were collected from both inter- and intra-predicted frames of these test sequences for QPs ranging from 0-51 [24]. The resulting decoding-complexity, rate, and distortion can thereafter be expressed for further analysis in terms of cycles per pixel (cpp), bits per pixel (bpp), and the mean squared error (MSE), respectively, as follows.  
\[\text{MSE}(p,\lambda_{r},\lambda_{c},q)=\frac{1}{N}\sum_{i=1}^{N}\left\{s_{i} -s^{\prime}_{i}(p,\lambda_{r},\lambda_{c},q)\right\}^{2}, \tag{3}\]  
where \(s_{i}\) and \(s^{\prime}_{i}\) correspond to the \(i\)th original and reconstructed pixel, respectively, \(q\) is the QP, and \(N\) is the number of pixels in the frame. Similarly, the bpp and cpp are defined as,  
\[\text{bpp}(p,\lambda_{r},\lambda_{c},q)=\frac{R(p,\lambda_{r},\lambda_{c},q)}{ \text{W}\times\text{H}}, \tag{4}\]  
and  
\[\text{cppp}(p,\lambda_{r},\lambda_{c},q)=\frac{C(p,\lambda_{r},\lambda_{c},q) }{\text{W}\times\text{H}}, \tag{5}\]  
where W and H correspond to the frame width and frame height, respectively. Further, \(R\) and \(C\) represent the total number of bits required to encode the frame and the estimated decoding-complexity of that frame once encoded. In the model used in this work, the relative decoding-complexity is expressed in terms of the number of CPU cycles used by the HM 16.0 reference decoder for each operation when executed on an Intel x86 CPU architecture platform.  
Figure 1 graphically illustrates the behaviors of decoding-complexity, rate, and distortion in the parameter space spanned by \(p\), \(\lambda_{r}\) and \(\lambda_{c}\) for the \"kimono 1080p\" sequence, including the discrete operating points that can be achieved by the encoder (i.e., each data point in Figure 1 represents the resultant bit rate, decoding-complexity and distortion for a particular combination of \(\lambda_{r}\) and \(\lambda_{c}\)). (The behaviors of decoding-complexity, rate, and distortion remain similar across QPs and sequences, albeit with different model parameters). It was observed that this general behavior can be modeled in a content-dependent manner using a 2-dimensional nth-power model given by  
\[\text{MSE}(p,\lambda_{r},\lambda_{c},q) = a(q)\times\text{bpp}^{b(q)}(p,\lambda_{r},\lambda_{c},q)+ \tag{6}\] \[c(q)\times\text{cpp}^{d(q)}(p,\lambda_{r},\lambda_{c},q),\]  
where \(a(q)\), \(b(q)\), \(c(q)\), and \(d(q)\) are QP and content-dependent model parameters. Naturally, this implies that a content-adaptive approach is necessary to compute the appropriate model parameters dynamically, in order to determine the optimum coding structure \(p\) for particular content. The next section describes an approach to build upon the model in (6) and to adaptively compute its model parameters (as described in Section 4.3), which leads to the novel joint decoding-complexity and rate-controlled encoding algorithm proposed in this manuscript.