**4.3 Experiment set 2**

In this section, the performance of the proposed methods has been demonstrated using image in MPEG database [39]. Fernandez [25] presents a technique to produce output polygon from a given digital boundary. Authors in [25] demonstrated the efficiency of their method by comparing their results with method [23], *Polygonal Approximation of Digital Planar Curve Using Novel Significant Measure DOI: http://dx.doi.org/10.5772/intechopen.92145*

#### **Figure 4.**

*Polygonal approximation of chromosome curve at varying amount of dominant points. (a) RDP2 [24] at 11 DPs, (b) RDP3 [24] at 6 DPs, (c) Masood [9] at 9 DPs, (d) Masood [9] at 6 DPs, (e) Prasad [23] Masood opt at 11 DPs, (f) Prasad [23] Carmona opt at 10 DPs, (g) Proposed method at 11 DPs, (h) Proposed method at 6 DPs.*

#### **Figure 5.**

*Polygonal approximation of leaf curve at varying amount of points. (a) Prasad [24] PRO 0.6 at 18 DPs, (b) Prasad [24] RDP2 at 16 DPs, (c) Masood [9] at 16 DPs, (d) Prasad [23] Masood\_opt at 18 DPs, (e) Carmona [11] at 20 DPs, (f) Prasad [23] Carmona\_opt at 18 DPs, (g) Proposed method at 16 DPs.*

which is capable of producing output polygon in non-parametric mode. So the better counterpart method to compare the proposed method is the one proposed in [25]. **Table 3** summarizes the results of the proposed method along with the results claimed as the best in [25] for the contours in MPEG database [39]. For the bell-7 contour, the snapshot at 23, 22, 20 and 7 number of dominant points, the proposed method produces a less approximation error in terms of ISE WE WE2 than others mentioned in [9, 11, 23, 25]. Especially the output approximation at 7 DPs, the

#### **Figure 6.**

*Polygonal approximation of infinity curve at varying amount of DPs. (a) Masood [9] at 8 DPs, (b) Prasad [23] Masood \_opt at 9 DPs, (c) Carmona [11] at 8 DPs, (d) Carmona [11] at 7 DPs, (e) Prasad [24] PRO 1.0 at 7 DPs, (f) Prasad [24]\_RDP 3 at 5 DPs, (g) Proposed method at 6 and 4 DPs.*



*Polygonal Approximation of Digital Planar Curve Using Novel Significant Measure DOI: http://dx.doi.org/10.5772/intechopen.92145*

#### **Table 3.**

*Comparative results for the MPEG database contours.*

#### **Figure 7.**

*The output approximation for the bell-7 contour by various methods: (a) Prasad [23] RDP at 28 DPs, (b) Fernandez [25] at 23 DPs, (c) Rosin [40] at 7 DPs, (d) Masood [9] at 15 DPs, (e) Carmona [11] at 15 DPs, (f) Proposed method 20 DPs, (g) Proposed method at 7 DPs.*

**Figure 8.**

*The output polygon from octopus-17 by various methods: (a) Carmona [11] at 43 DPS, (b) Prasad [23] RDP at 55 DPs, (c) Prasad [23] Carmona\_opt, (d) Fernandez [25] at 43 DPS, (e) Proposed method at 43 DPs.*

proposed method and Rosin [40] method produce the curve with the mandatory points compared to others, but the proposed method produces minimal error measure than Rosin [40], and the output can be found in **Figure 7(c)** and **(h)**.

For the octopus-14 contour, the proposed efficiently produces the output curve with minimal deviation from the original curve compared to others. By observing **Figure 8(e)**, the proposed produces an outlying approximation that is visibly

*Polygonal Approximation of Digital Planar Curve Using Novel Significant Measure DOI: http://dx.doi.org/10.5772/intechopen.92145*

**Figure 9.**

*Output approximated curve for ray-17 contour by various methods: (a) Carmona [11] at 14 DPs, (b) Prasad [23] RDP\_opt at 54 DPs, (c) Fernandez [25] at 24 DPs, (d) Rosin [40] at 14 DPs, (e) Proposed method results at 24 DPs, (f) Proposed method at 14 DPs.*

excellent than [25]. In order to support this claim, the output curve for octopus-14 can be found in **Figure 8** along with results of [11, 23, 25]. When the input is the ray-17 contour, at 14 number of DPs, the new proposal produces minimal error than the results of [9, 40], and then for the same curve at 35 DPs, the results are good than [25] in terms of ISE, WE and WE2. The graphic shots of the proposed method along with [11, 23, 25, 40] can be found in **Figure 9**. When the input for the proposed method is chicken-5 curve, the proposed method approximation error measures are compared with results produced by the techniques in [9, 11, 23, 25, 29, 30], and by using all the quantitative performance evaluators, the proposed work produces the output curve with minimal error possible, and the visual snapshots are shown in **Figure 10**. For the input curve device 6-9, the proposed method results are compared with the results in [9, 11, 23, 25, 29, 30], it is been conceived that the proposed one produces the minimal error (ISE ,WE) than the error produces by the methods in [9, 11, 23, 25]. The output curve for device 6–9 can be found in **Figure 11**. Then finally for the truck-07 curve, the results of the proposed method at 40, 12 and 11 dominant points are compared with the results of [9, 11, 23, 25]. In all iterations against the mentioned dominant points, the proposed method outperforms well than others. Especially output curve at 11 dominant points, the proposed method efficiently chooses the good curvature points in such a way that the output curve does not deviate much than the original input curve (please see the snapshot at **Figure 12(a)**, **(b)** with **(g)**).

#### **Figure 10.**

*Final approximation of chicken-5 contour by various methods: (a) Carmona [11] at 54 DPs, (b) Prasad [23] RDP\_opt at 218 DPs, (c) Prasad [23] Carmona\_opt 258 DPs, (d) Fernandez [25] at 54 DPs, (e) Proposed method at 54 DPs, (f) Proposed method at 29 DPs.*

#### **Figure 11.**

*Final approximation obtained from device 6–9 curve: (a) Carmona [11] at 22 DPs, (b) Prasad [23] RDP\_opt at 38 DPs, (c) Prasad [23] Carmona\_opt at 77 DPs, (d) Fernandez [25] at 33 DPs, (e) Proposed method at 22 DPs.*

#### **4.4 Rotation invariance**

To test the efficiency of the proposed method against rotation invariance, bell-7 contour is rotated using varying amount angle. Then, the rotated contour is given as an input to the proposed method as well as to the technique in [9]. The results are

*Polygonal Approximation of Digital Planar Curve Using Novel Significant Measure DOI: http://dx.doi.org/10.5772/intechopen.92145*

#### **Figure 12.**

*Final approximation obtained from truck-07 curve: (a) Masood [9] at 11 DPs, (b) Carmona [11] at 12 DPs, (c) Prasad [23] Carmona\_opt at 29 DPs, (d) Prasad [23] RDP\_opt at 33 DPs, (e) Fernandez [25] at 40 DPs, (f) Proposed method at 44 DPs, (g) Proposed method at 11 DPs.*


**Table 4.**

*Robustness of the proposed method against rotation using quantitative measurement.*

summarized for the reader's perusal. How do researcher determine a polygonal approximation is rotation invariant or not and what extent? The answer is the metrics such as area of polygon, perimeter and compactness may be suggested to use along with results from human perception. The authors in [41] use the abovementioned metrics to prove whether the technique is able to produce the polygon with the same positioned points before as well as after the rotation. This can be measured using compactness metric. Moreover, the authors in [41] demonstrated that the techniques proposed in [9, 11, 12] are scaling as well as translation invariant using compactness metric.

The mathematical interpretation of compactness metric (*COMP*) has been mentioned in Eq. (2). **Table 4** summarizes the value obtained by using COMP for the bell-7 contour by the proposed method.

$$comp = Area/Perimeter^2\tag{4}$$

To compare the robustness of the technique against rotation, the snapshots using bell-7 contour are displayed in **Figures 13** and **14**. The output polygon at 20

#### **Figure 13.**

*The output polygon at 20 DPs by proposed methods in varying amount of angles: (a) Polygon at 20 DPs, (b) Polygon at 20°, (c) Polygon at 30°, (d) Polygon at 45°, (e) Polygon at 70°, (f) Polygon at 80°, (g) Polygon at 180°.*

amounts of dominant points is used here to check if the technique is robust enough against rotation invariance. Most of the techniques considered in this chapter produce polygon in non-parametric mode. The best thing to compare the efficiency of rotation invariance is to compare the output at minimal possible amount of points since the input curve may contain more redundant points. So the result of the proposed method is compared with Masood [9]. By using [9], any researcher can produce a curve with specified number of dominant points. In **Table 4**, the value for geometric invariance assessment metrics (area of polygon, perimeter and compactness) reveals that the results by proposed method using rotated contours measure against compactness metric are more or less nearer to the value produced by the proposed method before rotation, and the visual snapshots in **Figure 13** also support the same. The results of Masood [9] in terms of quantitative measurements can be found in **Table 5**. Bell-7 at 30° value for compactness metric varies high while comparing the results obtained before rotation. In the remaining angles, the rotated contours compactness metric is more or less nearer to the value obtained by *Polygonal Approximation of Digital Planar Curve Using Novel Significant Measure DOI: http://dx.doi.org/10.5772/intechopen.92145*

#### **Figure 14.**

*The output polygon at 20 DPs by Masood [9] in varying amount of angles: (a) Polygon at 20 DPs, (b) Polygon at 20°, (c) Polygon at 30°, (d) Polygon at 45°, (e) Polygon at 70°, (f) Polygon at 80°, (g) Polygon at 180°.*


**Table 5.**

*Robustness of Masood [9] against rotation using quantitative measurement.*

the method before rotation. Masood [9] snapshots can be found in **Figure 14**. When the authors noticed that in the output curve produced by Masood [9], the position of the dominant point is heavily dislocated after rotation, whereas the proposed methods try to maintain the same positioned dominant points in the rotated contours too (see **Figure 13**).

#### **5. Conclusion**

The proposed significant measure computing metric predicts the position of a projection of every boundary point between its candidate line segment, thereby invoking suitable significant measure computing metric and accumulating its
