You can modify your filter function to add the condition:
function html_widget_title( $title ) {
if ( '' != $title ) {
$output="<img src="https://wordpress.stackexchange.com/images/title_".$title.'.png" alt="'.$title.'" title="'.$title.'"/>';
return $output;
} else {
return $title;
}
}
add_filter( 'widget_title', 'html_widget_title' );