Cache control headers for resized images

I have a function that resizes images on the fly in WordPress. I also have W3 Total Cache set up to add cache control to all my images, but the ones being resized by the function are served as private with no cache control.

The function looks like this:

public static function resize( $path, $max_width = null, $max_height = null, $crop = true, $args = '') {

    // determine base path for uploads
    $upload_dir = wp_upload_dir();

    // let's specify some common defaults
    $defaults = array(
        'quality' => 100,
        'suffix' => null,
        'uploads_url' => $upload_dir['baseurl'],
        'uploads_directory' => $upload_dir['basedir'],
        'crops_directory' => '/wpx/',
        'filetype' => null
    );

    // standard WP default arguments merge
    $parameters = wp_parse_args( $args, $defaults );
    extract( $parameters, EXTR_SKIP );

    // get the path
    $system_root = $_SERVER[ 'DOCUMENT_ROOT' ];
    $file_path = parse_url($path);
    $image = wp_get_image_editor($system_root.$file_path['path']);

    // only do stuff on success
    if ( ! is_wp_error( $image ) ) {
        // resize the image
        if ($max_width || $max_height) { $image->resize( $max_width, $max_height, $crop ); }
        // set the quality
        if ($parameters['quality']) $image->set_quality( $parameters['quality'] );
        // rotate the image
        if ($parameters['rotate']) $image->rotate( $parameters['rotate'] );
        // flip the image
        if ($parameters['flip']) $image->flip( true, false );
        // crop the image
        if ($parameters['crop']) $image->crop( $parameters['crop'] );
        // generate the filename

        $filename = $image->generate_filename($parameters['suffix'], $parameters['uploads_directory'].$parameters['crops_directory'], $parameters['filetype'] );

        // save the image
        $image_meta = $image->save($filename);
        // insert the URL path to the crop
        $image_meta['url'] = $parameters['uploads_url'].$parameters['crops_directory'].$image_meta['file'];
        // return the data
        return $image_meta;
    }
}

The website in question is http://dquinn.net. If you do a PageSpeedTest, you’ll see it complains “Specify a cache validator” for all the images in the /wpx/ folder, which is where this function saves them.

What am I doing wrong??