How do I tweak an image field markup?
For link, it's easy to provide in field--myfield.tpl.php:

$items[0]['#path']['options']['attributes'] = array('style'=&gt;'border-style: none !important;');


But image doesn't have this, so you can miss upper code and rewrite formatter in template.php (add attributes there).

/**
 * Returns HTML for an image field formatter.
 *
 * @param $variables
 *   An associative array containing:
 *   - item: An array of image data.
 *   - image_style: An optional image style.
 *   - path: An array containing the link 'path' and link 'options'.
 *
 * @ingroup themeable
 */
function YOURTHEMENAME_image_formatter($variables) {
  $item = $variables['item'];
  $image = array(
    'path' =&gt; $item['uri'],
    'alt' =&gt; $item['alt'],
  );
  // Do not output an empty 'title' attribute.
  if (drupal_strlen($item['title']) &gt; 0) {
    $image['title'] = $item['title'];
  }

  if ($variables['image_style']) {
    $image['style_name'] = $variables['image_style'];
    $output = theme('image_style', $image); // Here you should add attribute, check that it's your field.
  }
  else {
    $output = theme('image', $image);
  }

  if (!empty($variables['path']['path'])) {
    $path = $variables['path']['path'];
    $options = $variables['path']['options'];
    // When displaying an image inside a link, the html option must be TRUE.
    $options['html'] = TRUE;
    $output = l($output, $path, $options);  // Here you should add attribute, check that it's your field. Or just take upper code.
  }

  return $output;
}