Problem statementThis problem is about an image burring caused by uniform acceleration. Supposethat the motion of the object is only in x-direction. The initial speed of that objectat time t=0 is zero. It accelerates with a uniform acceleration a. The distance itmoves at time t follows x0(t) = at2/2 . Find the blurring function H(u, v) if theobject travels for a time T. In this problem we assume that the shutter openingand closing times are negligible.InterpretationThe motion blurring of an image is mainly caused by the relatively motionbetween the image capture and the object. If the process speed of thephotosensitive elements and the processor of the imaging system Vsystem is lessthan the moving speed of the object Vobject, there will be overlaps in a period oftime, which is the image blurring for perception. The problem of motion blurring can be regarded as a process that the inputimage f(x, y) is degraded by the degradation function H the added a noise termn(x, y), which can be showed in the following figure.If H is a linear, position-invariant process, the degraded process can be given inspatial domain byg(x, y) = h(x, y) f(x, y) + n(x, y)where h(x, y) is the degradation function in spatial domain. Then we can get the representation in frequency domain by G(u, v) = H(u, v) F(u, v) + N(u, v)DegradationFunctionHg(x, y)f(x, y)n(x, y)In this problem, we focus on the modeling of H(u, v). So we will neglect the effectof N(u, v) to simple the modeling.Next we will model the motion blurring in a general way. As we have mentionedat first, the blurring is caused by relative motion between the image and thesensor during image acquisition. The planar motion of the object can bedecomposed into two direction and we suppose that x0(t) and y0(t) are the time-verifying components of motion in x and y directions. Then the final image we getis a procedure to record a continuous exposure by integrating the instantaneousexposure over a specific time interval when the shutter is open. We have assumed that the shutter opening and closing times are negligible. Thenif the T is the total exposure time of the sensor. We can getx−x0(t), y − y0f [¿(t )]dtg(x , y)=∫0T¿where g(x,y) is the blurred image representation in spatial domain.If we compute the Fourier transform of g(x,y), we have G(u , v)=∫−∞+∞∫−∞+∞g(x , y)e− j 2 π(ux+uy)dxdyx−x0(t), y − y0∫0Tf [¿(t )]dt ¿ e− j 2 π(ux+uy)dxdy¿∫−∞+∞¿¿∫−∞+∞¿If we reverse the order of integration we will getx−x0(t), y − y0[∫−∞+∞∫−∞+∞f [¿(t)]e− j 2 π(ux+uy)dxdy ]dtG(u , v)=∫0T¿¿∫0TF(u , v)e− j 2 π(u x0(t )+u y0(t))dt ¿ F(u , v)∫0Te− j 2 π(u x0(t)+u y0(t))dtCompared with degradation model we have built, we can getH (u , v )=∫0Te−j 2 π(u x0(t)+u