You should first map your input image to cortical coordinates and then remap it to Cartesian. You also have to convert your image to grayscale if you want to take advantage of Adjacent mapping.
By changing your code to the following one, you might get what you are looking for.
cv::cvtColor(inputFrame,inputFrame,CV_BGR2GRAY);
cv::LogPolar_Adjacent *logP=new cv::LogPolar_Adjacent(inputFrame.cols,inputFrame.rows,cv::Point2i(inputFrame.cols/2,inputFrame.rows/2));
logPolar_out=logP->to_cortical(inputFrame);
logPolar_out=logP->to_cartesian(logPolar_out);
cv::imshow("Log Polar output",logPolar_out);
cv::imshow("Log Polar input",inputFrame);