5**Question:** A programmer is analyzing medical image data, where pixel intensity \( I(x) \) at position \( x \) is modeled as \( I(x) = ax^2 + bx + c \). Given that \( I(1) = 6 \), \( I(2) = 11 \), and \( I(3) = 18 \), determine the values of \( a \), \( b \), and \( c \).
["Finding the Coefficients (a), (b), and (c) in I(x) = ax² + bx + c Using Given Pixel Intensities", "When analyzing medical image data, accurate modeling of pixel intensity as a function of position is crucial for tasks like segmentation and anomaly detection. In this case, a quadratic model ( I(x) = ax^2 + bx + c ) fits intensity values at three spatial locations:", "- ( I(1) = 6 )\n- ( I(2) = 11 )\n- ( I(3) = 18 )", "By substituting these points into the quadratic equation, we form a system of equations to solve for the unknown coefficients (a), (b), and (c).", "Step 1: Set up equations from given data points", "At ( x = 1 ):\n[\na(1)^2 + b(1) + c = 6 \Rightarrow a + b + c = 6 \quad \ ext{(Equation 1)}\n]", "At ( x = 2 ):\n[\na(2)^2 + b(2) + c = 11 \Rightarrow 4a + 2b + c = 11 \quad \ ext{(Equation 2)}\n]", "At ( x = 3 ):\n[\na(3)^2 + b(3) + c = 18 \Rightarrow 9a + 3b + c = 18 \quad \ ext{(Equation 3)}\n]", "Step 2: Eliminate (c) to reduce the system", "Subtract Equation 1 from Equation 2:\n[\n(4a + 2b + c) - (a + b + c) = 11 - 6 \Rightarrow 3a + b = 5 \quad \ ext{(Equation 4)}\n]", "Subtract Equation 2 from Equation 3:\n[\n(9a + 3b + c) - (4a + 2b + c) = 18 - 11 \Rightarrow 5a + b = 7 \quad \ ext{(Equation 5)}\n]", "Step 3: Solve the simplified system", "Now subtract Equation 4 from Equation 5:\n[\n(5a + b) - (3a + b) = 7 - 5 \Rightarrow 2a = 2 \Rightarrow a = 1\n]", "Substitute ( a = 1 ) into Equation 4:\n[\n3(1) + b = 5 \Rightarrow b = 2\n]", "Finally, substitute ( a = 1 ) and ( b = 2 ) into Equation 1:\n[\n1 + 2 + c = 6 \Rightarrow c = 3\n]", "Conclusion: Final coefficients\nThe quadratic intensity model is\n[\nI(x) = x^2 + 2x + 3\n]\nwith coefficients:\n- ( a = 1 )\n- ( b = 2 )\n- ( c = 3 )", "This precise fit enables accurate reconstruction and analysis of spatial data in medical imaging, supporting improved diagnostic algorithms.", "By solving for these parameters directly from pixel measurements, programmers ensure robust mathematical foundations for downstream applications in radiology and image processing."]
