Formulation and evaluation of polarization-modulated triple-view information display with three TN-LCD layers

Abstract

We have derived formulation and evaluation of polarization-modulated triple-view information display with three TN-LCD layers. The proposed display is composed of three liquid–crystal display (LCD) panels that are placed at a certain gap between polarizers. The viewed image is generated after polarization modulations by the three LCD panels. The polarization modulated results depend on the view directions. Thus, information of three binary images is shared between the displayed images on the three LCD panels.

Appendix

The procedure for deriving the general solution is described. When the center viewpoint is included, Eqs. (16–18) are used as examples; when the center viewpoint is not included, the results of the study for all combinations are noted.

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\end{array}$$

(37)

$$\begin{array}{c}{{\text{C}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}}\end{array}$$

(38)

$${\text{R}}_{{{\text{k}},{\text{l}}}} = {\text{r}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}}}}$$

(39)

To solve the simultaneous equation of XOR, we make a sum of the same terms.

(37) $\oplus$(38)_l+1

$${\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{C}}_{{{\text{k}},{\text{l}} + 1}} = {\text{r}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 1}} \Rightarrow {\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{C}}_{{{\text{k}},{\text{l}} + 1}} = {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 1}}$$

(40)

(37) $\oplus$(38)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{C}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}}\end{array}$$

(41)

(37)_l+1 $\oplus$(38)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{C}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\end{array}$$

(42)

(39) $\oplus$(39)_l+1

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}+1}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+1}\end{array}$$

(43)

(42) $\oplus$(43)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{C}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}+1}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+1}\end{array}$$

(44)

(41)_{k+1, l+1} $\oplus$(44)

$${\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{C}}_{{{\text{k}},{\text{l}}}} \oplus {\text{C}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}},{\text{l}}}} \oplus {\text{R}}_{{{\text{k}},{\text{l}} + 1}} \begin{array}{*{20}c} { = {\text{r}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 2}} } \\ \end{array}$$

(45)

(39) $\oplus$(39)_l+2

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}+2}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}-1,{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+2}\end{array}$$

(46)

(40)_{k+1, l+1} $\oplus$(46)

$$\begin{gathered} {\text{U}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{C}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{R}}_{{{\text{k}},{\text{l}}}} \oplus {\text{R}}_{{{\text{k}},{\text{l}} + 2}} \hfill \\ = {\text{r}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} - 1,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 2}} \hfill \\ \end{gathered}$$

(47)

(42) $\oplus$(47)_k+1

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{C}}_{{{\text{k}},{\text{l}}}} \oplus {\text{C}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 2}} \hfill \\ = {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ \end{gathered}$$

(48)

(40) $\oplus$(43)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{C}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}+1}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+1}\end{array}$$

(49)

(41) $\oplus$(49)_k+1

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{C}}_{{{\text{k}},{\text{l}}}} \oplus {\text{C}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 1}} \hfill \\ = {\text{f}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 1}} \hfill \\ \end{gathered}$$

(50)

From these results, the general formula for triple-view display (upper, center, right) is as follows

$${{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}+2}={{\text{U}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{U}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{C}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{C}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+1}$$

$${{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+2}={{\text{U}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{U}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{C}}}_{{\text{k}},{\text{l}}}\oplus {{\text{C}}}_{{\text{k}}+2,{\text{l}}+2}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+2}$$

$${{\text{f}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+2}={{\text{U}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{U}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{C}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{C}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+2}$$

Solving simultaneous equations with ${{\text{U}}}_{{\text{k}},{\text{l}}}$, ${{\text{L}}}_{{\text{k}},{\text{l}}}$, and ${{\text{R}}}_{{\text{k}},{\text{l}}}$.

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\end{array}$$

(51)

$$\begin{array}{c}{{\text{L}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\end{array}$$

(52)

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(53)

Solve the above XOR simultaneous equations.

(51)_k+1 $\oplus$(52)_l+1

$$\begin{array}{c}{{\text{U}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{L}}}_{{\text{k}},{\text{l}}+1}={{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}-1}\end{array}$$

(54)

(51) $\oplus$(52)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{L}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\end{array}$$

(55)

(51)_l+1 $\oplus$(52)_k+1

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{L}}}_{{\text{k}}+1,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(56)

(53)_l+1 $\oplus$(53)_k+1

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}}\end{array}$$

(57)

(55)_{k+2, l+1} $\oplus$(57)

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} = {\text{r}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} } \\ \end{array}$$

(58)

(56) $\oplus$(58)

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \hfill \\ = {\text{r}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}} + 1}} \hfill \\ \end{gathered}$$

(59)

(53)_{k+1, l+2} $\oplus$(53)_k+3

$$\begin{array}{c}{{\text{R}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{R}}}_{{\text{k}}+3,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}}+3,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+4,{\text{l}}}\end{array}$$

(60)

(56) $\oplus$(60)

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}}}} = {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 4,{\text{l}}}} } \\ \end{array}$$

(61)

(54)_{k+3, l+1} $\oplus$(61)

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 4,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 3,{\text{l}} + 2}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}}}} \hfill \\ = {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 4,{\text{l}} + 1}} \hfill \\ \end{gathered}$$

(62)

(53)_l+1 $\oplus$(53)_k+1

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}}\end{array}$$

(63)

(54) $\oplus$(63)

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} = {\text{r}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} } \\ \end{array}$$

(64)

(55) $\oplus$(64)_k+1

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}},{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 2,{\text{l}}}} \hfill \\ = {\text{f}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 3,{\text{l}}}} \hfill \\ \end{gathered}$$

(65)

From these results, the general formula for triple-view display (upper, left, right) is as follows

$$\begin{gathered} {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ = {\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 2,{\text{l}} - 1}} \oplus {\text{L}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 2,{\text{l}} - 1}} \hfill \\ {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ = {\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 4,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{L}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}} - 1}} \hfill \\ {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ = {\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}}}} \hfill \\ \end{gathered}$$

Solving simultaneous equations with ${{\text{U}}}_{{\text{k}},{\text{l}}}$, ${{\text{L}}}_{{\text{k}},{\text{l}}}$, and ${{\text{B}}}_{{\text{k}},{\text{l}}}$.

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\end{array}$$

(66)

$$\begin{array}{c}{{\text{L}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\end{array}$$

(67)

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\end{array}$$

(68)

Solve the above XOR simultaneous equations.

(66)_k+1 $\oplus$(67)_l+1

$$\begin{array}{c}{{\text{U}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{L}}}_{{\text{k}},{\text{l}}+1}={{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}-1}\end{array}$$

(69)

(66) $\oplus$(67)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{L}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\end{array}$$

(70)

(66)_l+1 $\oplus$(67)_k+1

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{L}}}_{{\text{k}}+1,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(71)

(68)_l+1 $\oplus$(68)_k+1

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{B}}}_{{\text{k}}+1,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}-1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+1}\end{array}$$

(72)

(71) $\oplus$(72)

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} = {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 1}} } \\ \end{array}$$

(73)

(70)_{k+1, l+2} $\oplus$(73)

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} \hfill \\ = {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ \end{gathered}$$

(74)

(68)_l+2 $\oplus$(68)_k+2

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}-1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+3}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+1}\end{array}$$

(75)

(69)_{k+1, l+2} $\oplus$(75)

$$\begin{gathered} {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}}}} \hfill \\ = {\text{r}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ \end{gathered}$$

(76)

(71) $\oplus$(76)_l+1

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 3}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 4}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 3}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}} + 1}} } \\ { = {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 3}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 4}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 3}} } \\ \end{array}$$

(77)

(69) $\oplus$(72)

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} = {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 1}} } \\ \end{array}$$

(78)

(70) $\oplus$(78)_l+1

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{U}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{L}}}_{{\text{k}},{\text{l}}}\oplus {{\text{L}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{B}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{B}}}_{{\text{k}}+1,{\text{l}}+1}\\ ={{\text{f}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+3}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+2}\end{array}$$

(79)

From these results, the general formula for triple-view display (upper, left, bottom) is as follows

$$\begin{gathered} {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ = {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} \hfill \\ {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ = {\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}}}} \hfill \\ {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ = {\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}} + 1}} \hfill \\ \end{gathered}$$

Solving simultaneous equations with ${{\text{L}}}_{{\text{k}},{\text{l}}}$, ${{\text{R}}}_{{\text{k}},{\text{l}}}$, and ${{\text{B}}}_{{\text{k}},{\text{l}}}$.

$$\begin{array}{c}{{\text{L}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\end{array}$$

(80)

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(81)

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\end{array}$$

(82)

Solve the above XOR simultaneous equations.

(80) $\oplus$(81)_k+2

$$\begin{array}{c}{{\text{L}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}}+2,{\text{l}}}={{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+3,{\text{l}}}\end{array}$$

(83)

(80) $\oplus$(81)

$$\begin{array}{c}{{\text{L}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(84)

(80)_k+2 $\oplus$(81)

$$\begin{array}{c}{{\text{L}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+3,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}}\end{array}$$

(85)

(82) $\oplus$(82)_k+2

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}-1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+1}\end{array}$$

(86)

(84)_{k+1, l+1} $\oplus$(86)

$$\begin{array}{c}{{\text{L}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{B}}}_{{\text{k}},{\text{l}}}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}-1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}}\end{array}$$

(87)

(85) $\oplus$(87)

$$\begin{gathered} {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}},{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}},{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}}}} \hfill \\ \begin{array}{*{20}c} { = {\text{r}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}}}} } \\ \end{array} \hfill \\ \end{gathered}$$

(88)

(82)_l+1 $\oplus$(82)_{k+4, l+1}

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{B}}}_{{\text{k}}+4,{\text{l}}+1}={{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+4,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+4,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+4,{\text{l}}+2}\end{array}$$

(89)

(85)_k+1 $\oplus$(89)

$$\begin{array}{*{20}c} {{\text{L}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 4,{\text{l}} + 1}} = {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 4,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 4,{\text{l}} + 2}} } \\ \end{array}$$

(90)

(83)_{k+1, l+2} $\oplus$(90)

$$\begin{gathered} {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 4,{\text{l}} + 1}} \hfill \\ \begin{array}{*{20}c} { = {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 4,{\text{l}} + 1}} } \\ \end{array} \hfill \\ \end{gathered}$$

(91)

(82)_l+1 $\oplus$(82)_{k+2, l+1}

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}+1}={{\text{r}}}_{{\text{k}},{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+2}\end{array}$$

(92)

(84)_k+1 $\oplus$(92)

$$\begin{array}{c}{{\text{L}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{B}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}+1}={{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+2}\end{array}$$

(93)

(83)_l+1 $\oplus$(93)

$$\begin{gathered} {\text{L}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}} + 1}} \hfill \\ = {\text{f}}_{{{\text{k}} - 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 3,{\text{l}} + 1}} \hfill \\ \end{gathered}$$

(94)

From these results, the general formula for triple-view display (left, right, bottom) is as follows

$$\begin{gathered} {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{r}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ = {\text{L}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 3,{\text{l}}}} \hfill \\ {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ = {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}},{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 3,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 3,{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 4,{\text{l}}}} \hfill \\ = {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 2,{\text{l}} - 1}} \oplus {\text{R}}_{{{\text{k}} + 2,{\text{l}} - 1}} \oplus {\text{R}}_{{{\text{k}} + 3,{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 3,{\text{l}}}} \hfill \\ \end{gathered}$$

Solving simultaneous equations with ${{\text{U}}}_{{\text{k}},{\text{l}}}$, ${{\text{R}}}_{{\text{k}},\mathrm{ l}}$, and ${{\text{B}}}_{{\text{k}},{\text{l}}}$.

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\end{array}$$

(95)

$$\begin{array}{c}{{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(96)

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\end{array}$$

(97)

Solve the above XOR simultaneous equations.

(95) $\oplus$(96)_{k+1, l+1}

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+1}={{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+1}\end{array}$$

(98)

(95) $\oplus$(96)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},\mathrm{ l}-1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}}\end{array}$$

(99)

(95)_{k+1, l+1} $\oplus$(96)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}},{\text{l}}}={{\text{r}}}_{{\text{k}}-1,{\text{l}}}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+1}\end{array}$$

(100)

(97) $\oplus$(97)_{k+1, l+1}

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}}\oplus {{\text{B}}}_{{\text{k}}+1,{\text{l}}+1}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{r}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+1,{\text{l}}+2}\end{array}$$

(101)

(100) $\oplus$(101)

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{R}}_{{{\text{k}},{\text{l}}}} \oplus {\text{B}}_{{{\text{k}},{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}} + 1}} = {\text{r}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 2}} } \\ \end{array}$$

(102)

(99)_l+2 $\oplus$(102)

$$\begin{gathered} {\text{U}}_{{{\text{k}} + 1,{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{R}}_{{{\text{k}},{\text{l}}}} \oplus {\text{R}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}},{\text{l}}}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}} + 1}} \hfill \\ = {\text{r}}_{{{\text{k}} - 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} - 1,{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 3}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 2}} \hfill \\ \end{gathered}$$

(103)

(97) $\oplus$(97)_{k+2, l+2}

$$\begin{array}{c}{{\text{B}}}_{{\text{k}},{\text{l}}}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}+2}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+2}\oplus {{\text{f}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{f}}}_{{\text{k}}+2,{\text{l}}+3}\end{array}$$

(104)

(98)_l+2 $\oplus$(104)

$$\begin{array}{c}{{\text{U}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+3}\oplus {{\text{B}}}_{{\text{k}},{\text{l}}}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}+2}={{\text{r}}}_{{\text{k}},{\text{l}}-1}\oplus {{\text{r}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+2}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+3}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+2}\end{array}$$

(105)

(100)_k+1 $\oplus$(105)_l+1

$${{\text{U}}}_{{\text{k}},{\text{l}}+3}\oplus {{\text{U}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{R}}}_{{\text{k}}+1,{\text{l}}+4}\oplus {{\text{B}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{B}}}_{{\text{k}}+2,{\text{l}}+3}\begin{array}{c}={{\text{m}}}_{{\text{k}},{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}},{\text{l}}+3}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}}\oplus {{\text{m}}}_{{\text{k}}+1,{\text{l}}+4}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+1}\oplus {{\text{m}}}_{{\text{k}}+2,{\text{l}}+3}\end{array}$$

(106)

(99)_k+1 $\oplus$(101)_l+1

$$\begin{array}{*{20}c} {{\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}} + 2}} = {\text{m}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{ l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} } \\ \end{array}$$

(107)

(98)_l+1 $\oplus$(107)

$$\begin{gathered} {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{R}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}} + 2}} \hfill \\ = {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{ l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ \end{gathered}$$

(108)

From these results, the general formula for triple-view display (upper, left, bottom) is as follows

$$\begin{gathered} {\text{r}}_{{{\text{k}},{\text{l}}}} \oplus {\text{r}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{r}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{r}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ = {\text{U}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{U}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 1}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}}}} \hfill \\ {\text{m}}_{{{\text{k}},{\text{l}}}} \oplus {\text{m}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{m}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{m}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ = {\text{U}}_{{{\text{k}},{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 2}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{B}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}}}} \hfill \\ {\text{f}}_{{{\text{k}},{\text{l}}}} \oplus {\text{f}}_{{{\text{k}},{\text{l}} + 2}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} - 1}} \oplus {\text{f}}_{{{\text{k}} + 1,{\text{l}} + 3}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}}}} \oplus {\text{f}}_{{{\text{k}} + 2,{\text{l}} + 2}} \hfill \\ = {\text{U}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{U}}_{{{\text{k}} + 2,{\text{l}} + 1}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}}}} \oplus {\text{L}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 1,{\text{l}} + 2}} \oplus {\text{B}}_{{{\text{k}} + 2,{\text{l}} + 1}} \hfill \\ \end{gathered}$$