Dynamically add titles to post images if they are empty

I have a client’s site where they want to add image title attributes dynamically to images that don’t have one already. So if the title attribute is empty, add a title of whatever the post title is. How would I accomplish this?

Effectively:
<img src=img.jpg title=[wp-post-title-here] />

You can try the following test plugin, that uses the domDocument class, to see how it works on your HTML.

It assumes PHP 5.4+ with LibXML 2.7.8+.

<?php
/**
 * Plugin Name: Add Missing Image Title Attributes
 * Description: For posts in the main loop (Assumes PHP 5.4+ with LibXML 2.7.8+)
 * Plugin URI:  http://wordpress.stackexchange.com/a/188560/26350
 * Author:      Birgir Erlendsson (birgire)
 * Version:     0.0.1
 */

namespace wpse\birgire;

add_action( 'init', function()
{
    $o = new AddMissingImgTitle;
    $o->activate( new \domDocument );
} );

class AddMissingImgTitle
{
    private $dom;

    public function activate( \domDocument $dom )
    {
        $this->dom = $dom;
        add_filter( 'the_content', [ $this, 'the_content' ] );
    }

    public function the_content( $html )
    {
        if( ! in_the_loop() )
            return $html;

        if( false === strpos( $html, '<img' ) )
            return $html;

        return $this->process( $html );
    }                                                               

    private function process( $html )
    {
        // Handle utf8 strings
        // See http://php.net/manual/en/domdocument.loadhtml.php#95251
        $html="<?xml encoding="UTF-8">" . $html;

        // Load without HTML wrapper
        // See https://stackoverflow.com/a/22490902/2078474
        $this->dom->loadHTML( $html, LIBXML_HTML_NOIMPLIED | LIBXML_HTML_NODEFDTD );

        // Fetch all image tags:
        $images = $this->dom->getElementsByTagName( 'img' );
        foreach ( $images as $image )
        {
            // Add the title attribute if it's missing (using the post title):
            if( '' === $image->getAttribute( 'title' ) )
                $image->setAttribute( 'title', esc_attr( get_the_title() ) );
        }
        return str_replace( '<?xml encoding="UTF-8">', '', $this->dom->saveHTML() );
    }

} // end class

For older versions of LibXML, you can check out the answers to this question for alternative ways without the LIBXML_HTML_NOIMPLIED and LIBXML_HTML_NODEFDTD options.