Let's solve the questions step by step: --- ### **d. Briefly explain the terms in the equation. (4 marks)** The formula for accuracy is: \[ \text{Accuracy} = \frac{TP + TN}{P + N} \] **Explanation of Terms:** - **TP (True Positives):** The number of defect pixels correctly identified as defects by the detection system. - **TN (True Negatives):** The number of background pixels correctly identified as background by the detection system. - **P (Total Positives):** The total number of actual defect pixels (TP + FN, where FN = False Negatives). - **N (Total Negatives):** The total number of actual background pixels (TN + FP, where FP = False Positives). So, accuracy is the proportion of correctly identified pixels (both defects and background) out of the total number of pixels. --- ### **e.** #### **i. At approximately what pixel value should the threshold be placed for the best accuracy? (1 mark)** To maximize accuracy, the threshold should be placed at the pixel value where the overlap between the two distributions (background and defect pixels) is minimized. This is typically at the crossing point of the two curves. **From the graph, the crossing point is about at pixel value ≈ 140.** --- #### **ii. Sketch an approximate graph of accuracy vs pixel value for the threshold. (3 marks)** **Description of the sketch:** - For very low threshold values (left side), almost all pixels are classified as defects (high FP, low TN, high FN, low TP), so accuracy is low. - As the threshold increases and approaches the crossing point (around 140), accuracy increases and reaches a maximum because most background pixels are correctly classified as background and most defect pixels as defects. - After the crossing point, as the threshold gets very high, almost all pixels are classified as background (high FN, low TP, high TN, low FP), causing accuracy to drop again. - The graph should have a peak at the threshold that best separates the two distributions (around pixel value 140). **Sketch:** ``` Accuracy | | /\ | / \ | / \ |___________/ \_____________ | +------------------------------------> Threshold (pixel value) 0 ~140 300 ``` - The peak is at the crossing point (~140). - Accuracy is low at both extremes and highest at the best threshold. --- ## **Summary Table** | Sub-question | Answer | |--------------|------------------------------------------------------------------------------------------------------------| | d | TP = True Positives, TN = True Negatives, P = Total Positives, N = Total Negatives. | | e.i | The threshold should be placed at approximately **pixel value 140** for the best accuracy. | | e.ii | Sketch is a bell-shaped curve with a peak at pixel value ~140 (see above for the ASCII diagram). | If you need a hand-drawn sketch or further explanation, let me know!

**d.** TP (True Positives): Number of defect pixels correctly identified as defects. TN (True Negatives): Number of background pixels correctly identified as background. P (Total Positives): Total actual defect pixels (TP + false negatives). N (Total Negatives): Total actual background pixels (TN + false positives). **e.** i. Approximately at pixel value **140**. ii. The accuracy vs. pixel value graph peaks at around pixel value 140, rising from low accuracy at low thresholds, reaching maximum at the crossing point, then decreasing again at higher thresholds.