Index: src/wp-includes/theme.php
===================================================================
--- src/wp-includes/theme.php	(revision 25448)
+++ src/wp-includes/theme.php	(working copy)
@@ -897,7 +897,7 @@
 function get_header_image() {
 	$url = get_theme_mod( 'header_image', get_theme_support( 'custom-header', 'default-image' ) );
 
-	if ( 'remove-header' == $url )
+	if ( 'remove-header' == $url || ( ! is_random_header_image() && empty( $url ) ) )
 		return false;
 
 	if ( is_random_header_image() )
@@ -1036,12 +1036,17 @@
  *
  * @since 3.4.0
  *
- * @return object
+ * @return mixed False if there is no header, or an object with the properties
+ * 	'url', 'thumbnail_url', 'width', and 'height'.
  */
 function get_custom_header() {
 	global $_wp_default_headers;
 
-	if ( is_random_header_image() ) {
+	$url = get_theme_mod( 'header_image', get_theme_support( 'custom-header', 'default-image' ) );
+
+	if ( 'remove-header' == $url || ( ! is_random_header_image() && empty( $url ) ) ) {
+		return false;
+	} elseif ( is_random_header_image() ) {
 		$data = _get_random_header_data();
 	} else {
 		$data = get_theme_mod( 'header_image_data' );