It is a bad idea here to inherit from QImage for this functionality because you will get your class having all the public interface methods that Image has.
This means your end users will have a bunch of methods available to call, whereas it seems your desire is just to allow them to call methods to execute algorithms, e.g. the aforementioned Fourier transformation. I would suggest to have a QImage member instead. This is also called "has-a" instead of "is-a" relation ship.
Either way, you will need to have the constructor or a method to set the QImage filename. I would write something like this:
imagehandler.h
#include <QImage>
#include <QDebug>
class ImageHandler
{
public:
ImageHandler(const QString &fileName)
: m_image(fileName)
{
...
}
// Or just use an explicit method like this
void loadImage(const QString &fileName)
{
if (!m_image.load(fileName)
qDebug() << "Failed to load the image from file:" << fileName;
}
...
private:
QImage m_image;
}